--------------------------------------- k is , 3 Counting the Occurrences of Coefficients that Appear in the Expansion of, n P(x, y) , modolu , 3 For all Polynomials that are Sums of, 3, Monomials taken from, 2 2 2 2 2 2 {1, x, y, x , y , x y, x y , x y, x y } By Shalosh B. Ekhad In this webbook, we will consider the sequences described in the title, that\ after normalization and weeding out obvious symmetry, concerns the following set of, 28, polynomials 2 2 2 {1 + x + y, x + x + 1, y + x + 1, y + x + y, x y + y + 1, x y + x + y, 2 2 2 2 2 2 2 2 x y + y + 1, x y + y + x, x + y + 1, x + y + y, x + x y + y , 2 2 2 2 2 2 2 x y + y + 1, x y + x + 1, x y + x + y, x y + y + 1, x y + y + x, 2 2 2 2 2 2 2 2 x y + x y + 1, x y + x + 1, x y + x + y, x y + x + y , 2 2 2 2 2 2 2 2 2 x y + x y + 1, x y + y + 1, x y + x + y, x y + y + 1, 2 2 2 2 2 2 2 2 2 2 2 2 x y + y + x, x y + x y + 1, x y + x + y , x y + x y + 1} by finding enumerative automata with at most, 500, states . ----------------------------------------------------------------------------\ ---- n Theorem number, 1, :consider the sequence, (1 + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.022, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 2, :consider the sequence, (x + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 2, 3, 9, 2, 2, 6, 6, 3, 9, 6, 9, 27, 2, 2, 6, 6, 2, 6, 8, 6, 18, 6, 6, 18, 18, 3, 9, 6, 9, 27, 6, 6, 18, 18, 9, 27, 18, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 2, 0, 0, 2, 2, 6, 6, 0, 0, 6, 0, 0, 2, 2, 6, 6, 2, 6, 8, 6, 18, 6, 6, 18 , 18, 0, 0, 6, 0, 0, 6, 6, 18, 18, 0, 0, 18, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [293242067884135544935936513642647623193965101056, 293242067884135544935936513642647623193965101056] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886 , 5083731656658, 15251194969974, 45753584909922, 137260754729766, 411782264189298, 1235346792567894, 3706040377703682, 11118121133111046, 33354363399333138, 100063090197999414, 300189270593998242, 900567811781994726, 2701703435345984178, 8105110306037952534] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886 , 5083731656658, 15251194969974, 45753584909922, 137260754729766, 411782264189298, 1235346792567894, 3706040377703682, 11118121133111046, 33354363399333138, 100063090197999414, 300189270593998242, 900567811781994726, 2701703435345984178, 8105110306037952534] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is t - 1 ------- 3 t - 1 and in Maple notation (t-1)/(3*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t - ------- 3 t - 1 and in Maple notation -2*t/(3*t-1) This theorem took, 0.028, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 3, :consider the sequence, (y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.018, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 4, :consider the sequence, (y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.017, seconds. to state and prove ----------------------------------------------------------------------------\ ---- n Theorem number, 5, :consider the sequence, (x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.018, seconds. to state and prove ----------------------------------------------------------------------------\ ---- n Theorem number, 6, :consider the sequence, (x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.016, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 7, :consider the sequence, (x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.018, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 8, :consider the sequence, (x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.018, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 9, :consider the sequence, (x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.019, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 10, :consider the sequence, (x + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.030, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 11, :consider the sequence, (x + x y + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 2, 3, 9, 2, 2, 6, 6, 3, 9, 6, 9, 27, 2, 2, 6, 6, 2, 6, 8, 6, 18, 6, 6, 18, 18, 3, 9, 6, 9, 27, 6, 6, 18, 18, 9, 27, 18, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 2, 0, 0, 2, 2, 6, 6, 0, 0, 6, 0, 0, 2, 2, 6, 6, 2, 6, 8, 6, 18, 6, 6, 18 , 18, 0, 0, 6, 0, 0, 6, 6, 18, 18, 0, 0, 18, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [293242067884135544935936513642647623193965101056, 293242067884135544935936513642647623193965101056] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886 , 5083731656658, 15251194969974, 45753584909922, 137260754729766, 411782264189298, 1235346792567894, 3706040377703682, 11118121133111046, 33354363399333138, 100063090197999414, 300189270593998242, 900567811781994726, 2701703435345984178, 8105110306037952534] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886 , 5083731656658, 15251194969974, 45753584909922, 137260754729766, 411782264189298, 1235346792567894, 3706040377703682, 11118121133111046, 33354363399333138, 100063090197999414, 300189270593998242, 900567811781994726, 2701703435345984178, 8105110306037952534] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is t - 1 ------- 3 t - 1 and in Maple notation (t-1)/(3*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t - ------- 3 t - 1 and in Maple notation -2*t/(3*t-1) This theorem took, 0.032, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 12, :consider the sequence, (x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.018, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 13, :consider the sequence, (x y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.018, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 14, :consider the sequence, (x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.017, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 15, :consider the sequence, (x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.019, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 16, :consider the sequence, (x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.018, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 17, :consider the sequence, (x y + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.018, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 18, :consider the sequence, (x y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.021, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 19, :consider the sequence, (x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 8, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.063, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 20, :consider the sequence, (x y + x + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.020, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 21, :consider the sequence, (x y + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 8, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.063, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 22, :consider the sequence, (x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.020, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 23, :consider the sequence, (x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 8, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.064, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 24, :consider the sequence, (x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.019, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 25, :consider the sequence, (x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.019, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 26, :consider the sequence, (x y + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 2, 3, 9, 2, 2, 6, 6, 3, 9, 6, 9, 27, 2, 2, 6, 6, 2, 6, 8, 6, 18, 6, 6, 18, 18, 3, 9, 6, 9, 27, 6, 6, 18, 18, 9, 27, 18, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 2, 0, 0, 2, 2, 6, 6, 0, 0, 6, 0, 0, 2, 2, 6, 6, 2, 6, 8, 6, 18, 6, 6, 18 , 18, 0, 0, 6, 0, 0, 6, 6, 18, 18, 0, 0, 18, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [293242067884135544935936513642647623193965101056, 293242067884135544935936513642647623193965101056] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886 , 5083731656658, 15251194969974, 45753584909922, 137260754729766, 411782264189298, 1235346792567894, 3706040377703682, 11118121133111046, 33354363399333138, 100063090197999414, 300189270593998242, 900567811781994726, 2701703435345984178, 8105110306037952534] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886 , 5083731656658, 15251194969974, 45753584909922, 137260754729766, 411782264189298, 1235346792567894, 3706040377703682, 11118121133111046, 33354363399333138, 100063090197999414, 300189270593998242, 900567811781994726, 2701703435345984178, 8105110306037952534] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is t - 1 ------- 3 t - 1 and in Maple notation (t-1)/(3*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t - ------- 3 t - 1 and in Maple notation -2*t/(3*t-1) This theorem took, 0.032, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 27, :consider the sequence, (x y + x + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.020, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 28, :consider the sequence, (x y + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 3, 3, 3, 9, 9, 3, 9, 18, 3, 9, 9, 9, 27, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54 , 18, 54, 108, 3, 9, 9, 9, 27, 27, 9, 27, 54, 9, 27, 27, 27, 81] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 3, 0, 0, 9, 3, 9, 18, 0, 0, 9, 0, 0, 27, 9, 27, 54, 3, 9, 18, 9, 27, 54, 18, 54, 108, 0, 0, 9, 0, 0, 27, 9, 27, 54, 0, 0, 27, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [637179250451194222801004569825051668810489887334716025385586038727874415951872 , 637179250451194222801004569825051668810489887334716025385586038727874415951872] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368 , 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408, 2369190669160808448, 14215144014964850688, 85290864089789104128, 511745184538734624768, 3070471107232407748608, 18422826643394446491648, 110536959860366678949888, 663221759162200073699328, 3979330554973200442195968, 23875983329839202653175808 , 143255899979035215919054848, 859535399874211295514329088, 5157212399245267773085974528, 30943274395471606638515847168, 185659646372829639831095083008, 1113957878236977838986570498048, 6683747269421867033919422988288] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 6 t - 1 and in Maple notation (3*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 t - ------- 6 t - 1 and in Maple notation -3*t/(6*t-1) This theorem took, 0.019, seconds. to state and prove ------------------------------------------------------------------ This concludes this webbook, that took, 1.155, seconds. to generate. k is , 4 Counting the Occurrences of Coefficients that Appear in the Expansion of, n P(x, y) , modolu , 3 For all Polynomials that are Sums of, 4, Monomials taken from, 2 2 2 2 2 2 {1, x, y, x , y , x y, x y , x y, x y } By Shalosh B. Ekhad In this webbook, we will consider the sequences described in the title, that\ after normalization and weeding out obvious symmetry, concerns the following set of, 54, polynomials 2 2 2 {y + x + y + 1, x y + x + y + 1, x y + y + y + 1, x y + y + x + 1, 2 2 2 2 2 2 2 x y + y + x + y, x + y + y + 1, x + y + x + y, x + x y + y + 1, 2 2 2 2 2 2 2 x + x y + y + y, x y + x + y + 1, x y + y + y + 1, x y + y + x + 1, 2 2 2 2 x y + y + x + y, x y + x y + y + 1, x y + x y + x + 1, 2 2 2 2 2 x y + x y + x + y, x y + x y + y + 1, x y + x y + y + x, 2 2 2 2 2 2 2 2 2 x y + x + y + 1, x y + x + x + 1, x y + x + x + y, x y + x + y + 1, 2 2 2 2 2 2 2 2 x y + x + y + y, x y + x + y + x, x y + x + x y + 1, 2 2 2 2 2 2 2 x y + x + x y + y, x y + x + x y + y , x y + x y + y + 1, 2 2 2 2 2 2 2 2 x y + x y + x + y, x y + x y + y + 1, x y + x y + y + x, 2 2 2 2 2 2 2 2 x y + x y + x y + 1, x y + x y + x + y , x y + x + y + 1, 2 2 2 2 2 2 2 2 2 x y + y + y + 1, x y + y + x + 1, x y + y + x + y, 2 2 2 2 2 2 2 x y + x y + y + 1, x y + x y + x + y, x y + x y + y + 1, 2 2 2 2 2 2 2 2 2 2 2 x y + x y + y + x, x y + x + y + 1, x y + x + y + y, 2 2 2 2 2 2 2 2 2 2 x y + x + x y + y , x y + x y + y + 1, x y + x y + x + 1, 2 2 2 2 2 2 2 2 2 2 2 x y + x y + x + y, x y + x y + y + 1, x y + x y + y + x, 2 2 2 2 2 2 2 2 2 2 2 x y + x y + x y + 1, x y + x y + x + 1, x y + x y + x + y, 2 2 2 2 2 2 2 2 2 x y + x y + x + y , x y + x y + x y + 1} by finding enumerative automata with at most, 500, states . ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 1, :consider the sequence, (y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.055, seconds. to state and prove ----------------------------------------------------------------------------\ ---- n Theorem number, 2, :consider the sequence, (x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 5, 4, 16, 20, 5, 20, 41, 4, 16, 20, 16, 64, 80, 20, 80, 164, 5, 20, 41, 20, 80, 164, 41, 164, 365, 4, 16, 20, 16, 64, 80, 20, 80, 164, 16, 64, 80, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 4, 0, 0, 16, 4, 16, 40, 0, 0, 16, 0, 0, 64, 16, 64, 160, 4, 16, 40, 16, 64, 160, 40, 160, 364, 0, 0, 16, 0, 0, 64, 16, 64, 160, 0, 0, 64, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [542934428254863952002380212403205877340118100850670114517356747234881622893159\ 1747510754240102400, 5429344282548639520023802124032058773401181008506700788361\ 644295858845964360099385136970144415744] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401, 54709494565756179605, 492385451091805616441, 4431469059826250547965, 39883221538436254931681, 358948993845926294385125, 3230540944613336649466121, 29074868501520029845195085, 261673816513680268606755761, 2355064348623122417460801845, 21195579137608101757147216601, 190760212238472915814324949405, 1716841910146256242328924544641, 15451577191316306180960320901765, 139064194721846755628642888115881, 1251577752496620800657785993042925, 11264199772469587205920073937386321, 101377797952226284853280665436476885, 912400181570036563679525988928291961, 8211601634130329073115733900354627645, 73904414707172961658041605103191648801] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044, 6078832729528464400, 54709494565756179604, 492385451091805616440, 4431469059826250547964, 39883221538436254931680, 358948993845926294385124, 3230540944613336649466120, 29074868501520029845195084, 261673816513680268606755760, 2355064348623122417460801844, 21195579137608101757147216600, 190760212238472915814324949404, 1716841910146256242328924544640, 15451577191316306180960320901764, 139064194721846755628642888115880, 1251577752496620800657785993042924, 11264199772469587205920073937386320, 101377797952226284853280665436476884, 912400181570036563679525988928291960, 8211601634130329073115733900354627644, 73904414707172961658041605103191648800] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 t - 1 - ----------------- (9 t - 1) (t - 1) and in Maple notation -(5*t-1)/(9*t-1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 t ----------------- (9 t - 1) (t - 1) and in Maple notation 4*t/(9*t-1)/(t-1) This theorem took, 0.018, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 3, :consider the sequence, (x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.034, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 4, :consider the sequence, (x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.054, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 5, :consider the sequence, (x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 5, 4, 16, 20, 5, 20, 41, 4, 16, 20, 16, 64, 80, 20, 80, 164, 5, 20, 41, 20, 80, 164, 41, 164, 365, 4, 16, 20, 16, 64, 80, 20, 80, 164, 16, 64, 80, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 4, 0, 0, 16, 4, 16, 40, 0, 0, 16, 0, 0, 64, 16, 64, 160, 4, 16, 40, 16, 64, 160, 40, 160, 364, 0, 0, 16, 0, 0, 64, 16, 64, 160, 0, 0, 64, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [542934428254863952002380212403205877340118100850670114517356747234881622893159\ 1747510754240102400, 5429344282548639520023802124032058773401181008506700788361\ 644295858845964360099385136970144415744] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401, 54709494565756179605, 492385451091805616441, 4431469059826250547965, 39883221538436254931681, 358948993845926294385125, 3230540944613336649466121, 29074868501520029845195085, 261673816513680268606755761, 2355064348623122417460801845, 21195579137608101757147216601, 190760212238472915814324949405, 1716841910146256242328924544641, 15451577191316306180960320901765, 139064194721846755628642888115881, 1251577752496620800657785993042925, 11264199772469587205920073937386321, 101377797952226284853280665436476885, 912400181570036563679525988928291961, 8211601634130329073115733900354627645, 73904414707172961658041605103191648801] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044, 6078832729528464400, 54709494565756179604, 492385451091805616440, 4431469059826250547964, 39883221538436254931680, 358948993845926294385124, 3230540944613336649466120, 29074868501520029845195084, 261673816513680268606755760, 2355064348623122417460801844, 21195579137608101757147216600, 190760212238472915814324949404, 1716841910146256242328924544640, 15451577191316306180960320901764, 139064194721846755628642888115880, 1251577752496620800657785993042924, 11264199772469587205920073937386320, 101377797952226284853280665436476884, 912400181570036563679525988928291960, 8211601634130329073115733900354627644, 73904414707172961658041605103191648800] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 t - 1 - ----------------- (9 t - 1) (t - 1) and in Maple notation -(5*t-1)/(9*t-1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 t ----------------- (9 t - 1) (t - 1) and in Maple notation 4*t/(9*t-1)/(t-1) This theorem took, 0.019, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 6, :consider the sequence, (x + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.058, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 7, :consider the sequence, (x + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.059, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 8, :consider the sequence, (x + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.038, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 9, :consider the sequence, (x + x y + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.038, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 10, :consider the sequence, (x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.081, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 11, :consider the sequence, (x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.036, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 12, :consider the sequence, (x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 5, 4, 16, 20, 5, 20, 41, 4, 16, 20, 16, 64, 80, 20, 80, 164, 5, 20, 41, 20, 80, 164, 41, 164, 365, 4, 16, 20, 16, 64, 80, 20, 80, 164, 16, 64, 80, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 4, 0, 0, 16, 4, 16, 40, 0, 0, 16, 0, 0, 64, 16, 64, 160, 4, 16, 40, 16, 64, 160, 40, 160, 364, 0, 0, 16, 0, 0, 64, 16, 64, 160, 0, 0, 64, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [542934428254863952002380212403205877340118100850670114517356747234881622893159\ 1747510754240102400, 5429344282548639520023802124032058773401181008506700788361\ 644295858845964360099385136970144415744] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401, 54709494565756179605, 492385451091805616441, 4431469059826250547965, 39883221538436254931681, 358948993845926294385125, 3230540944613336649466121, 29074868501520029845195085, 261673816513680268606755761, 2355064348623122417460801845, 21195579137608101757147216601, 190760212238472915814324949405, 1716841910146256242328924544641, 15451577191316306180960320901765, 139064194721846755628642888115881, 1251577752496620800657785993042925, 11264199772469587205920073937386321, 101377797952226284853280665436476885, 912400181570036563679525988928291961, 8211601634130329073115733900354627645, 73904414707172961658041605103191648801] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044, 6078832729528464400, 54709494565756179604, 492385451091805616440, 4431469059826250547964, 39883221538436254931680, 358948993845926294385124, 3230540944613336649466120, 29074868501520029845195084, 261673816513680268606755760, 2355064348623122417460801844, 21195579137608101757147216600, 190760212238472915814324949404, 1716841910146256242328924544640, 15451577191316306180960320901764, 139064194721846755628642888115880, 1251577752496620800657785993042924, 11264199772469587205920073937386320, 101377797952226284853280665436476884, 912400181570036563679525988928291960, 8211601634130329073115733900354627644, 73904414707172961658041605103191648800] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 t - 1 - ----------------- (9 t - 1) (t - 1) and in Maple notation -(5*t-1)/(9*t-1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 t ----------------- (9 t - 1) (t - 1) and in Maple notation 4*t/(9*t-1)/(t-1) This theorem took, 0.020, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 13, :consider the sequence, (x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.052, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 14, :consider the sequence, (x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 5, 4, 16, 20, 5, 20, 41, 4, 16, 20, 16, 64, 80, 20, 80, 164, 5, 20, 41, 20, 80, 164, 41, 164, 365, 4, 16, 20, 16, 64, 80, 20, 80, 164, 16, 64, 80, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 4, 0, 0, 16, 4, 16, 40, 0, 0, 16, 0, 0, 64, 16, 64, 160, 4, 16, 40, 16, 64, 160, 40, 160, 364, 0, 0, 16, 0, 0, 64, 16, 64, 160, 0, 0, 64, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [542934428254863952002380212403205877340118100850670114517356747234881622893159\ 1747510754240102400, 5429344282548639520023802124032058773401181008506700788361\ 644295858845964360099385136970144415744] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401, 54709494565756179605, 492385451091805616441, 4431469059826250547965, 39883221538436254931681, 358948993845926294385125, 3230540944613336649466121, 29074868501520029845195085, 261673816513680268606755761, 2355064348623122417460801845, 21195579137608101757147216601, 190760212238472915814324949405, 1716841910146256242328924544641, 15451577191316306180960320901765, 139064194721846755628642888115881, 1251577752496620800657785993042925, 11264199772469587205920073937386321, 101377797952226284853280665436476885, 912400181570036563679525988928291961, 8211601634130329073115733900354627645, 73904414707172961658041605103191648801] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044, 6078832729528464400, 54709494565756179604, 492385451091805616440, 4431469059826250547964, 39883221538436254931680, 358948993845926294385124, 3230540944613336649466120, 29074868501520029845195084, 261673816513680268606755760, 2355064348623122417460801844, 21195579137608101757147216600, 190760212238472915814324949404, 1716841910146256242328924544640, 15451577191316306180960320901764, 139064194721846755628642888115880, 1251577752496620800657785993042924, 11264199772469587205920073937386320, 101377797952226284853280665436476884, 912400181570036563679525988928291960, 8211601634130329073115733900354627644, 73904414707172961658041605103191648800] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 t - 1 - ----------------- (9 t - 1) (t - 1) and in Maple notation -(5*t-1)/(9*t-1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 t ----------------- (9 t - 1) (t - 1) and in Maple notation 4*t/(9*t-1)/(t-1) This theorem took, 0.021, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 15, :consider the sequence, (x y + x y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.062, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 16, :consider the sequence, (x y + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.035, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 17, :consider the sequence, (x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.055, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 18, :consider the sequence, (x y + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.034, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 19, :consider the sequence, (x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 15, 4, 16, 49, 4, 16, 16, 16, 64, 59, 15, 60, 184, 4, 16, 52, 16, 64, 196, 49, 196, 369, 4, 16, 16, 16, 64, 60, 16, 64, 196, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 23, 6, 24, 42, 0, 0, 24, 0, 0, 91, 23, 92, 150, 6, 24, 48, 24, 96, 177, 42, 168, 404, 0, 0, 24, 0, 0, 92, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [747339094487699982121564378687277951577525511881252912369088384795115785338052\ 69945077610210918400, 747339094487699982121564378687282727623242254678730129199\ 87712946666296423893935093002389789081600] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 49, 369, 3210, 25840, 212665, 1724976, 14041578, 114019654, 926439241, 7524495357, 61120594098, 496442507344, 4032360311650, 32752537815387, 266030943070788, 2160819811742896, 17551135597159069, 142558052571733767, 1157919405158141736, 9405131824430543218, 76392627861689872558, 620494603082112039414, 5039930742018999473403, 40936539526388788133704, 332504622592612951392712, 2700749142345287121789600, 21936675266822335897710654 , 178179348165167763938082373, 1447251223209246674987990287, 11755212512751057641364570132, 95481018778399952240234702283, 775538931096371392155172792609, 6299269125335721474501977651527, 51165441117576203279911260647883, 415588271062611423754874475461130, 3375591166074677961605757104597557, 27418039713553112130111943812207241, 222701406879241761802628836736176599, 1808879013384671796641747886704306286] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 404, 3176, 26352, 212277, 1730087, 14034245, 114066915, 926322978, 7524929753, 61119038741, 496446865281, 4032342032217, 32752586605502, 266030743040072, 2160820389386700, 17551133463188112, 142558059420112892, 1157919382180563701, 9405131903735812263, 76392627608212572648, 620494603978481836934, 5039930739164022367049, 40936539536373286186533, 332504622560158676462946, 2700749142456086838143531, 21936675266453677071164585 , 178179348166401517994198868, 1447251223205080807525721682, 11755212512764872120504260924, 95481018778353119610830333096, 775538931096526722707846888670, 6299269125335196502736012168205, 51165441117577952677922350208321, 415588271062605542107712797361339, 3375591166074697657193597931652176, 27418039713553046174860582973889609, 222701406879241983285685560773757116, 1808879013384671056147886699508179612] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 3.897, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 20, :consider the sequence, (x y + x + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.040, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 21, :consider the sequence, (x y + x + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 14, 4, 16, 43, 4, 16, 16, 16, 64, 56, 14, 56, 152, 4, 16, 52, 16, 64, 185, 43, 172, 349, 4, 16, 16, 16, 64, 56, 16, 64, 172, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 42, 0, 0, 24, 0, 0, 88, 22, 88, 148, 6, 24, 48, 24, 96, 172, 42, 168, 339, 0, 0, 24, 0, 0, 88, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [116563615066120903652891837234710881448427048780539622565378717461555241643347\ 55549568012606504960, 116563615066120903652891837234710881448427036365241000233\ 98679273181765473225564450431987393495040] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 43, 349, 2725, 22012, 174463, 1389685, 11059369, 87988684, 700214287, 5571580405, 44334719065, 352781310064, 2807158177723, 22337207138749, 177742163416045, 1414334703690856, 11254181031837547, 89552063528976493, 712586027367532285, 5670208180984550404, 45119129008579140295, 359023114495874158597, 2856828125609676390337, 22732427551596193661092, 180887067631073776895671, 1439359310066138202733669, 11453307583528500522710161 , 91136558943739928322056440, 725194212722622449725017379, 5770534374585076614825317965, 45917447194258042787620761253, 365375512902819142817084818816, 2907375596561773314359283478771, 23134645210148962474982680469725, 184087604516046173128308070261141, 1464826705947916780584489522161740, 11655957412771864930381705692147919, 92749089470233943609379176035328917, 738025482842919374038021847620081945] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 339, 2778, 21852, 174756, 1389591, 11057124, 88000188, 700176366, 5571663435, 44334639678, 352780951656, 2807160623016, 22337198161047, 177742185900360, 1414334671882848, 11254180988430258, 89552064023793003, 712586025302177778, 5670208186786551540, 45119128998215742300, 359023114495505928855, 2856828125703378620892, 22732427551135974405348, 180887067632515512032262, 1439359310063097366919563, 11453307583530722233306230 , 91136558943755941463531808, 725194212722523548088096192, 5770534374585423185852872599, 45917447194257207994097657376, 365375512902820190809911412296, 2907375596561775547322046949098, 23134645210148942129723108032683, 184087604516046253853462451982410, 1464826705947916562259864784651500, 11655957412771865292566744909987796, 92749089470233943758049629307175831, 738025482842919370086639203881416852] Using the found enumerative automaton with, 22, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (2 t - 1) (888 t + 704 t + 546 t + 25 t - 318 t - 113 t - 91 t + 36 t 3 2 / + 20 t - 5 t - t + 1) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation (2*t-1)*(888*t^11+704*t^10+546*t^9+25*t^8-318*t^7-113*t^6-91*t^5+36*t^4+20*t^3-\ 5*t^2-t+1)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^ 4-18*t^3-14*t^2+10*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 3 - 3 (648 t + 280 t + 22 t - 322 t - 291 t + 89 t + 50 t + 68 t - 9 t 2 / - 21 t + 2) t / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation -3*(648*t^11+280*t^10+22*t^9-322*t^8-291*t^7+89*t^6+50*t^5+68*t^4-9*t^3-21*t^2+ 2)*t/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^4-18*t ^3-14*t^2+10*t-1) This theorem took, 0.313, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 22, :consider the sequence, (x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.059, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 23, :consider the sequence, (x y + x + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 14, 4, 16, 43, 4, 16, 16, 16, 64, 56, 14, 56, 152, 4, 16, 52, 16, 64, 185, 43, 172, 349, 4, 16, 16, 16, 64, 56, 16, 64, 172, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 42, 0, 0, 24, 0, 0, 88, 22, 88, 148, 6, 24, 48, 24, 96, 172, 42, 168, 339, 0, 0, 24, 0, 0, 88, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [116563615066120903652891837234710881448427048780539622565378717461555241643347\ 55549568012606504960, 116563615066120903652891837234710881448427036365241000233\ 98679273181765473225564450431987393495040] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 43, 349, 2725, 22012, 174463, 1389685, 11059369, 87988684, 700214287, 5571580405, 44334719065, 352781310064, 2807158177723, 22337207138749, 177742163416045, 1414334703690856, 11254181031837547, 89552063528976493, 712586027367532285, 5670208180984550404, 45119129008579140295, 359023114495874158597, 2856828125609676390337, 22732427551596193661092, 180887067631073776895671, 1439359310066138202733669, 11453307583528500522710161 , 91136558943739928322056440, 725194212722622449725017379, 5770534374585076614825317965, 45917447194258042787620761253, 365375512902819142817084818816, 2907375596561773314359283478771, 23134645210148962474982680469725, 184087604516046173128308070261141, 1464826705947916780584489522161740, 11655957412771864930381705692147919, 92749089470233943609379176035328917, 738025482842919374038021847620081945] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 339, 2778, 21852, 174756, 1389591, 11057124, 88000188, 700176366, 5571663435, 44334639678, 352780951656, 2807160623016, 22337198161047, 177742185900360, 1414334671882848, 11254180988430258, 89552064023793003, 712586025302177778, 5670208186786551540, 45119128998215742300, 359023114495505928855, 2856828125703378620892, 22732427551135974405348, 180887067632515512032262, 1439359310063097366919563, 11453307583530722233306230 , 91136558943755941463531808, 725194212722523548088096192, 5770534374585423185852872599, 45917447194257207994097657376, 365375512902820190809911412296, 2907375596561775547322046949098, 23134645210148942129723108032683, 184087604516046253853462451982410, 1464826705947916562259864784651500, 11655957412771865292566744909987796, 92749089470233943758049629307175831, 738025482842919370086639203881416852] Using the found enumerative automaton with, 22, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (2 t - 1) (888 t + 704 t + 546 t + 25 t - 318 t - 113 t - 91 t + 36 t 3 2 / + 20 t - 5 t - t + 1) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation (2*t-1)*(888*t^11+704*t^10+546*t^9+25*t^8-318*t^7-113*t^6-91*t^5+36*t^4+20*t^3-\ 5*t^2-t+1)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^ 4-18*t^3-14*t^2+10*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 - 3 t (648 t + 280 t + 22 t - 322 t - 291 t + 89 t + 50 t + 68 t 3 2 / - 9 t - 21 t + 2) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation -3*t*(648*t^11+280*t^10+22*t^9-322*t^8-291*t^7+89*t^6+50*t^5+68*t^4-9*t^3-21*t^ 2+2)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^4-18*t ^3-14*t^2+10*t-1) This theorem took, 0.347, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 24, :consider the sequence, (x y + x + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 5, 4, 16, 20, 5, 20, 41, 4, 16, 20, 16, 64, 80, 20, 80, 164, 5, 20, 41, 20, 80, 164, 41, 164, 365, 4, 16, 20, 16, 64, 80, 20, 80, 164, 16, 64, 80, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 4, 0, 0, 16, 4, 16, 40, 0, 0, 16, 0, 0, 64, 16, 64, 160, 4, 16, 40, 16, 64, 160, 40, 160, 364, 0, 0, 16, 0, 0, 64, 16, 64, 160, 0, 0, 64, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [542934428254863952002380212403205877340118100850670114517356747234881622893159\ 1747510754240102400, 5429344282548639520023802124032058773401181008506700788361\ 644295858845964360099385136970144415744] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401, 54709494565756179605, 492385451091805616441, 4431469059826250547965, 39883221538436254931681, 358948993845926294385125, 3230540944613336649466121, 29074868501520029845195085, 261673816513680268606755761, 2355064348623122417460801845, 21195579137608101757147216601, 190760212238472915814324949405, 1716841910146256242328924544641, 15451577191316306180960320901765, 139064194721846755628642888115881, 1251577752496620800657785993042925, 11264199772469587205920073937386321, 101377797952226284853280665436476885, 912400181570036563679525988928291961, 8211601634130329073115733900354627645, 73904414707172961658041605103191648801] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044, 6078832729528464400, 54709494565756179604, 492385451091805616440, 4431469059826250547964, 39883221538436254931680, 358948993845926294385124, 3230540944613336649466120, 29074868501520029845195084, 261673816513680268606755760, 2355064348623122417460801844, 21195579137608101757147216600, 190760212238472915814324949404, 1716841910146256242328924544640, 15451577191316306180960320901764, 139064194721846755628642888115880, 1251577752496620800657785993042924, 11264199772469587205920073937386320, 101377797952226284853280665436476884, 912400181570036563679525988928291960, 8211601634130329073115733900354627644, 73904414707172961658041605103191648800] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 t - 1 - ----------------- (9 t - 1) (t - 1) and in Maple notation -(5*t-1)/(9*t-1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 t ----------------- (9 t - 1) (t - 1) and in Maple notation 4*t/(9*t-1)/(t-1) This theorem took, 0.022, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 25, :consider the sequence, (x y + x + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 46, 4, 16, 16, 16, 64, 68, 17, 68, 179, 4, 16, 52, 16, 64, 200, 46, 184, 434, 4, 16, 16, 16, 64, 68, 16, 64, 184, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 48, 0, 0, 24, 0, 0, 88, 22, 88, 184, 6, 24, 48, 24, 96, 190, 48, 192, 436, 0, 0, 24, 0, 0, 88, 24, 96, 192, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [379580757157229078897899883886434775259298623761228624785450749345153966119653\ 206371860480000000000, 37958075715722907889789988388643477525929862374142158415\ 6884664946767978535653206371860480000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 46, 434, 3976, 36122, 326788, 2949668, 26590156, 239528108, 2156837704, 19416962732, 174779748592, 1573152922052, 14159050770688, 127434821338964, 1146930171864064, 10322455228350980, 92902514355311920, 836124710107083380, 7525132767412429696, 67726246648201614020, 609536477837385420496, 5485829587038388242068, 49372472698304325181792, 444352286271960688250852, 3999170735946909299281264, 35992537418839687904043956, 323932840735278727355212672, 2915395586391929570194549892, 26238560376129262385869047184, 236147043876825441652901468756, 2125323397343020631758708354720, 19127910588311641600737044818724, 172151195355759998536807772610736, 1549360758505783108805464730220788, 13944246828067609965818627438809024, 125498221460165588229866576665803716, 1129483993179172512690483368876425168, 10165355938800448747929023951299847060, 91488203450140951639119332713295182048] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 48, 436, 3978, 36124, 326790, 2949670, 26590158, 239528110, 2156837706, 19416962734, 174779748594, 1573152922054, 14159050770690, 127434821338966, 1146930171864066, 10322455228350982, 92902514355311922, 836124710107083382, 7525132767412429698, 67726246648201614022, 609536477837385420498, 5485829587038388242070, 49372472698304325181794, 444352286271960688250854, 3999170735946909299281266, 35992537418839687904043958, 323932840735278727355212674, 2915395586391929570194549894, 26238560376129262385869047186, 236147043876825441652901468758, 2125323397343020631758708354722, 19127910588311641600737044818726, 172151195355759998536807772610738, 1549360758505783108805464730220790, 13944246828067609965818627438809026, 125498221460165588229866576665803718, 1129483993179172512690483368876425170, 10165355938800448747929023951299847062, 91488203450140951639119332713295182050] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 8 7 6 5 4 3 2 162 t - 26 t + 502 t - 752 t + 226 t + 81 t - 61 t + 13 t - 1 -------------------------------------------------------------------- 6 5 4 3 2 (t - 1) (9 t - 1) (12 t + 4 t + 38 t - 38 t - 4 t + 7 t - 1) and in Maple notation (162*t^8-26*t^7+502*t^6-752*t^5+226*t^4+81*t^3-61*t^2+13*t-1)/(t-1)/(9*t-1)/(12 *t^6+4*t^5+38*t^4-38*t^3-4*t^2+7*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 2 (81 t - 5 t + 244 t - 363 t + 80 t + 59 t - 27 t + 3) t - ----------------------------------------------------------------- 6 5 4 3 2 (t - 1) (9 t - 1) (12 t + 4 t + 38 t - 38 t - 4 t + 7 t - 1) and in Maple notation -2*(81*t^7-5*t^6+244*t^5-363*t^4+80*t^3+59*t^2-27*t+3)*t/(t-1)/(9*t-1)/(12*t^6+ 4*t^5+38*t^4-38*t^3-4*t^2+7*t-1) This theorem took, 0.430, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 26, :consider the sequence, (x y + x + x y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 13, 4, 16, 46, 4, 16, 16, 16, 64, 52, 13, 52, 154, 4, 16, 52, 16, 64, 178, 46, 184, 313, 4, 16, 16, 16, 64, 52, 16, 64, 184, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 21, 6, 24, 33, 0, 0, 24, 0, 0, 84, 21, 84, 120, 6, 24, 48, 24, 96, 162, 33, 132, 303, 0, 0, 24, 0, 0, 84, 24, 96, 132, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [162316748584659891246977075565457621824621635325509435509837698244858683853609\ 7047665544334934016, 1623167485846598912469770755654522568268735722082864926800\ 406568834215379011455736718455665065984] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 46, 313, 2443, 18694, 145024, 1125067, 8740333, 67913800, 527787922, 4101833917, 31879096495, 247763224426, 1925612884612, 14965856546191, 116314634546257, 903997464560236, 7025869711149910, 54605071718117185, 424390717158130387, 3298365440996264782, 25634902361689748488, 199234509093056834515, 1548450977567390015029, 12034563896767430833168, 93532653140488837738138, 726935955357163089306949, 5649747606322486182613111, 43909848976338613186014898, 341267428472119909955878156, 2652331093162315137596620183, 20613922222967257715049565081, 160211442119739691655800933684, 1245163628175966378458605912414, 9677414049950157770671994517193, 75212880118706767580448756539803, 584554645130647914431328107495830, 4543159796626774790736103092993424, 35309446447171307374439733164896987, 274425083909959789423598476388910205] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 33, 303, 2346, 18528, 144267, 1123305, 8734020, 67896906, 527733141, 4101677763, 31878611646, 247761802644, 1925608553727, 14965843684605, 116314595699352, 903997348543806, 7025869362052041, 54605070672920727, 424390714018346706, 3298365431585302344, 25634902333440083955, 199234509008341395345, 1548450977313176588652, 12034563896004924771762, 93532653138201051118461, 726935955350300266318827, 5649747606301896639906918, 43909848976276846705379964, 341267428471934606219006055, 2652331093161759234975938469, 20613922222965589990007650752, 160211442119734688515034929062, 1245163628175951368967588421809, 9677414049950112742336380998847, 75212880118706632495167038077818, 584554645130647509176032707923760, 4543159796626773574969117382649435, 35309446447171303727140975057120569, 274425083909959778481697804019069844] Using the found enumerative automaton with, 14, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 36 t - 13 t + 5 t - 1 -------------------------------------------- 2 (t - 1) (3 t - 1) (2 t + 1) (6 t + 7 t - 1) and in Maple notation (36*t^4+36*t^3-13*t^2+5*t-1)/(t-1)/(3*t-1)/(2*t+1)/(6*t^2+7*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 3 t (12 t + 8 t - 7 t + 2) - -------------------------------------------- 2 (t - 1) (3 t - 1) (2 t + 1) (6 t + 7 t - 1) and in Maple notation -3*t*(12*t^3+8*t^2-7*t+2)/(t-1)/(3*t-1)/(2*t+1)/(6*t^2+7*t-1) This theorem took, 0.154, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 27, :consider the sequence, (x y + x + x y + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.041, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 28, :consider the sequence, (x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 14, 4, 16, 43, 4, 16, 16, 16, 64, 56, 14, 56, 152, 4, 16, 52, 16, 64, 185, 43, 172, 349, 4, 16, 16, 16, 64, 56, 16, 64, 172, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 42, 0, 0, 24, 0, 0, 88, 22, 88, 148, 6, 24, 48, 24, 96, 172, 42, 168, 339, 0, 0, 24, 0, 0, 88, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [116563615066120903652891837234710881448427048780539622565378717461555241643347\ 55549568012606504960, 116563615066120903652891837234710881448427036365241000233\ 98679273181765473225564450431987393495040] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 43, 349, 2725, 22012, 174463, 1389685, 11059369, 87988684, 700214287, 5571580405, 44334719065, 352781310064, 2807158177723, 22337207138749, 177742163416045, 1414334703690856, 11254181031837547, 89552063528976493, 712586027367532285, 5670208180984550404, 45119129008579140295, 359023114495874158597, 2856828125609676390337, 22732427551596193661092, 180887067631073776895671, 1439359310066138202733669, 11453307583528500522710161 , 91136558943739928322056440, 725194212722622449725017379, 5770534374585076614825317965, 45917447194258042787620761253, 365375512902819142817084818816, 2907375596561773314359283478771, 23134645210148962474982680469725, 184087604516046173128308070261141, 1464826705947916780584489522161740, 11655957412771864930381705692147919, 92749089470233943609379176035328917, 738025482842919374038021847620081945] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 339, 2778, 21852, 174756, 1389591, 11057124, 88000188, 700176366, 5571663435, 44334639678, 352780951656, 2807160623016, 22337198161047, 177742185900360, 1414334671882848, 11254180988430258, 89552064023793003, 712586025302177778, 5670208186786551540, 45119128998215742300, 359023114495505928855, 2856828125703378620892, 22732427551135974405348, 180887067632515512032262, 1439359310063097366919563, 11453307583530722233306230 , 91136558943755941463531808, 725194212722523548088096192, 5770534374585423185852872599, 45917447194257207994097657376, 365375512902820190809911412296, 2907375596561775547322046949098, 23134645210148942129723108032683, 184087604516046253853462451982410, 1464826705947916562259864784651500, 11655957412771865292566744909987796, 92749089470233943758049629307175831, 738025482842919370086639203881416852] Using the found enumerative automaton with, 22, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (2 t - 1) (888 t + 704 t + 546 t + 25 t - 318 t - 113 t - 91 t + 36 t 3 2 / + 20 t - 5 t - t + 1) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation (2*t-1)*(888*t^11+704*t^10+546*t^9+25*t^8-318*t^7-113*t^6-91*t^5+36*t^4+20*t^3-\ 5*t^2-t+1)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^ 4-18*t^3-14*t^2+10*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 - 3 t (648 t + 280 t + 22 t - 322 t - 291 t + 89 t + 50 t + 68 t 3 2 / - 9 t - 21 t + 2) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation -3*t*(648*t^11+280*t^10+22*t^9-322*t^8-291*t^7+89*t^6+50*t^5+68*t^4-9*t^3-21*t^ 2+2)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^4-18*t ^3-14*t^2+10*t-1) This theorem took, 0.336, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 29, :consider the sequence, (x y + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 5, 4, 16, 20, 5, 20, 41, 4, 16, 20, 16, 64, 80, 20, 80, 164, 5, 20, 41, 20, 80, 164, 41, 164, 365, 4, 16, 20, 16, 64, 80, 20, 80, 164, 16, 64, 80, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 4, 0, 0, 16, 4, 16, 40, 0, 0, 16, 0, 0, 64, 16, 64, 160, 4, 16, 40, 16, 64, 160, 40, 160, 364, 0, 0, 16, 0, 0, 64, 16, 64, 160, 0, 0, 64, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [542934428254863952002380212403205877340118100850670114517356747234881622893159\ 1747510754240102400, 5429344282548639520023802124032058773401181008506700788361\ 644295858845964360099385136970144415744] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401, 54709494565756179605, 492385451091805616441, 4431469059826250547965, 39883221538436254931681, 358948993845926294385125, 3230540944613336649466121, 29074868501520029845195085, 261673816513680268606755761, 2355064348623122417460801845, 21195579137608101757147216601, 190760212238472915814324949405, 1716841910146256242328924544641, 15451577191316306180960320901765, 139064194721846755628642888115881, 1251577752496620800657785993042925, 11264199772469587205920073937386321, 101377797952226284853280665436476885, 912400181570036563679525988928291961, 8211601634130329073115733900354627645, 73904414707172961658041605103191648801] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044, 6078832729528464400, 54709494565756179604, 492385451091805616440, 4431469059826250547964, 39883221538436254931680, 358948993845926294385124, 3230540944613336649466120, 29074868501520029845195084, 261673816513680268606755760, 2355064348623122417460801844, 21195579137608101757147216600, 190760212238472915814324949404, 1716841910146256242328924544640, 15451577191316306180960320901764, 139064194721846755628642888115880, 1251577752496620800657785993042924, 11264199772469587205920073937386320, 101377797952226284853280665436476884, 912400181570036563679525988928291960, 8211601634130329073115733900354627644, 73904414707172961658041605103191648800] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 t - 1 - ----------------- (9 t - 1) (t - 1) and in Maple notation -(5*t-1)/(9*t-1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 t ----------------- (9 t - 1) (t - 1) and in Maple notation 4*t/(9*t-1)/(t-1) This theorem took, 0.021, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 30, :consider the sequence, (x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 15, 4, 16, 49, 4, 16, 16, 16, 64, 59, 15, 60, 184, 4, 16, 52, 16, 64, 196, 49, 196, 369, 4, 16, 16, 16, 64, 60, 16, 64, 196, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 23, 6, 24, 42, 0, 0, 24, 0, 0, 91, 23, 92, 150, 6, 24, 48, 24, 96, 177, 42, 168, 404, 0, 0, 24, 0, 0, 92, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [747339094487699982121564378687277951577525511881252912369088384795115785338052\ 69945077610210918400, 747339094487699982121564378687282727623242254678730129199\ 87712946666296423893935093002389789081600] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 49, 369, 3210, 25840, 212665, 1724976, 14041578, 114019654, 926439241, 7524495357, 61120594098, 496442507344, 4032360311650, 32752537815387, 266030943070788, 2160819811742896, 17551135597159069, 142558052571733767, 1157919405158141736, 9405131824430543218, 76392627861689872558, 620494603082112039414, 5039930742018999473403, 40936539526388788133704, 332504622592612951392712, 2700749142345287121789600, 21936675266822335897710654 , 178179348165167763938082373, 1447251223209246674987990287, 11755212512751057641364570132, 95481018778399952240234702283, 775538931096371392155172792609, 6299269125335721474501977651527, 51165441117576203279911260647883, 415588271062611423754874475461130, 3375591166074677961605757104597557, 27418039713553112130111943812207241, 222701406879241761802628836736176599, 1808879013384671796641747886704306286] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 404, 3176, 26352, 212277, 1730087, 14034245, 114066915, 926322978, 7524929753, 61119038741, 496446865281, 4032342032217, 32752586605502, 266030743040072, 2160820389386700, 17551133463188112, 142558059420112892, 1157919382180563701, 9405131903735812263, 76392627608212572648, 620494603978481836934, 5039930739164022367049, 40936539536373286186533, 332504622560158676462946, 2700749142456086838143531, 21936675266453677071164585 , 178179348166401517994198868, 1447251223205080807525721682, 11755212512764872120504260924, 95481018778353119610830333096, 775538931096526722707846888670, 6299269125335196502736012168205, 51165441117577952677922350208321, 415588271062605542107712797361339, 3375591166074697657193597931652176, 27418039713553046174860582973889609, 222701406879241983285685560773757116, 1808879013384671056147886699508179612] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 3.854, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 31, :consider the sequence, (x y + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 14, 4, 16, 43, 4, 16, 16, 16, 64, 56, 14, 56, 152, 4, 16, 52, 16, 64, 185, 43, 172, 349, 4, 16, 16, 16, 64, 56, 16, 64, 172, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 42, 0, 0, 24, 0, 0, 88, 22, 88, 148, 6, 24, 48, 24, 96, 172, 42, 168, 339, 0, 0, 24, 0, 0, 88, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [116563615066120903652891837234710881448427048780539622565378717461555241643347\ 55549568012606504960, 116563615066120903652891837234710881448427036365241000233\ 98679273181765473225564450431987393495040] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 43, 349, 2725, 22012, 174463, 1389685, 11059369, 87988684, 700214287, 5571580405, 44334719065, 352781310064, 2807158177723, 22337207138749, 177742163416045, 1414334703690856, 11254181031837547, 89552063528976493, 712586027367532285, 5670208180984550404, 45119129008579140295, 359023114495874158597, 2856828125609676390337, 22732427551596193661092, 180887067631073776895671, 1439359310066138202733669, 11453307583528500522710161 , 91136558943739928322056440, 725194212722622449725017379, 5770534374585076614825317965, 45917447194258042787620761253, 365375512902819142817084818816, 2907375596561773314359283478771, 23134645210148962474982680469725, 184087604516046173128308070261141, 1464826705947916780584489522161740, 11655957412771864930381705692147919, 92749089470233943609379176035328917, 738025482842919374038021847620081945] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 339, 2778, 21852, 174756, 1389591, 11057124, 88000188, 700176366, 5571663435, 44334639678, 352780951656, 2807160623016, 22337198161047, 177742185900360, 1414334671882848, 11254180988430258, 89552064023793003, 712586025302177778, 5670208186786551540, 45119128998215742300, 359023114495505928855, 2856828125703378620892, 22732427551135974405348, 180887067632515512032262, 1439359310063097366919563, 11453307583530722233306230 , 91136558943755941463531808, 725194212722523548088096192, 5770534374585423185852872599, 45917447194257207994097657376, 365375512902820190809911412296, 2907375596561775547322046949098, 23134645210148942129723108032683, 184087604516046253853462451982410, 1464826705947916562259864784651500, 11655957412771865292566744909987796, 92749089470233943758049629307175831, 738025482842919370086639203881416852] Using the found enumerative automaton with, 22, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (2 t - 1) (888 t + 704 t + 546 t + 25 t - 318 t - 113 t - 91 t + 36 t 3 2 / + 20 t - 5 t - t + 1) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation (2*t-1)*(888*t^11+704*t^10+546*t^9+25*t^8-318*t^7-113*t^6-91*t^5+36*t^4+20*t^3-\ 5*t^2-t+1)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^ 4-18*t^3-14*t^2+10*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 3 - 3 (648 t + 280 t + 22 t - 322 t - 291 t + 89 t + 50 t + 68 t - 9 t 2 / - 21 t + 2) t / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation -3*(648*t^11+280*t^10+22*t^9-322*t^8-291*t^7+89*t^6+50*t^5+68*t^4-9*t^3-21*t^2+ 2)*t/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^4-18*t ^3-14*t^2+10*t-1) This theorem took, 0.271, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 32, :consider the sequence, (x y + x y + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 13, 4, 16, 46, 4, 16, 16, 16, 64, 52, 13, 52, 154, 4, 16, 52, 16, 64, 178, 46, 184, 313, 4, 16, 16, 16, 64, 52, 16, 64, 184, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 21, 6, 24, 33, 0, 0, 24, 0, 0, 84, 21, 84, 120, 6, 24, 48, 24, 96, 162, 33, 132, 303, 0, 0, 24, 0, 0, 84, 24, 96, 132, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [162316748584659891246977075565457621824621635325509435509837698244858683853609\ 7047665544334934016, 1623167485846598912469770755654522568268735722082864926800\ 406568834215379011455736718455665065984] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 46, 313, 2443, 18694, 145024, 1125067, 8740333, 67913800, 527787922, 4101833917, 31879096495, 247763224426, 1925612884612, 14965856546191, 116314634546257, 903997464560236, 7025869711149910, 54605071718117185, 424390717158130387, 3298365440996264782, 25634902361689748488, 199234509093056834515, 1548450977567390015029, 12034563896767430833168, 93532653140488837738138, 726935955357163089306949, 5649747606322486182613111, 43909848976338613186014898, 341267428472119909955878156, 2652331093162315137596620183, 20613922222967257715049565081, 160211442119739691655800933684, 1245163628175966378458605912414, 9677414049950157770671994517193, 75212880118706767580448756539803, 584554645130647914431328107495830, 4543159796626774790736103092993424, 35309446447171307374439733164896987, 274425083909959789423598476388910205] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 33, 303, 2346, 18528, 144267, 1123305, 8734020, 67896906, 527733141, 4101677763, 31878611646, 247761802644, 1925608553727, 14965843684605, 116314595699352, 903997348543806, 7025869362052041, 54605070672920727, 424390714018346706, 3298365431585302344, 25634902333440083955, 199234509008341395345, 1548450977313176588652, 12034563896004924771762, 93532653138201051118461, 726935955350300266318827, 5649747606301896639906918, 43909848976276846705379964, 341267428471934606219006055, 2652331093161759234975938469, 20613922222965589990007650752, 160211442119734688515034929062, 1245163628175951368967588421809, 9677414049950112742336380998847, 75212880118706632495167038077818, 584554645130647509176032707923760, 4543159796626773574969117382649435, 35309446447171303727140975057120569, 274425083909959778481697804019069844] Using the found enumerative automaton with, 14, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 36 t - 13 t + 5 t - 1 -------------------------------------------- 2 (t - 1) (3 t - 1) (2 t + 1) (6 t + 7 t - 1) and in Maple notation (36*t^4+36*t^3-13*t^2+5*t-1)/(t-1)/(3*t-1)/(2*t+1)/(6*t^2+7*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 3 t (12 t + 8 t - 7 t + 2) - -------------------------------------------- 2 (t - 1) (3 t - 1) (2 t + 1) (6 t + 7 t - 1) and in Maple notation -3*t*(12*t^3+8*t^2-7*t+2)/(t-1)/(3*t-1)/(2*t+1)/(6*t^2+7*t-1) This theorem took, 0.145, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 33, :consider the sequence, (x y + x y + x + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.131, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 34, :consider the sequence, (x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 14, 4, 16, 43, 4, 16, 16, 16, 64, 56, 14, 56, 152, 4, 16, 52, 16, 64, 185, 43, 172, 349, 4, 16, 16, 16, 64, 56, 16, 64, 172, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 42, 0, 0, 24, 0, 0, 88, 22, 88, 148, 6, 24, 48, 24, 96, 172, 42, 168, 339, 0, 0, 24, 0, 0, 88, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [116563615066120903652891837234710881448427048780539622565378717461555241643347\ 55549568012606504960, 116563615066120903652891837234710881448427036365241000233\ 98679273181765473225564450431987393495040] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 43, 349, 2725, 22012, 174463, 1389685, 11059369, 87988684, 700214287, 5571580405, 44334719065, 352781310064, 2807158177723, 22337207138749, 177742163416045, 1414334703690856, 11254181031837547, 89552063528976493, 712586027367532285, 5670208180984550404, 45119129008579140295, 359023114495874158597, 2856828125609676390337, 22732427551596193661092, 180887067631073776895671, 1439359310066138202733669, 11453307583528500522710161 , 91136558943739928322056440, 725194212722622449725017379, 5770534374585076614825317965, 45917447194258042787620761253, 365375512902819142817084818816, 2907375596561773314359283478771, 23134645210148962474982680469725, 184087604516046173128308070261141, 1464826705947916780584489522161740, 11655957412771864930381705692147919, 92749089470233943609379176035328917, 738025482842919374038021847620081945] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 339, 2778, 21852, 174756, 1389591, 11057124, 88000188, 700176366, 5571663435, 44334639678, 352780951656, 2807160623016, 22337198161047, 177742185900360, 1414334671882848, 11254180988430258, 89552064023793003, 712586025302177778, 5670208186786551540, 45119128998215742300, 359023114495505928855, 2856828125703378620892, 22732427551135974405348, 180887067632515512032262, 1439359310063097366919563, 11453307583530722233306230 , 91136558943755941463531808, 725194212722523548088096192, 5770534374585423185852872599, 45917447194257207994097657376, 365375512902820190809911412296, 2907375596561775547322046949098, 23134645210148942129723108032683, 184087604516046253853462451982410, 1464826705947916562259864784651500, 11655957412771865292566744909987796, 92749089470233943758049629307175831, 738025482842919370086639203881416852] Using the found enumerative automaton with, 22, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (2 t - 1) (888 t + 704 t + 546 t + 25 t - 318 t - 113 t - 91 t + 36 t 3 2 / + 20 t - 5 t - t + 1) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation (2*t-1)*(888*t^11+704*t^10+546*t^9+25*t^8-318*t^7-113*t^6-91*t^5+36*t^4+20*t^3-\ 5*t^2-t+1)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^ 4-18*t^3-14*t^2+10*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 - 3 t (648 t + 280 t + 22 t - 322 t - 291 t + 89 t + 50 t + 68 t 3 2 / - 9 t - 21 t + 2) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation -3*t*(648*t^11+280*t^10+22*t^9-322*t^8-291*t^7+89*t^6+50*t^5+68*t^4-9*t^3-21*t^ 2+2)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^4-18*t ^3-14*t^2+10*t-1) This theorem took, 0.287, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 35, :consider the sequence, (x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is t (6 t - 5) - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -t*(6*t-5)/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.040, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 36, :consider the sequence, (x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.058, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 37, :consider the sequence, (x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 15, 4, 16, 49, 4, 16, 16, 16, 64, 59, 15, 60, 184, 4, 16, 52, 16, 64, 196, 49, 196, 369, 4, 16, 16, 16, 64, 60, 16, 64, 196, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 23, 6, 24, 42, 0, 0, 24, 0, 0, 91, 23, 92, 150, 6, 24, 48, 24, 96, 177, 42, 168, 404, 0, 0, 24, 0, 0, 92, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [747339094487699982121564378687277951577525511881252912369088384795115785338052\ 69945077610210918400, 747339094487699982121564378687282727623242254678730129199\ 87712946666296423893935093002389789081600] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 49, 369, 3210, 25840, 212665, 1724976, 14041578, 114019654, 926439241, 7524495357, 61120594098, 496442507344, 4032360311650, 32752537815387, 266030943070788, 2160819811742896, 17551135597159069, 142558052571733767, 1157919405158141736, 9405131824430543218, 76392627861689872558, 620494603082112039414, 5039930742018999473403, 40936539526388788133704, 332504622592612951392712, 2700749142345287121789600, 21936675266822335897710654 , 178179348165167763938082373, 1447251223209246674987990287, 11755212512751057641364570132, 95481018778399952240234702283, 775538931096371392155172792609, 6299269125335721474501977651527, 51165441117576203279911260647883, 415588271062611423754874475461130, 3375591166074677961605757104597557, 27418039713553112130111943812207241, 222701406879241761802628836736176599, 1808879013384671796641747886704306286] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 404, 3176, 26352, 212277, 1730087, 14034245, 114066915, 926322978, 7524929753, 61119038741, 496446865281, 4032342032217, 32752586605502, 266030743040072, 2160820389386700, 17551133463188112, 142558059420112892, 1157919382180563701, 9405131903735812263, 76392627608212572648, 620494603978481836934, 5039930739164022367049, 40936539536373286186533, 332504622560158676462946, 2700749142456086838143531, 21936675266453677071164585 , 178179348166401517994198868, 1447251223205080807525721682, 11755212512764872120504260924, 95481018778353119610830333096, 775538931096526722707846888670, 6299269125335196502736012168205, 51165441117577952677922350208321, 415588271062605542107712797361339, 3375591166074697657193597931652176, 27418039713553046174860582973889609, 222701406879241983285685560773757116, 1808879013384671056147886699508179612] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 3.870, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 38, :consider the sequence, (x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.042, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 39, :consider the sequence, (x y + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 13, 4, 16, 46, 4, 16, 16, 16, 64, 52, 13, 52, 154, 4, 16, 52, 16, 64, 178, 46, 184, 313, 4, 16, 16, 16, 64, 52, 16, 64, 184, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 21, 6, 24, 33, 0, 0, 24, 0, 0, 84, 21, 84, 120, 6, 24, 48, 24, 96, 162, 33, 132, 303, 0, 0, 24, 0, 0, 84, 24, 96, 132, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [162316748584659891246977075565457621824621635325509435509837698244858683853609\ 7047665544334934016, 1623167485846598912469770755654522568268735722082864926800\ 406568834215379011455736718455665065984] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 46, 313, 2443, 18694, 145024, 1125067, 8740333, 67913800, 527787922, 4101833917, 31879096495, 247763224426, 1925612884612, 14965856546191, 116314634546257, 903997464560236, 7025869711149910, 54605071718117185, 424390717158130387, 3298365440996264782, 25634902361689748488, 199234509093056834515, 1548450977567390015029, 12034563896767430833168, 93532653140488837738138, 726935955357163089306949, 5649747606322486182613111, 43909848976338613186014898, 341267428472119909955878156, 2652331093162315137596620183, 20613922222967257715049565081, 160211442119739691655800933684, 1245163628175966378458605912414, 9677414049950157770671994517193, 75212880118706767580448756539803, 584554645130647914431328107495830, 4543159796626774790736103092993424, 35309446447171307374439733164896987, 274425083909959789423598476388910205] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 33, 303, 2346, 18528, 144267, 1123305, 8734020, 67896906, 527733141, 4101677763, 31878611646, 247761802644, 1925608553727, 14965843684605, 116314595699352, 903997348543806, 7025869362052041, 54605070672920727, 424390714018346706, 3298365431585302344, 25634902333440083955, 199234509008341395345, 1548450977313176588652, 12034563896004924771762, 93532653138201051118461, 726935955350300266318827, 5649747606301896639906918, 43909848976276846705379964, 341267428471934606219006055, 2652331093161759234975938469, 20613922222965589990007650752, 160211442119734688515034929062, 1245163628175951368967588421809, 9677414049950112742336380998847, 75212880118706632495167038077818, 584554645130647509176032707923760, 4543159796626773574969117382649435, 35309446447171303727140975057120569, 274425083909959778481697804019069844] Using the found enumerative automaton with, 14, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 36 t - 13 t + 5 t - 1 -------------------------------------------- 2 (t - 1) (3 t - 1) (2 t + 1) (6 t + 7 t - 1) and in Maple notation (36*t^4+36*t^3-13*t^2+5*t-1)/(t-1)/(3*t-1)/(2*t+1)/(6*t^2+7*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 3 t (12 t + 8 t - 7 t + 2) - -------------------------------------------- 2 (t - 1) (3 t - 1) (2 t + 1) (6 t + 7 t - 1) and in Maple notation -3*t*(12*t^3+8*t^2-7*t+2)/(t-1)/(3*t-1)/(2*t+1)/(6*t^2+7*t-1) This theorem took, 0.146, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 40, :consider the sequence, (x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.040, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 41, :consider the sequence, (x y + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 46, 4, 16, 16, 16, 64, 68, 17, 68, 179, 4, 16, 52, 16, 64, 200, 46, 184, 434, 4, 16, 16, 16, 64, 68, 16, 64, 184, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 48, 0, 0, 24, 0, 0, 88, 22, 88, 184, 6, 24, 48, 24, 96, 190, 48, 192, 436, 0, 0, 24, 0, 0, 88, 24, 96, 192, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [379580757157229078897899883886434775259298623761228624785450749345153966119653\ 206371860480000000000, 37958075715722907889789988388643477525929862374142158415\ 6884664946767978535653206371860480000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 46, 434, 3976, 36122, 326788, 2949668, 26590156, 239528108, 2156837704, 19416962732, 174779748592, 1573152922052, 14159050770688, 127434821338964, 1146930171864064, 10322455228350980, 92902514355311920, 836124710107083380, 7525132767412429696, 67726246648201614020, 609536477837385420496, 5485829587038388242068, 49372472698304325181792, 444352286271960688250852, 3999170735946909299281264, 35992537418839687904043956, 323932840735278727355212672, 2915395586391929570194549892, 26238560376129262385869047184, 236147043876825441652901468756, 2125323397343020631758708354720, 19127910588311641600737044818724, 172151195355759998536807772610736, 1549360758505783108805464730220788, 13944246828067609965818627438809024, 125498221460165588229866576665803716, 1129483993179172512690483368876425168, 10165355938800448747929023951299847060, 91488203450140951639119332713295182048] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 48, 436, 3978, 36124, 326790, 2949670, 26590158, 239528110, 2156837706, 19416962734, 174779748594, 1573152922054, 14159050770690, 127434821338966, 1146930171864066, 10322455228350982, 92902514355311922, 836124710107083382, 7525132767412429698, 67726246648201614022, 609536477837385420498, 5485829587038388242070, 49372472698304325181794, 444352286271960688250854, 3999170735946909299281266, 35992537418839687904043958, 323932840735278727355212674, 2915395586391929570194549894, 26238560376129262385869047186, 236147043876825441652901468758, 2125323397343020631758708354722, 19127910588311641600737044818726, 172151195355759998536807772610738, 1549360758505783108805464730220790, 13944246828067609965818627438809026, 125498221460165588229866576665803718, 1129483993179172512690483368876425170, 10165355938800448747929023951299847062, 91488203450140951639119332713295182050] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 8 7 6 5 4 3 2 162 t - 26 t + 502 t - 752 t + 226 t + 81 t - 61 t + 13 t - 1 -------------------------------------------------------------------- 6 5 4 3 2 (t - 1) (9 t - 1) (12 t + 4 t + 38 t - 38 t - 4 t + 7 t - 1) and in Maple notation (162*t^8-26*t^7+502*t^6-752*t^5+226*t^4+81*t^3-61*t^2+13*t-1)/(t-1)/(9*t-1)/(12 *t^6+4*t^5+38*t^4-38*t^3-4*t^2+7*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 2 (81 t - 5 t + 244 t - 363 t + 80 t + 59 t - 27 t + 3) t - ----------------------------------------------------------------- 6 5 4 3 2 (t - 1) (9 t - 1) (12 t + 4 t + 38 t - 38 t - 4 t + 7 t - 1) and in Maple notation -2*(81*t^7-5*t^6+244*t^5-363*t^4+80*t^3+59*t^2-27*t+3)*t/(t-1)/(9*t-1)/(12*t^6+ 4*t^5+38*t^4-38*t^3-4*t^2+7*t-1) This theorem took, 0.488, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 42, :consider the sequence, (x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 5, 4, 16, 20, 5, 20, 41, 4, 16, 20, 16, 64, 80, 20, 80, 164, 5, 20, 41, 20, 80, 164, 41, 164, 365, 4, 16, 20, 16, 64, 80, 20, 80, 164, 16, 64, 80, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 4, 0, 0, 16, 4, 16, 40, 0, 0, 16, 0, 0, 64, 16, 64, 160, 4, 16, 40, 16, 64, 160, 40, 160, 364, 0, 0, 16, 0, 0, 64, 16, 64, 160, 0, 0, 64, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [542934428254863952002380212403205877340118100850670114517356747234881622893159\ 1747510754240102400, 5429344282548639520023802124032058773401181008506700788361\ 644295858845964360099385136970144415744] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401, 54709494565756179605, 492385451091805616441, 4431469059826250547965, 39883221538436254931681, 358948993845926294385125, 3230540944613336649466121, 29074868501520029845195085, 261673816513680268606755761, 2355064348623122417460801845, 21195579137608101757147216601, 190760212238472915814324949405, 1716841910146256242328924544641, 15451577191316306180960320901765, 139064194721846755628642888115881, 1251577752496620800657785993042925, 11264199772469587205920073937386321, 101377797952226284853280665436476885, 912400181570036563679525988928291961, 8211601634130329073115733900354627645, 73904414707172961658041605103191648801] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044, 6078832729528464400, 54709494565756179604, 492385451091805616440, 4431469059826250547964, 39883221538436254931680, 358948993845926294385124, 3230540944613336649466120, 29074868501520029845195084, 261673816513680268606755760, 2355064348623122417460801844, 21195579137608101757147216600, 190760212238472915814324949404, 1716841910146256242328924544640, 15451577191316306180960320901764, 139064194721846755628642888115880, 1251577752496620800657785993042924, 11264199772469587205920073937386320, 101377797952226284853280665436476884, 912400181570036563679525988928291960, 8211601634130329073115733900354627644, 73904414707172961658041605103191648800] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 t - 1 - ----------------- (9 t - 1) (t - 1) and in Maple notation -(5*t-1)/(9*t-1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 t ----------------- (9 t - 1) (t - 1) and in Maple notation 4*t/(9*t-1)/(t-1) This theorem took, 0.021, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 43, :consider the sequence, (x y + x + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.061, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 44, :consider the sequence, (x y + x + x y + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.042, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 45, :consider the sequence, (x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.060, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 46, :consider the sequence, (x y + x y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 5, 4, 16, 20, 5, 20, 41, 4, 16, 20, 16, 64, 80, 20, 80, 164, 5, 20, 41, 20, 80, 164, 41, 164, 365, 4, 16, 20, 16, 64, 80, 20, 80, 164, 16, 64, 80, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 4, 0, 0, 16, 4, 16, 40, 0, 0, 16, 0, 0, 64, 16, 64, 160, 4, 16, 40, 16, 64, 160, 40, 160, 364, 0, 0, 16, 0, 0, 64, 16, 64, 160, 0, 0, 64, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [542934428254863952002380212403205877340118100850670114517356747234881622893159\ 1747510754240102400, 5429344282548639520023802124032058773401181008506700788361\ 644295858845964360099385136970144415744] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401, 54709494565756179605, 492385451091805616441, 4431469059826250547965, 39883221538436254931681, 358948993845926294385125, 3230540944613336649466121, 29074868501520029845195085, 261673816513680268606755761, 2355064348623122417460801845, 21195579137608101757147216601, 190760212238472915814324949405, 1716841910146256242328924544641, 15451577191316306180960320901765, 139064194721846755628642888115881, 1251577752496620800657785993042925, 11264199772469587205920073937386321, 101377797952226284853280665436476885, 912400181570036563679525988928291961, 8211601634130329073115733900354627645, 73904414707172961658041605103191648801] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044, 6078832729528464400, 54709494565756179604, 492385451091805616440, 4431469059826250547964, 39883221538436254931680, 358948993845926294385124, 3230540944613336649466120, 29074868501520029845195084, 261673816513680268606755760, 2355064348623122417460801844, 21195579137608101757147216600, 190760212238472915814324949404, 1716841910146256242328924544640, 15451577191316306180960320901764, 139064194721846755628642888115880, 1251577752496620800657785993042924, 11264199772469587205920073937386320, 101377797952226284853280665436476884, 912400181570036563679525988928291960, 8211601634130329073115733900354627644, 73904414707172961658041605103191648800] Using the found enumerative automaton with, 2, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 t - 1 - ----------------- (9 t - 1) (t - 1) and in Maple notation -(5*t-1)/(9*t-1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 t ----------------- (9 t - 1) (t - 1) and in Maple notation 4*t/(9*t-1)/(t-1) This theorem took, 0.022, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 47, :consider the sequence, (x y + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 14, 4, 16, 43, 4, 16, 16, 16, 64, 56, 14, 56, 152, 4, 16, 52, 16, 64, 185, 43, 172, 349, 4, 16, 16, 16, 64, 56, 16, 64, 172, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 42, 0, 0, 24, 0, 0, 88, 22, 88, 148, 6, 24, 48, 24, 96, 172, 42, 168, 339, 0, 0, 24, 0, 0, 88, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [116563615066120903652891837234710881448427048780539622565378717461555241643347\ 55549568012606504960, 116563615066120903652891837234710881448427036365241000233\ 98679273181765473225564450431987393495040] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 43, 349, 2725, 22012, 174463, 1389685, 11059369, 87988684, 700214287, 5571580405, 44334719065, 352781310064, 2807158177723, 22337207138749, 177742163416045, 1414334703690856, 11254181031837547, 89552063528976493, 712586027367532285, 5670208180984550404, 45119129008579140295, 359023114495874158597, 2856828125609676390337, 22732427551596193661092, 180887067631073776895671, 1439359310066138202733669, 11453307583528500522710161 , 91136558943739928322056440, 725194212722622449725017379, 5770534374585076614825317965, 45917447194258042787620761253, 365375512902819142817084818816, 2907375596561773314359283478771, 23134645210148962474982680469725, 184087604516046173128308070261141, 1464826705947916780584489522161740, 11655957412771864930381705692147919, 92749089470233943609379176035328917, 738025482842919374038021847620081945] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 339, 2778, 21852, 174756, 1389591, 11057124, 88000188, 700176366, 5571663435, 44334639678, 352780951656, 2807160623016, 22337198161047, 177742185900360, 1414334671882848, 11254180988430258, 89552064023793003, 712586025302177778, 5670208186786551540, 45119128998215742300, 359023114495505928855, 2856828125703378620892, 22732427551135974405348, 180887067632515512032262, 1439359310063097366919563, 11453307583530722233306230 , 91136558943755941463531808, 725194212722523548088096192, 5770534374585423185852872599, 45917447194257207994097657376, 365375512902820190809911412296, 2907375596561775547322046949098, 23134645210148942129723108032683, 184087604516046253853462451982410, 1464826705947916562259864784651500, 11655957412771865292566744909987796, 92749089470233943758049629307175831, 738025482842919370086639203881416852] Using the found enumerative automaton with, 22, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (2 t - 1) (888 t + 704 t + 546 t + 25 t - 318 t - 113 t - 91 t + 36 t 3 2 / + 20 t - 5 t - t + 1) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation (2*t-1)*(888*t^11+704*t^10+546*t^9+25*t^8-318*t^7-113*t^6-91*t^5+36*t^4+20*t^3-\ 5*t^2-t+1)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^ 4-18*t^3-14*t^2+10*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 3 - 3 (648 t + 280 t + 22 t - 322 t - 291 t + 89 t + 50 t + 68 t - 9 t 2 / - 21 t + 2) t / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation -3*(648*t^11+280*t^10+22*t^9-322*t^8-291*t^7+89*t^6+50*t^5+68*t^4-9*t^3-21*t^2+ 2)*t/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^4-18*t ^3-14*t^2+10*t-1) This theorem took, 0.361, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 48, :consider the sequence, (x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.039, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 49, :consider the sequence, (x y + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.038, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 50, :consider the sequence, (x y + x y + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.040, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 51, :consider the sequence, (x y + x y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 17, 4, 16, 44, 4, 16, 16, 16, 64, 69, 17, 68, 170, 4, 16, 52, 16, 64, 192, 44, 176, 410, 4, 16, 16, 16, 64, 68, 16, 64, 176, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 49, 0, 0, 24, 0, 0, 86, 22, 88, 184, 6, 24, 48, 24, 96, 191, 49, 196, 418, 0, 0, 24, 0, 0, 88, 24, 96, 196, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [217875889884372449244187889686377705298141545000054378271632246665473249434371\ 241279488000000000000, 21787588988437244924418788968637770529814154500005443389\ 7947109279913286459141822349312000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 44, 410, 3617, 31448, 272192, 2351906, 20307377, 175287308, 1512813860, 13055475818, 112664563601, 972246266144, 8390013411224, 72401549603378, 624787800346241, 5391591523460948, 46526599779321308, 401500045812176954, 3464733708282150209, 29898824466429240296, 258011082648913641776, 2226499524444295508546, 19213516221415151448209, 165802508009305393094684, 1430788166693458308879572, 12346946992084781366929418, 106547638260126504987698993, 919449903334746713385460016, 7934367561136018861352916488, 68469405854911787985048994130, 590854847848243505723120902625, 5098765599995855453576775491684, 43999657002659278604666159633420, 379693825570754862906404801076698, 3276557386977332434386489343966049, 28274961527269740899280951625118456, 243998000018575327530620920794955808, 2105574006021028942267908547469880674, 18169992764259958127701080578922627185] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 49, 418, 3628, 31462, 272209, 2351926, 20307400, 175287334, 1512813889, 13055475850, 112664563636, 972246266182, 8390013411265, 72401549603422, 624787800346288, 5391591523460998, 46526599779321361, 401500045812177010, 3464733708282150268, 29898824466429240358, 258011082648913641841, 2226499524444295508614, 19213516221415151448280, 165802508009305393094758, 1430788166693458308879649, 12346946992084781366929498, 106547638260126504987699076, 919449903334746713385460102, 7934367561136018861352916577, 68469405854911787985048994222, 590854847848243505723120902720, 5098765599995855453576775491782, 43999657002659278604666159633521, 379693825570754862906404801076802, 3276557386977332434386489343966156, 28274961527269740899280951625118566, 243998000018575327530620920794955921, 2105574006021028942267908547469880790, 18169992764259958127701080578922627304] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 36 t + 46 t - 40 t + 10 t - 1 - ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation -(36*t^4+46*t^3-40*t^2+10*t-1)/(18*t^3+27*t^2-12*t+1)/(t-1)^2 The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (36 t + 44 t - 35 t + 6) t ----------------------------------- 3 2 2 (18 t + 27 t - 12 t + 1) (t - 1) and in Maple notation (36*t^3+44*t^2-35*t+6)*t/(18*t^3+27*t^2-12*t+1)/(t-1)^2 This theorem took, 0.060, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 52, :consider the sequence, (x y + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 15, 4, 16, 49, 4, 16, 16, 16, 64, 59, 15, 60, 184, 4, 16, 52, 16, 64, 196, 49, 196, 369, 4, 16, 16, 16, 64, 60, 16, 64, 196, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 23, 6, 24, 42, 0, 0, 24, 0, 0, 91, 23, 92, 150, 6, 24, 48, 24, 96, 177, 42, 168, 404, 0, 0, 24, 0, 0, 92, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [747339094487699982121564378687277951577525511881252912369088384795115785338052\ 69945077610210918400, 747339094487699982121564378687282727623242254678730129199\ 87712946666296423893935093002389789081600] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 49, 369, 3210, 25840, 212665, 1724976, 14041578, 114019654, 926439241, 7524495357, 61120594098, 496442507344, 4032360311650, 32752537815387, 266030943070788, 2160819811742896, 17551135597159069, 142558052571733767, 1157919405158141736, 9405131824430543218, 76392627861689872558, 620494603082112039414, 5039930742018999473403, 40936539526388788133704, 332504622592612951392712, 2700749142345287121789600, 21936675266822335897710654 , 178179348165167763938082373, 1447251223209246674987990287, 11755212512751057641364570132, 95481018778399952240234702283, 775538931096371392155172792609, 6299269125335721474501977651527, 51165441117576203279911260647883, 415588271062611423754874475461130, 3375591166074677961605757104597557, 27418039713553112130111943812207241, 222701406879241761802628836736176599, 1808879013384671796641747886704306286] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 404, 3176, 26352, 212277, 1730087, 14034245, 114066915, 926322978, 7524929753, 61119038741, 496446865281, 4032342032217, 32752586605502, 266030743040072, 2160820389386700, 17551133463188112, 142558059420112892, 1157919382180563701, 9405131903735812263, 76392627608212572648, 620494603978481836934, 5039930739164022367049, 40936539536373286186533, 332504622560158676462946, 2700749142456086838143531, 21936675266453677071164585 , 178179348166401517994198868, 1447251223205080807525721682, 11755212512764872120504260924, 95481018778353119610830333096, 775538931096526722707846888670, 6299269125335196502736012168205, 51165441117577952677922350208321, 415588271062605542107712797361339, 3375591166074697657193597931652176, 27418039713553046174860582973889609, 222701406879241983285685560773757116, 1808879013384671056147886699508179612] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.106, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 53, :consider the sequence, (x y + x y + x + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 3, 4, 16, 8, 3, 12, 28, 4, 16, 12, 16, 64, 28, 8, 32, 80, 3, 12, 34, 12, 48, 98, 28, 112, 156, 4, 16, 12, 16, 64, 32, 12, 48, 112, 16, 64, 48, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 16, 5, 20, 24, 0, 0, 20, 0, 0, 60, 16, 64, 64, 5, 20, 30, 20, 80, 82, 24, 96, 164, 0, 0, 20, 0, 0, 64, 20, 80, 96, 0, 0, 80, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [381424223450415153668548275559857072730819144022276737014291399115211270269358\ 21810794496, 381424223450415153668548275688412116271538366065633704401584407316\ 98750977198164493205504] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 28, 156, 976, 5808, 35008, 209856, 1259776, 7557888, 45349888, 272096256 , 1632587776, 9795514368, 58773127168, 352638713856, 2115832446976, 12694994485248, 76169967566848, 457019804614656, 2742118830309376, 16452712978710528, 98716277882748928, 592297667283910656, 3553786003745406976, 21322716022422110208, 127936296134700433408, 767617776808001273856, 4605706660848678731776, 27634239965091267084288, 165805439790550286860288, 994832638743298499936256, 5968995832459801737035776, 35813974994758797537312768 , 214883849968552828173549568, 1289303099811316917501689856, 7735818598867901676808830976, 46414911593207409854694555648, 278489469559244459815362101248, 1670936817355466758067538886656, 10025620904132800551154012389376] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 24, 164, 960, 5840, 34944, 209984, 1259520, 7558400, 45348864, 272098304 , 1632583680, 9795522560, 58773110784, 352638746624, 2115832381440, 12694994616320, 76169967304704, 457019805138944, 2742118829260800, 16452712980807680, 98716277878554624, 592297667292299264, 3553786003728629760, 21322716022455664640, 127936296134633324544, 767617776808135491584, 4605706660848410296320, 27634239965091803955200, 165805439790549213118464, 994832638743300647419904, 5968995832459797442068480, 35813974994758806127247360 , 214883849968552810993680384, 1289303099811316951861428224, 7735818598867901608089354240, 46414911593207409992133509120, 278489469559244459540484194304, 1670936817355466758617294700544, 10025620904132800550054500761600] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 3 t + 1 ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation (6*t^2-3*t+1)/(6*t-1)/(2*t-1)/(2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is (6 t - 5) t - ----------------------------- (6 t - 1) (2 t - 1) (2 t + 1) and in Maple notation -(6*t-5)*t/(6*t-1)/(2*t-1)/(2*t+1) This theorem took, 0.040, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 54, :consider the sequence, (x y + x y + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 4, 4, 4, 16, 14, 4, 16, 43, 4, 16, 16, 16, 64, 56, 14, 56, 152, 4, 16, 52, 16, 64, 185, 43, 172, 349, 4, 16, 16, 16, 64, 56, 16, 64, 172, 16, 64, 64, 64, 256] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 22, 6, 24, 42, 0, 0, 24, 0, 0, 88, 22, 88, 148, 6, 24, 48, 24, 96, 172, 42, 168, 339, 0, 0, 24, 0, 0, 88, 24, 96, 168, 0, 0, 96, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [116563615066120903652891837234710881448427048780539622565378717461555241643347\ 55549568012606504960, 116563615066120903652891837234710881448427036365241000233\ 98679273181765473225564450431987393495040] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 43, 349, 2725, 22012, 174463, 1389685, 11059369, 87988684, 700214287, 5571580405, 44334719065, 352781310064, 2807158177723, 22337207138749, 177742163416045, 1414334703690856, 11254181031837547, 89552063528976493, 712586027367532285, 5670208180984550404, 45119129008579140295, 359023114495874158597, 2856828125609676390337, 22732427551596193661092, 180887067631073776895671, 1439359310066138202733669, 11453307583528500522710161 , 91136558943739928322056440, 725194212722622449725017379, 5770534374585076614825317965, 45917447194258042787620761253, 365375512902819142817084818816, 2907375596561773314359283478771, 23134645210148962474982680469725, 184087604516046173128308070261141, 1464826705947916780584489522161740, 11655957412771864930381705692147919, 92749089470233943609379176035328917, 738025482842919374038021847620081945] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 339, 2778, 21852, 174756, 1389591, 11057124, 88000188, 700176366, 5571663435, 44334639678, 352780951656, 2807160623016, 22337198161047, 177742185900360, 1414334671882848, 11254180988430258, 89552064023793003, 712586025302177778, 5670208186786551540, 45119128998215742300, 359023114495505928855, 2856828125703378620892, 22732427551135974405348, 180887067632515512032262, 1439359310063097366919563, 11453307583530722233306230 , 91136558943755941463531808, 725194212722523548088096192, 5770534374585423185852872599, 45917447194257207994097657376, 365375512902820190809911412296, 2907375596561775547322046949098, 23134645210148942129723108032683, 184087604516046253853462451982410, 1464826705947916562259864784651500, 11655957412771865292566744909987796, 92749089470233943758049629307175831, 738025482842919370086639203881416852] Using the found enumerative automaton with, 22, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (2 t - 1) (888 t + 704 t + 546 t + 25 t - 318 t - 113 t - 91 t + 36 t 3 2 / + 20 t - 5 t - t + 1) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation (2*t-1)*(888*t^11+704*t^10+546*t^9+25*t^8-318*t^7-113*t^6-91*t^5+36*t^4+20*t^3-\ 5*t^2-t+1)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^ 4-18*t^3-14*t^2+10*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 - 3 t (648 t + 280 t + 22 t - 322 t - 291 t + 89 t + 50 t + 68 t 3 2 / - 9 t - 21 t + 2) / ( / 7 6 5 4 3 2 (28 t + 72 t + 13 t - 17 t - 26 t - 2 t + 3 t + 1) 6 5 4 3 2 (60 t + 20 t - 3 t - 18 t - 14 t + 10 t - 1)) and in Maple notation -3*t*(648*t^11+280*t^10+22*t^9-322*t^8-291*t^7+89*t^6+50*t^5+68*t^4-9*t^3-21*t^ 2+2)/(28*t^7+72*t^6+13*t^5-17*t^4-26*t^3-2*t^2+3*t+1)/(60*t^6+20*t^5-3*t^4-18*t ^3-14*t^2+10*t-1) This theorem took, 0.388, seconds. to state and prove ------------------------------------------------------------------ This concludes this webbook, that took, 25.693, seconds. to generate. k is , 5 Counting the Occurrences of Coefficients that Appear in the Expansion of, n P(x, y) , modolu , 3 For all Polynomials that are Sums of, 5, Monomials taken from, 2 2 2 2 2 2 {1, x, y, x , y , x y, x y , x y, x y } By Shalosh B. Ekhad In this webbook, we will consider the sequences described in the title, that\ after normalization and weeding out obvious symmetry, concerns the following set of, 63, polynomials 2 2 2 2 2 {x y + y + x + y + 1, x + y + x + y + 1, x + x y + y + y + 1, 2 2 2 2 2 x + x y + y + x + y, x y + y + x + y + 1, x y + x y + x + y + 1, 2 2 2 2 2 2 x y + x y + y + y + 1, x y + x y + y + x + 1, x y + x y + y + x + y, 2 2 2 2 2 2 2 2 x y + x + x + y + 1, x y + x + y + y + 1, x y + x + y + x + 1, 2 2 2 2 2 2 2 x y + x + y + x + y, x y + x + x y + y + 1, x y + x + x y + x + 1, 2 2 2 2 2 2 2 2 x y + x + x y + x + y, x y + x + x y + y + 1, x y + x + x y + y + y, 2 2 2 2 2 2 2 2 x y + x + x y + y + x, x y + x y + x + y + 1, x y + x y + y + y + 1, 2 2 2 2 2 2 x y + x y + y + x + 1, x y + x y + y + x + y, 2 2 2 2 x y + x y + x y + y + 1, x y + x y + x y + x + y, 2 2 2 2 2 2 x y + x y + x y + y + 1, x y + x y + x y + y + x, 2 2 2 2 2 2 2 2 x y + x y + x + y + 1, x y + x y + x + y + y, 2 2 2 2 2 2 2 x y + x y + x + x y + y , x y + y + x + y + 1, 2 2 2 2 2 2 2 2 x y + x y + x + y + 1, x y + x y + y + y + 1, x y + x y + y + x + 1, 2 2 2 2 2 2 2 2 2 2 2 x y + x y + y + x + y, x y + x + y + y + 1, x y + x + y + x + y, 2 2 2 2 2 2 2 2 x y + x + x y + y + 1, x y + x + x y + y + y, 2 2 2 2 2 2 2 x y + x y + x + y + 1, x y + x y + y + y + 1, 2 2 2 2 2 2 2 2 x y + x y + y + x + 1, x y + x y + y + x + y, 2 2 2 2 2 2 x y + x y + x y + y + 1, x y + x y + x y + x + 1, 2 2 2 2 2 2 2 x y + x y + x y + x + y, x y + x y + x y + y + 1, 2 2 2 2 2 2 2 2 x y + x y + x y + y + x, x y + x y + x + y + 1, 2 2 2 2 2 2 2 2 x y + x y + x + x + 1, x y + x y + x + x + y, 2 2 2 2 2 2 2 2 2 2 x y + x y + x + y + 1, x y + x y + x + y + y, 2 2 2 2 2 2 2 2 2 x y + x y + x + y + x, x y + x y + x + x y + 1, 2 2 2 2 2 2 2 2 2 x y + x y + x + x y + y, x y + x y + x + x y + y , 2 2 2 2 2 2 2 2 x y + x y + x y + y + 1, x y + x y + x y + x + y, 2 2 2 2 2 2 2 2 2 2 x y + x y + x y + y + 1, x y + x y + x y + y + x, 2 2 2 2 2 2 2 2 2 2 x y + x y + x y + x y + 1, x y + x y + x y + x + y } by finding enumerative automata with at most, 500, states . ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 1, :consider the sequence, (x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 1) (2 t - 1) (3 t - 5 t + 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t-1)*(2*t-1)*(3*t^2-5*t+1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.060, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 2, :consider the sequence, (x + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 3, 5, 25, 8, 3, 15, 55, 5, 25, 15, 25, 125, 33, 8, 40, 185, 3, 15, 73, 15, 75, 246, 55, 275, 295, 5, 25, 15, 25, 125, 40, 15, 75, 275, 25, 125, 75, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 36, 0, 0, 40, 0, 0, 140, 30, 150, 90, 8, 40, 48, 40, 200, 142, 36, 180, 390, 0, 0, 40, 0, 0, 150, 40, 200, 180, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [524569192196997746788106616626123185942745576537979614773144931748694289126433\ 432102203369140625000000, 52456919219700138958117688024303396302753498057758144\ 1643466733014767014537937939167022705078125000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 55, 295, 2673, 16743, 128731, 859309, 6238833, 42802095, 304010935, 2110214689, 14858636997, 103646493867, 727261783723, 5083451774845, 35618882099625, 249182951147607, 1744976448013279, 12211823754516121, 85496729635183533, 598416407769202275, 4189195324948036723, 29323144664212423381, 205267649402496635409, 1436848931416694748303, 10058055844725589746151, 70405895503727705064625, 492843547389970393038549, 3449894862545730230943963, 24149309870287589242579771, 169044968504025294738106669, 1183315701392208340895080185, 8283205874047572754907206215, 57982459661544859718162586991, 405877136438958994551756961609, 2841140328080995352953019847549, 19887980663162536336836380255187, 139215872145600280436663382805891, 974511072159238107645626501834245, 6821577656057012519383321246582113] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 36, 390, 2294, 18638, 121116, 897384, 6085670, 43567910, 300929856, 2125620084, 14796655634, 103956400682, 726014914956, 5089686118680, 35593799037038, 249308366460542, 1744471855962264, 12214346714771196, 85486578835640234, 598467161766918770, 4188991122897499716, 29324165674465108416, 205263541501707795590, 1436869470920638947398, 10057973206726377576720, 70406308693723765911780, 492841884974234265851138, 3449903174624410866881018, 24149276427729612623096316, 169045135716815177835523944, 1183315028633487574023830558, 8283209237841176589263454350, 57982446127762644724053885960, 405877204107870069522300466764, 2841140055824010212567052202202, 19887982024447462038766218481922, 139215866668648990677444431622708, 974511099543994556441721257750160, 6821577545878037993063671492799030] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 58 t - 140 t - 214 t + 144 t + 23 t - 21 t + 7 t - 1 - ---------------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -(58*t^7-140*t^6-214*t^5+144*t^4+23*t^3-21*t^2+7*t-1)/(7*t-1)/(2*t^2+3*t-1)/(58 *t^4-23*t^2+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 2 t (29 t + 70 t + 55 t - 101 t + t + 22 t - 4) - ---------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -2*t*(29*t^6+70*t^5+55*t^4-101*t^3+t^2+22*t-4)/(7*t-1)/(2*t^2+3*t-1)/(58*t^4-23 *t^2+1) This theorem took, 0.132, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 3, :consider the sequence, (x + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 3, 5, 25, 8, 3, 15, 55, 5, 25, 15, 25, 125, 33, 8, 40, 185, 3, 15, 73, 15, 75, 246, 55, 275, 295, 5, 25, 15, 25, 125, 40, 15, 75, 275, 25, 125, 75, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 36, 0, 0, 40, 0, 0, 140, 30, 150, 90, 8, 40, 48, 40, 200, 142, 36, 180, 390, 0, 0, 40, 0, 0, 150, 40, 200, 180, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [524569192196997746788106616626123185942745576537979614773144931748694289126433\ 432102203369140625000000, 52456919219700138958117688024303396302753498057758144\ 1643466733014767014537937939167022705078125000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 55, 295, 2673, 16743, 128731, 859309, 6238833, 42802095, 304010935, 2110214689, 14858636997, 103646493867, 727261783723, 5083451774845, 35618882099625, 249182951147607, 1744976448013279, 12211823754516121, 85496729635183533, 598416407769202275, 4189195324948036723, 29323144664212423381, 205267649402496635409, 1436848931416694748303, 10058055844725589746151, 70405895503727705064625, 492843547389970393038549, 3449894862545730230943963, 24149309870287589242579771, 169044968504025294738106669, 1183315701392208340895080185, 8283205874047572754907206215, 57982459661544859718162586991, 405877136438958994551756961609, 2841140328080995352953019847549, 19887980663162536336836380255187, 139215872145600280436663382805891, 974511072159238107645626501834245, 6821577656057012519383321246582113] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 36, 390, 2294, 18638, 121116, 897384, 6085670, 43567910, 300929856, 2125620084, 14796655634, 103956400682, 726014914956, 5089686118680, 35593799037038, 249308366460542, 1744471855962264, 12214346714771196, 85486578835640234, 598467161766918770, 4188991122897499716, 29324165674465108416, 205263541501707795590, 1436869470920638947398, 10057973206726377576720, 70406308693723765911780, 492841884974234265851138, 3449903174624410866881018, 24149276427729612623096316, 169045135716815177835523944, 1183315028633487574023830558, 8283209237841176589263454350, 57982446127762644724053885960, 405877204107870069522300466764, 2841140055824010212567052202202, 19887982024447462038766218481922, 139215866668648990677444431622708, 974511099543994556441721257750160, 6821577545878037993063671492799030] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 58 t - 140 t - 214 t + 144 t + 23 t - 21 t + 7 t - 1 - ---------------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -(58*t^7-140*t^6-214*t^5+144*t^4+23*t^3-21*t^2+7*t-1)/(7*t-1)/(2*t^2+3*t-1)/(58 *t^4-23*t^2+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 2 t (29 t + 70 t + 55 t - 101 t + t + 22 t - 4) - ---------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -2*t*(29*t^6+70*t^5+55*t^4-101*t^3+t^2+22*t-4)/(7*t-1)/(2*t^2+3*t-1)/(58*t^4-23 *t^2+1) This theorem took, 0.126, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 4, :consider the sequence, (x + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 1) (2 t - 1) (3 t - 5 t + 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t-1)*(2*t-1)*(3*t^2-5*t+1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.061, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 5, :consider the sequence, (x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 25, 5, 25, 50, 5, 25, 25, 25, 125, 125, 25, 125, 202, 5, 25, 74, 25, 125, 238, 50, 250, 440, 5, 25, 25, 25, 125, 125, 25, 125, 250, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 23, 7, 35, 58, 0, 0, 35, 0, 0, 91, 23, 115, 194, 7, 35, 70, 35, 175, 266, 58, 290, 472, 0, 0, 35, 0, 0, 115, 35, 175, 290, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [136162413819432278698506963161122032529321126599757500889853390225408000000000\ 0000000000000000000000000000, 1361624138194322786985069631611220325302089628744\ 064983416503711825920000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 50, 440, 3680, 30080, 243200, 1955840, 15687680, 125665280, 1005977600, 8050442240, 64414023680, 515354132480, 4123000832000, 32984677744640, 263880106311680, 2111051587911680, 16888455652966400, 135107817022423040, 1080863223374151680, 8646908535772282880, 69175279281294540800, 553402278230821437440, 4427218401768431943680, 35417747917834897326080, 283341986157428945715200, 2266735900518430634147840, 18133887249183441346887680 , 145071098173611515869921280, 1160568786109468067338649600, 9284550291758048300226314240, 74276402345593601447878983680, 594211218810865671767305748480, 4753689750671392814875541504000, 38029518006109012281952714096640, 304236144051823577307415241031680, 2433889152426394534666496041287680, 19471113219458379942160664782438400, 155768905755855934196600104068055040, 1246151246047603052210059975778631680] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 58, 472, 3808, 30592, 245248, 1964032, 15720448, 125796352, 1006501888, 8052539392, 64422412288, 515387686912, 4123135049728, 32985214615552, 263882253795328, 2111060177846272, 16888490012704768, 135107954461376512, 1080863773129965568, 8646910734795538432, 69175288077387563008, 553402313415193526272, 4427218542505920299008, 35417748480784850747392, 283341988409228759400448, 2266735909525629888888832, 18133887285212238365851648 , 145071098317726703945777152, 1160568786685928819642073088, 9284550294063891309440008192, 74276402354816973484733759488, 594211218847759159914724851712, 4753689750818966767465217916928, 38029518006699308092311419748352, 304236144054184760548850063638528, 2433889152435839267632235331715072, 19471113219496158874023621944147968, 155768905756007049924051932714893312, 1246151246048207515119867290365984768] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 22 t - 7 t + 1 ------------------- (8 t - 1) (4 t - 1) and in Maple notation (22*t^2-7*t+1)/(8*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is t (26 t - 7) - ------------------- (8 t - 1) (4 t - 1) and in Maple notation -t*(26*t-7)/(8*t-1)/(4*t-1) This theorem took, 0.057, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 n Theorem number, 6, :consider the sequence, (x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 1) (2 t - 1) (3 t - 5 t + 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t-1)*(2*t-1)*(3*t^2-5*t+1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.054, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 7, :consider the sequence, (x y + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 1) (2 t - 1) (3 t - 5 t + 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t-1)*(2*t-1)*(3*t^2-5*t+1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.055, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 8, :consider the sequence, (x y + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 25, 5, 25, 50, 5, 25, 25, 25, 125, 125, 25, 125, 202, 5, 25, 74, 25, 125, 238, 50, 250, 440, 5, 25, 25, 25, 125, 125, 25, 125, 250, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 23, 7, 35, 58, 0, 0, 35, 0, 0, 91, 23, 115, 194, 7, 35, 70, 35, 175, 266, 58, 290, 472, 0, 0, 35, 0, 0, 115, 35, 175, 290, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [136162413819432278698506963161122032529321126599757500889853390225408000000000\ 0000000000000000000000000000, 1361624138194322786985069631611220325302089628744\ 064983416503711825920000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 50, 440, 3680, 30080, 243200, 1955840, 15687680, 125665280, 1005977600, 8050442240, 64414023680, 515354132480, 4123000832000, 32984677744640, 263880106311680, 2111051587911680, 16888455652966400, 135107817022423040, 1080863223374151680, 8646908535772282880, 69175279281294540800, 553402278230821437440, 4427218401768431943680, 35417747917834897326080, 283341986157428945715200, 2266735900518430634147840, 18133887249183441346887680 , 145071098173611515869921280, 1160568786109468067338649600, 9284550291758048300226314240, 74276402345593601447878983680, 594211218810865671767305748480, 4753689750671392814875541504000, 38029518006109012281952714096640, 304236144051823577307415241031680, 2433889152426394534666496041287680, 19471113219458379942160664782438400, 155768905755855934196600104068055040, 1246151246047603052210059975778631680] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 58, 472, 3808, 30592, 245248, 1964032, 15720448, 125796352, 1006501888, 8052539392, 64422412288, 515387686912, 4123135049728, 32985214615552, 263882253795328, 2111060177846272, 16888490012704768, 135107954461376512, 1080863773129965568, 8646910734795538432, 69175288077387563008, 553402313415193526272, 4427218542505920299008, 35417748480784850747392, 283341988409228759400448, 2266735909525629888888832, 18133887285212238365851648 , 145071098317726703945777152, 1160568786685928819642073088, 9284550294063891309440008192, 74276402354816973484733759488, 594211218847759159914724851712, 4753689750818966767465217916928, 38029518006699308092311419748352, 304236144054184760548850063638528, 2433889152435839267632235331715072, 19471113219496158874023621944147968, 155768905756007049924051932714893312, 1246151246048207515119867290365984768] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 22 t - 7 t + 1 ------------------- (8 t - 1) (4 t - 1) and in Maple notation (22*t^2-7*t+1)/(8*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is t (26 t - 7) - ------------------- (8 t - 1) (4 t - 1) and in Maple notation -t*(26*t-7)/(8*t-1)/(4*t-1) This theorem took, 0.163, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 9, :consider the sequence, (x y + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (2 t - 1) (3 t - 1) (3 t - 5 t + 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (2*t-1)*(3*t-1)*(3*t^2-5*t+1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.056, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 10, :consider the sequence, (x y + x + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 13, 4, 20, 70, 5, 25, 20, 25, 125, 58, 13, 65, 256, 4, 20, 97, 20, 100, 347, 70, 350, 419, 5, 25, 20, 25, 125, 65, 20, 100, 350, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 35, 9, 45, 45, 0, 0, 45, 0, 0, 168, 35, 175, 143, 9, 45, 72, 45 , 225, 234, 45, 225, 526, 0, 0, 45, 0, 0, 175, 45, 225, 225, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267896524511317242722702136297257202930175362507071549601832518100467000060234\ 56808179616928100585937500000, 267896524511317243420256499444604614847475565661\ 72061890155641714073341721996257547289133071899414062500000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 70, 419, 4089, 29964, 254810, 1994140, 16312963, 130200467, 1053041769, 8454480642, 68148008738, 548106635614, 4413612943480, 35517131370215, 285914797108143, 2301176444858019, 18522931032077429, 149088421286304895, 1200029882993312707, 9659008455747134855, 77745848735926487139, 625776973122396328941, 5036898499829037784091, 40542090017523583991989, 326324320951145176414549, 2626591573435280454033953, 21141498307393145731709844 , 170168400675891567386020059, 1369689433209436619579814392, 11024661783793074775730027692, 88737758592232342740280410043, 714252269520383160124513756598, 5749033062799057816752594533214, 46274100120634991807830744306257, 372461303152855925853960097938263, 2997949648427038991013899409910615, 24130566112606310911001586682555006, 194227485115114895481448982103860894, 1563341523160241100076175055900336708] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 45, 526, 3590, 32063, 245344, 2035011, 16131234, 130995094, 1049527331, 8469919742, 68079882400, 548406442929, 4412291323632, 35522951332834, 285889151640917, 2301289406557217, 18522433347575473, 149090613660887226, 1200020224434273258, 9659051004601910053, 77745661289893631906, 625777798888770010901, 5036894861999550621247, 40542106043524364034663, 326324250350410412032395, 2626591884458387301564568, 21141496937217068921860226 , 170168406712041161477229740, 1369689406617888001670541496, 11024661900939026515155881862, 88737758076159618777682921536, 714252271793881090069601821174, 5749033052783429777004848956949, 46274100164757667296961306425737, 372461302958478659979995467240933, 2997949649283345244641617795843832, 24130566108833954223783048973451097, 194227485131733570422619952438571134, 1563341523087029477893471616820464186] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.340, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 11, :consider the sequence, (x y + x + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 14, 4, 20, 77, 5, 25, 20, 25, 125, 62, 14, 70, 309, 4, 20, 97, 20, 100, 382, 77, 385, 572, 5, 25, 20, 25, 125, 70, 20, 100, 385, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 39, 9, 45, 55, 0, 0, 45, 0, 0, 187, 39, 195, 190, 9, 45, 72, 45 , 225, 260, 55, 275, 682, 0, 0, 45, 0, 0, 195, 45, 225, 275, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [149705283111816902022749552413030261336338518382644366607174502686306120737075\ 8056640625000000000000000000000, 1497052831118169020996123785262971954827181470\ 694272562053092186112769263820648193359375000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 77, 572, 6095, 51503, 489074, 4303691, 39430124, 351999521, 3187156502, 28600703204, 257936265680, 2319001958492, 20885745717140, 187901874798527, 1691527824959939, 15221747449215083, 137007230138151521, 1233007814224183466, 11097393122576030276, 99874905451533009401, 898883224867843165532, 8089902557481179799065, 72809378580796430316710, 655283086643773913875208, 5897554984486051157304938, 53077957369271298701124143, 477701819613893150965109225, 4299315313019049609289974380, 38693843557176451529855216630, 348244561864609442112896023268, 3134201218936842320723142353762, 28207810116084828704069285306003, 253870295627453966870074707273464, 2284832636446257388227133175275373, 20563493857566817405453136664685973, 185071444032752012720185980804713291, 1665642999957928133145955093713824417, 14990786980216637147546113583065840763, 134917082925546749171586461162039209295] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 55, 682, 5515, 54553, 472897, 4389418, 38974327, 354421024, 3174275635, 28669191040, 257571928768, 2320939692619, 20875437648319, 187956704039146, 1691236158567028, 15223298899237321, 136998977244962458, 1233051714185466535, 11097159599636697145, 99876147647625334303, 898876617110295305122, 8089937706715270110775, 72809191607700213963475, 655284081227601660624991, 5897549693893301023152568, 53077985512040593111590940, 477701669911195247715124744, 4299316109347487900246763109, 38693839321186144163398570549, 348244584397535979767212488235, 3134201099075177910263851940419, 28207810753676987256245861953270, 253870292235845916727855784077774, 2284832654487578214669819344547688, 20563493761597807592700520767757159, 185071444543249595544748987480932055, 1665642997242387086690556558066894946, 14990786994661687156323408908041711471, 134917082848707744582845418333241390786] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 21 20 19 18 17 16 - (9072 t - 34884 t + 94851 t - 92004 t - 436641 t + 807473 t 15 14 13 12 11 + 381103 t - 1220119 t + 202509 t + 948007 t - 256030 t 10 9 8 7 6 5 4 - 363108 t + 100243 t + 68583 t - 21040 t - 5850 t + 2515 t + 61 t 3 2 / - 158 t + 22 t + 4 t - 1) / ((9 t - 1) / 8 7 6 5 4 3 2 (171 t - 246 t + 104 t + 103 t - 131 t + 13 t + 27 t - 10 t + 1) ( 11 10 9 8 7 6 5 4 1008 t - 12 t - 1981 t - 2597 t - 2554 t - 1019 t + 564 t + 576 t 3 2 + 112 t - 25 t - 11 t - 1)) and in Maple notation -(9072*t^21-34884*t^20+94851*t^19-92004*t^18-436641*t^17+807473*t^16+381103*t^ 15-1220119*t^14+202509*t^13+948007*t^12-256030*t^11-363108*t^10+100243*t^9+ 68583*t^8-21040*t^7-5850*t^6+2515*t^5+61*t^4-158*t^3+22*t^2+4*t-1)/(9*t-1)/(171 *t^8-246*t^7+104*t^6+103*t^5-131*t^4+13*t^3+27*t^2-10*t+1)/(1008*t^11-12*t^10-\ 1981*t^9-2597*t^8-2554*t^7-1019*t^6+564*t^5+576*t^4+112*t^3-25*t^2-11*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 20 19 18 17 16 15 - t (9072 t - 34884 t - 292977 t + 448995 t + 851340 t - 868390 t 14 13 12 11 10 - 507251 t + 1049613 t - 179830 t - 616508 t + 417408 t 9 8 7 6 5 4 3 + 232439 t - 211817 t - 57833 t + 49003 t + 8327 t - 5880 t - 604 t 2 / + 361 t + 17 t - 9) / ((9 t - 1) / 8 7 6 5 4 3 2 (171 t - 246 t + 104 t + 103 t - 131 t + 13 t + 27 t - 10 t + 1) ( 11 10 9 8 7 6 5 4 1008 t - 12 t - 1981 t - 2597 t - 2554 t - 1019 t + 564 t + 576 t 3 2 + 112 t - 25 t - 11 t - 1)) and in Maple notation -t*(9072*t^20-34884*t^19-292977*t^18+448995*t^17+851340*t^16-868390*t^15-507251 *t^14+1049613*t^13-179830*t^12-616508*t^11+417408*t^10+232439*t^9-211817*t^8-\ 57833*t^7+49003*t^6+8327*t^5-5880*t^4-604*t^3+361*t^2+17*t-9)/(9*t-1)/(171*t^8-\ 246*t^7+104*t^6+103*t^5-131*t^4+13*t^3+27*t^2-10*t+1)/(1008*t^11-12*t^10-1981*t ^9-2597*t^8-2554*t^7-1019*t^6+564*t^5+576*t^4+112*t^3-25*t^2-11*t-1) This theorem took, 0.523, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 12, :consider the sequence, (x y + x + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 5 t + 1) (3 t - 1) (2 t - 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t^2-5*t+1)*(3*t-1)*(2*t-1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.065, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 13, :consider the sequence, (x y + x + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 28, 6, 30, 66, 5, 25, 30, 25, 125, 140, 28, 140, 262, 6, 30, 100, 30, 150, 394, 66, 330, 651, 5, 25, 30, 25, 125, 140, 30, 150, 330, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 32, 8, 40, 77, 0, 0, 40, 0, 0, 154, 32, 160, 311, 8, 40, 96, 40 , 200, 404, 77, 385, 665, 0, 0, 40, 0, 0, 160, 40, 200, 385, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267472790943687754713045839179456699788223350084558470644928980995853850468750\ 000000000000000000000000000000000, 26747279094368775471304583917945669978822334\ 9911969458421259195445479157343750000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 66, 651, 5973, 52866, 467838, 4153845, 36917445, 328087866, 2915204382, 25901506647, 230135669685, 2044782871338, 18168187221804, 161426859510441, 1434299236695075, 12743938962713700, 113231591516223276, 1006077769355418447, 8939135009787587061, 79425405274544254272, 705705304433212164522, 6270285620201477521311, 55712322858527558034285, 495011408817065983739508, 4398242297026386902181258, 39078968602851831534695949, 347221840890572644884915903, 3085112302134291192548948076, 27411633704758373646510075396, 243556016370680989436453180013, 2164027644220976459349399444525, 19227673841672401095642365956188, 170840442980948006608464455697966, 1517940089802767494889392038275745, 13487099869481098744232321728369329, 119834678661785966012895741566618796, 1064747080465268292233048978076378048, 9460419621593504740170051979636841847, 84057088353341425342543117371967154421] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 77, 665, 5858, 52400, 467621, 4158083, 36934214, 328091786, 2915033849, 25900915739, 230135710868, 2044789537190, 18168207997361, 161426850899369, 1434298978727900, 12743938239783212, 113231592099185237, 1006077779243383547, 8939135034650340932, 79425405243245508710, 705705304057358994197, 6270285619358526038363, 55712322860039974615418, 495011408831240837961224, 4398242297054477624721671, 39078968602783076172667931, 347221840890042088053707834, 3085112302133375023315674224, 27411633704761375636255038749, 243556016370700700773997973617, 2164027644221005514548965147368, 19227673841672273776193659617740, 170840442980947279706099940477185, 1517940089802766608341576229876335, 13487099869481104025506971448981640, 119834678661785992618133734419724052, 1064747080465268317766145146712180803, 9460419621593504524955223210634180325, 84057088353341424376285267435024177166] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (1620 t + 8316 t - 1044 t - 53274 t + 52291 t - 25068 t + 39517 t 10 9 8 7 6 5 4 - 13477 t - 2487 t - 5401 t + 1945 t + 141 t - 410 t - 13 t 3 2 / + 23 t + 26 t - 6 t + 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation (1620*t^17+8316*t^16-1044*t^15-53274*t^14+52291*t^13-25068*t^12+39517*t^11-\ 13477*t^10-2487*t^9-5401*t^8+1945*t^7+141*t^6-410*t^5-13*t^4+23*t^3+26*t^2-6*t+ 1)/(t-1)/(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t ^8+409*t^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 t (1620 t + 2160 t - 59688 t + 66174 t - 39995 t + 41227 t 10 9 8 7 6 5 4 - 18547 t + 10687 t - 3203 t - 219 t - 2867 t - 437 t + 284 t 3 2 / + 118 t - 3 t - 19 t + 8) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation t*(1620*t^16+2160*t^15-59688*t^14+66174*t^13-39995*t^12+41227*t^11-18547*t^10+ 10687*t^9-3203*t^8-219*t^7-2867*t^6-437*t^5+284*t^4+118*t^3-3*t^2-19*t+8)/(t-1) /(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t^8+409*t ^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) This theorem took, 0.572, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 14, :consider the sequence, (x y + x + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 29, 6, 30, 67, 5, 25, 30, 25, 125, 143, 29, 145, 278, 6, 30, 100, 30, 150, 395, 67, 335, 611, 5, 25, 30, 25, 125, 145, 30, 150, 335, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 72, 0, 0, 40, 0, 0, 145, 30, 150, 273, 8, 40, 96, 40 , 200, 399, 72, 360, 649, 0, 0, 40, 0, 0, 150, 40, 200, 360, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [129220123879748073779379580591391094634430281282713001766865300651387463073486\ 328125000000000000000000000000000, 12922012387974807377937958059139109463443028\ 1282713001766865159073169429473632812500000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 67, 611, 5494, 48314, 422874, 3689558, 32174370, 280452859, 2444397429, 21303655546, 185664827159, 1618081633231, 14101653642728, 122896329972807, 1071044665871792, 9334179172772070, 81347582127413449, 708945982875028536, 6178479884426814199, 53845588145390805065, 469265484432470598487, 4089659008461396323969, 35641468110821559426351, 310616177589646433390771, 2707026811408276293973830, 23591798129505189614198473, 205603038961941389789381319, 1791835001220079964620582162, 15615881398452881048052303257, 136092749435239089859430623948, 1186051300996155095402806677731, 10336463143202698867240248018027, 90082503363088702122984404598209, 785070995730009914324607464002251, 6841910974124274155772136730948818, 59627404441700699159162100427897389, 519654139596462248393001267980260222, 4528797242277500134540531342085187119, 39468567454475281659687849459075748831] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 72, 649, 5592, 48715, 423917, 3694237, 32188379, 280514385, 2444587103, 21304444473, 185667322645, 1618091694822, 14101686298195, 122896458796310, 1071045093416068, 9334180829484185, 81347587717332819, 708946004204392340, 6178479957329249577, 53845588420266594790, 469265485381983306849, 4089659012008081106053, 35641468123176617388062, 310616177635450543312960, 2707026811568906401488974, 23591798130097137948662265, 205603038964028360348779070, 1791835001227734226162795212, 15615881398479981654561809833, 136092749435338108973064592668, 1186051300996506868907059293769, 10336463143203980285577289191521, 90082503363093266740026826929316, 785070995730026502055853965647386, 6841910974124333370804964966062837, 59627404441700913933464380499431163, 519654139596463016407568462973453801, 4528797242277502915898809751999011405, 39468567454475291619135831790978866159] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.138, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 15, :consider the sequence, (x y + x + x y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 3, 5, 25, 9, 3, 15, 64, 5, 25, 15, 25, 125, 39, 9, 45, 248, 3, 15, 73, 15, 75, 290, 64, 320, 342, 5, 25, 15, 25, 125, 45, 15, 75, 320, 25, 125, 75, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 34, 8, 40, 36, 0, 0, 40, 0, 0, 164, 34, 170, 108, 8, 40, 48, 40 , 200, 162, 36, 180, 500, 0, 0, 40, 0, 0, 170, 40, 200, 180, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [137935242201087480368791871580266240368460514124667269449590593414306640625000\ 000000000000000000000000000, 13793524220109436812498031593450345369172511734358\ 0347757696062164306640625000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 64, 342, 3862, 25695, 238153, 1757157, 15056914, 116552439, 965317699, 7638222204, 62312428795, 497999413212, 4035224920261, 32395336701435, 261696215806606, 2105222124849390, 16983080526892576, 136746668333945013, 1102469189443493719, 8880698698716458313, 71577351363003644746, 576683696298528852057, 4647415430764955152615, 37446415406823070942746, 301758435040983910631329, 2431502552048629236052194, 19593534156560847569625805 , 157883067821238610826170323, 1272238664653485402082115764, 10251672796306872066189437358, 82608646766980963899405252082, 665661092128298747676289323963, 5363927814152791043982966518101, 43222638542569672783522380909813, 348289647972587435439853788312562, 2806527006960345615775054734183939, 22615089433353459319734378500993059, 182233035054771847675378530044690736, 1468439636259198162437202361619648215] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 36, 500, 3033, 30233, 213804, 1889318, 14343231, 120416639, 944416611, 7751337887, 61700361651, 501311680805, 4017300668040, 32492335703936, 261171296038926, 2108062796604320, 16967707847633655, 136829859793228487, 1102018986677129394, 8883135038076890198, 71564166746347903263, 576755046844825610117, 4647029306461169978259, 37448504977180689608465, 301747127013005786149809, 2431563747164154620500667, 19593202989869434128875724 , 157884859980259863937195736, 1272228966111766678778274480, 10251725281443860817983315204, 82608362735660651252692129341, 665662629207091547534491422869, 5363919496016558141922688723128, 43222683557432111691917537653502, 348289404367796154123612820469199, 2806528325264815036792357094694995, 22615082299147826076503513673205803, 182233073662613359730499821756114939, 1468439427326977408762078918936040655] Using the found enumerative automaton with, 18, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 12 11 10 9 8 7 6 5 (6912 t + 1344 t - 1680 t - 2724 t - 1122 t + 2089 t - 549 t + 283 t 4 3 2 / - 182 t + 28 t + 10 t - 6 t + 1) / ((t + 1) / 6 5 4 3 2 (144 t - 72 t + 45 t - 110 t + 69 t - 15 t + 1) 6 5 4 3 2 (144 t + 8 t + 37 t - 40 t - 11 t + 5 t + 1)) and in Maple notation (6912*t^12+1344*t^11-1680*t^10-2724*t^9-1122*t^8+2089*t^7-549*t^6+283*t^5-182*t ^4+28*t^3+10*t^2-6*t+1)/(t+1)/(144*t^6-72*t^5+45*t^4-110*t^3+69*t^2-15*t+1)/( 144*t^6+8*t^5+37*t^4-40*t^3-11*t^2+5*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 10 9 8 7 6 5 4 3 t (2496 t + 2352 t + 900 t - 739 t - 2370 t + 1136 t - 352 t + 305 t 2 / - 40 t - 36 t + 8) / ((t + 1) / 6 5 4 3 2 (144 t - 72 t + 45 t - 110 t + 69 t - 15 t + 1) 6 5 4 3 2 (144 t + 8 t + 37 t - 40 t - 11 t + 5 t + 1)) and in Maple notation t*(2496*t^10+2352*t^9+900*t^8-739*t^7-2370*t^6+1136*t^5-352*t^4+305*t^3-40*t^2-\ 36*t+8)/(t+1)/(144*t^6-72*t^5+45*t^4-110*t^3+69*t^2-15*t+1)/(144*t^6+8*t^5+37*t ^4-40*t^3-11*t^2+5*t+1) This theorem took, 0.226, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 16, :consider the sequence, (x y + x + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 15, 5, 25, 64, 5, 25, 25, 25, 125, 69, 15, 75, 250, 5, 25, 74, 25, 125, 286, 64, 320, 429, 5, 25, 25, 25, 125, 75, 25, 125, 320, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 31, 7, 35, 48, 0, 0, 35, 0, 0, 155, 31, 155, 158, 7, 35, 70, 35 , 175, 248, 48, 240, 503, 0, 0, 35, 0, 0, 155, 35, 175, 240, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [101836551193649006065079686140632373943739732244404333325412600905728000000000\ 00000000000000000000000000000, 101836551193649006065079691975600565036310342269\ 14140499006773316812800000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 64, 429, 4006, 30798, 266692, 2153646, 18119446, 148777242, 1239214174, 10234774206, 84953304130, 703073716854, 5828783273998, 48273448966206, 400039073259730, 3313910922564594, 27458142093378550, 227482090002795558, 1884756729471575818, 15615089755609261482, 129373352272110257806, 1071861290620359999630, 8880475772390349306226, 73575222340947702672258, 609576602307635586043894, 5050381598001600261155046, 41842783913517064989833914 , 346670327091033676855209738, 2872188464542741235787436510, 23796281606585450560211998638, 197153877668618906087417346178, 1633433705647574361069557016018, 13533113450615314186001200535782, 112122796321976102809788236672694, 928945264730756183024908675705738, 7696376874186098727853955502247578, 63765024237824341130886783476027758, 528297713206739211064083533462846142, 4376983740854466777597514852721001874] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 48, 503, 3660, 32582, 258246, 2195576, 17915400, 149776664, 1234326804, 10258678844, 84836370444, 703645648796, 5825985709296, 48287132620124, 399972142010436, 3314238304285592, 27456540765929064, 227489922602220572, 1884718417754902956, 15615277150331001536, 129372435665311345440, 1071865774033937259812, 8880453842597538150660, 73575329606495486966168, 609576077637938931929640, 5050384164326913590424380, 41842771360808843314865100 , 346670388490297000128410048, 2872188164219538450840512016, 23796283075561155614025417860, 197153870483394447793375324884, 1633433740792778150649698188616, 13533113278709014448212238308248, 112122797162824147580740025067980, 928945260617902637381240831868444, 7696376894303365329243710324597840, 63765024139424431914560015701903584, 528297713688044267344070354224380308, 4376983738500251625920886974311288548] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (2736 t - 6240 t + 4344 t - 3460 t + 3220 t - 4488 t + 6856 t 10 9 8 7 6 5 4 - 5716 t + 3972 t - 2228 t + 1017 t - 626 t + 385 t - 237 t 3 2 / + 107 t - 32 t + 7 t - 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation (2736*t^17-6240*t^16+4344*t^15-3460*t^14+3220*t^13-4488*t^12+6856*t^11-5716*t^ 10+3972*t^9-2228*t^8+1017*t^7-626*t^6+385*t^5-237*t^4+107*t^3-32*t^2+7*t-1)/(t-\ 1)/(48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t ^7-608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 10 - (3312 t - 8304 t + 6144 t - 4700 t + 5876 t - 6784 t + 8420 t 9 8 7 6 5 4 3 - 7868 t + 7094 t - 6268 t + 4345 t - 2408 t + 1051 t - 388 t 2 / + 123 t - 36 t + 7) t / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation -(3312*t^16-8304*t^15+6144*t^14-4700*t^13+5876*t^12-6784*t^11+8420*t^10-7868*t^ 9+7094*t^8-6268*t^7+4345*t^6-2408*t^5+1051*t^4-388*t^3+123*t^2-36*t+7)*t/(t-1)/ (48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t^7-\ 608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) This theorem took, 0.552, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 17, :consider the sequence, (x y + x + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 19, 4, 20, 63, 5, 25, 20, 25, 125, 92, 19, 95, 252, 4, 20, 97, 20, 100, 348, 63, 315, 560, 5, 25, 20, 25, 125, 95, 20, 100, 315, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 35, 9, 45, 68, 0, 0, 45, 0, 0, 163, 35, 175, 259, 9, 45, 72, 45 , 225, 298, 68, 340, 613, 0, 0, 45, 0, 0, 175, 45, 225, 340, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [221956190468470136620174544998346999366245202257902488545310481507228122711181\ 6406250000000000000000000000000, 2219561904684701366201745449983469984352863726\ 659528815253365020937187538146972656250000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 63, 560, 5141, 44375, 384191, 3306131, 28493651, 245347625, 2112983225, 18193228205, 156647667200, 1348698513428, 11611920684335, 99974279596235, 860739631674608, 7410614344295480, 63802307922148484, 549311017502408027, 4729335896085973187, 40717579338739403726, 350561099676969390011, 3018182380974736902233, 25985269848255786282509, 223722147447079509844889, 1926152754522913296739271, 16583357833535059481404427, 142775673504831946854951140, 1229237958883696536185240243, 10583217169520270948455114730, 91117008577579560459960932747, 784478776034624115614195830034, 6754029347944443607629234268211, 58149326439158853778219502023739, 500641023470063586946040291883320, 4310306752090451345580011162659046, 37109911945103171144074538195795384, 319500593293992663581894682418514594, 2750764519898926791240627489140204393, 23682915158064631416417905788505314397] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 68, 613, 5119, 44482, 382912, 3307108, 28483630, 245388772, 2112905668, 18193758520, 156646107526, 1348703680921, 11611896541468, 99974351650279, 860739346505599, 7410615438436942, 63802304284444897, 549311031889362733, 4729335844438968634, 40717579523743611097, 350561098971864925102, 3018182383493047945126, 25985269838960629804549, 223722147481553991108298, 1926152754398189989565671, 16583357833997413401654538, 142775673503135856937321663, 1229237958889898709634286422, 10583217169497356909662552747, 91117008577663482275472757981, 784478776034315681047061579887, 6754029347945579372027267507335, 58149326439154687661419463968159, 500641023470078919312706553059060, 4310306752090394969151265097854690, 37109911945103378275513381300376938, 319500593293991901313178188396808761, 2750764519898929593525201595472424502, 23682915158064621110924748614088306625] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.670, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 18, :consider the sequence, (x y + x + x y + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 15, 5, 25, 64, 5, 25, 25, 25, 125, 69, 15, 75, 250, 5, 25, 74, 25, 125, 286, 64, 320, 429, 5, 25, 25, 25, 125, 75, 25, 125, 320, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 31, 7, 35, 48, 0, 0, 35, 0, 0, 155, 31, 155, 158, 7, 35, 70, 35 , 175, 248, 48, 240, 503, 0, 0, 35, 0, 0, 155, 35, 175, 240, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [101836551193649006065079686140632373943739732244404333325412600905728000000000\ 00000000000000000000000000000, 101836551193649006065079691975600565036310342269\ 14140499006773316812800000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 64, 429, 4006, 30798, 266692, 2153646, 18119446, 148777242, 1239214174, 10234774206, 84953304130, 703073716854, 5828783273998, 48273448966206, 400039073259730, 3313910922564594, 27458142093378550, 227482090002795558, 1884756729471575818, 15615089755609261482, 129373352272110257806, 1071861290620359999630, 8880475772390349306226, 73575222340947702672258, 609576602307635586043894, 5050381598001600261155046, 41842783913517064989833914 , 346670327091033676855209738, 2872188464542741235787436510, 23796281606585450560211998638, 197153877668618906087417346178, 1633433705647574361069557016018, 13533113450615314186001200535782, 112122796321976102809788236672694, 928945264730756183024908675705738, 7696376874186098727853955502247578, 63765024237824341130886783476027758, 528297713206739211064083533462846142, 4376983740854466777597514852721001874] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 48, 503, 3660, 32582, 258246, 2195576, 17915400, 149776664, 1234326804, 10258678844, 84836370444, 703645648796, 5825985709296, 48287132620124, 399972142010436, 3314238304285592, 27456540765929064, 227489922602220572, 1884718417754902956, 15615277150331001536, 129372435665311345440, 1071865774033937259812, 8880453842597538150660, 73575329606495486966168, 609576077637938931929640, 5050384164326913590424380, 41842771360808843314865100 , 346670388490297000128410048, 2872188164219538450840512016, 23796283075561155614025417860, 197153870483394447793375324884, 1633433740792778150649698188616, 13533113278709014448212238308248, 112122797162824147580740025067980, 928945260617902637381240831868444, 7696376894303365329243710324597840, 63765024139424431914560015701903584, 528297713688044267344070354224380308, 4376983738500251625920886974311288548] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (2736 t - 6240 t + 4344 t - 3460 t + 3220 t - 4488 t + 6856 t 10 9 8 7 6 5 4 - 5716 t + 3972 t - 2228 t + 1017 t - 626 t + 385 t - 237 t 3 2 / + 107 t - 32 t + 7 t - 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation (2736*t^17-6240*t^16+4344*t^15-3460*t^14+3220*t^13-4488*t^12+6856*t^11-5716*t^ 10+3972*t^9-2228*t^8+1017*t^7-626*t^6+385*t^5-237*t^4+107*t^3-32*t^2+7*t-1)/(t-\ 1)/(48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t ^7-608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 10 - t (3312 t - 8304 t + 6144 t - 4700 t + 5876 t - 6784 t + 8420 t 9 8 7 6 5 4 3 - 7868 t + 7094 t - 6268 t + 4345 t - 2408 t + 1051 t - 388 t 2 / + 123 t - 36 t + 7) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation -t*(3312*t^16-8304*t^15+6144*t^14-4700*t^13+5876*t^12-6784*t^11+8420*t^10-7868* t^9+7094*t^8-6268*t^7+4345*t^6-2408*t^5+1051*t^4-388*t^3+123*t^2-36*t+7)/(t-1)/ (48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t^7-\ 608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) This theorem took, 0.544, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 19, :consider the sequence, (x y + x + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 12, 4, 20, 69, 5, 25, 20, 25, 125, 52, 12, 60, 281, 4, 20, 97, 20, 100, 353, 69, 345, 448, 5, 25, 20, 25, 125, 60, 20, 100, 345, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 37, 9, 45, 44, 0, 0, 45, 0, 0, 177, 37, 185, 156, 9, 45, 72, 45 , 225, 228, 44, 220, 573, 0, 0, 45, 0, 0, 185, 45, 225, 220, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [732833319837921614402774463453741284334138041426903611089765648122778819121644\ 15407460182905197143554687500, 732833319837921629096453848732335133943344756704\ 97458382308510777387283798104048255481757223606109619140625] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 69, 448, 4905, 39772, 379821, 3309016, 30328017, 270217780, 2445631893, 21942327664, 197822745849, 1778695728268, 16016806476285, 144108533677192, 1297190426141601, 11673645720040036, 105068152056532197, 945586665627930400, 8510413505055670473, 76593053973479549884, 689340823621423370829, 6204050723292273228088, 55836539956133144599665, 502528442372684873662612, 4522758067516731001635381, 40704812176837743321359056, 366343361745603868359028377, 3297089994940113922897271020, 29673811258312629767745361053, 267064294805555645601358640104, 2403578685846290921953975807809, 21632208009635167739877042035908, 194689872901623762447437079495045, 1752208852040077598084215209596032, 15769879688733379702471529415661161, 141928917096737010723675802094466076, 1277360254379950129505922032082616557, 11496242286872966000589099222581439640, 103466180594589619830122888334043503633] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 44, 573, 4280, 42897, 364196, 3387141, 29937392, 272170905, 2435866268, 21991155789, 197578605224, 1779916431393, 16010702960660, 144139051255317, 1297037838250976, 11674408659493161, 105064337359266572, 945605739114258525, 8510318137624029848, 76593530810637753009, 689338439435632355204, 6204062644221228306213, 55836480351488369209040, 502528740395908750615737, 4522756577400611616869756, 40704819627418340245187181, 366343324492700883739887752, 3297090181204628845992974145, 29673810326990055152266845428, 267064299462168518678751218229, 2403578662563226556567012917184, 21632208126050489566811856489033, 194689872319547153312763007229420, 1752208854950460643757585570924157, 15769879674181464474104677609020536, 141928917169496586865510061127669201, 1277360254016152248796750736916600932, 11496242288691955404134955698411517765, 103466180585494672812393605954893113008] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 8 t + t + 1 - --------------------------- (t + 1) (9 t - 1) (5 t + 1) and in Maple notation -(8*t^2+t+1)/(t+1)/(9*t-1)/(5*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is t (17 t + 9) - --------------------------- (t + 1) (9 t - 1) (5 t + 1) and in Maple notation -t*(17*t+9)/(t+1)/(9*t-1)/(5*t+1) This theorem took, 0.185, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 20, :consider the sequence, (x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 28, 6, 30, 66, 5, 25, 30, 25, 125, 140, 28, 140, 262, 6, 30, 100, 30, 150, 394, 66, 330, 651, 5, 25, 30, 25, 125, 140, 30, 150, 330, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 32, 8, 40, 77, 0, 0, 40, 0, 0, 154, 32, 160, 311, 8, 40, 96, 40 , 200, 404, 77, 385, 665, 0, 0, 40, 0, 0, 160, 40, 200, 385, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267472790943687754713045839179456699788223350084558470644928980995853850468750\ 000000000000000000000000000000000, 26747279094368775471304583917945669978822334\ 9911969458421259195445479157343750000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 66, 651, 5973, 52866, 467838, 4153845, 36917445, 328087866, 2915204382, 25901506647, 230135669685, 2044782871338, 18168187221804, 161426859510441, 1434299236695075, 12743938962713700, 113231591516223276, 1006077769355418447, 8939135009787587061, 79425405274544254272, 705705304433212164522, 6270285620201477521311, 55712322858527558034285, 495011408817065983739508, 4398242297026386902181258, 39078968602851831534695949, 347221840890572644884915903, 3085112302134291192548948076, 27411633704758373646510075396, 243556016370680989436453180013, 2164027644220976459349399444525, 19227673841672401095642365956188, 170840442980948006608464455697966, 1517940089802767494889392038275745, 13487099869481098744232321728369329, 119834678661785966012895741566618796, 1064747080465268292233048978076378048, 9460419621593504740170051979636841847, 84057088353341425342543117371967154421] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 77, 665, 5858, 52400, 467621, 4158083, 36934214, 328091786, 2915033849, 25900915739, 230135710868, 2044789537190, 18168207997361, 161426850899369, 1434298978727900, 12743938239783212, 113231592099185237, 1006077779243383547, 8939135034650340932, 79425405243245508710, 705705304057358994197, 6270285619358526038363, 55712322860039974615418, 495011408831240837961224, 4398242297054477624721671, 39078968602783076172667931, 347221840890042088053707834, 3085112302133375023315674224, 27411633704761375636255038749, 243556016370700700773997973617, 2164027644221005514548965147368, 19227673841672273776193659617740, 170840442980947279706099940477185, 1517940089802766608341576229876335, 13487099869481104025506971448981640, 119834678661785992618133734419724052, 1064747080465268317766145146712180803, 9460419621593504524955223210634180325, 84057088353341424376285267435024177166] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (1620 t + 8316 t - 1044 t - 53274 t + 52291 t - 25068 t + 39517 t 10 9 8 7 6 5 4 - 13477 t - 2487 t - 5401 t + 1945 t + 141 t - 410 t - 13 t 3 2 / + 23 t + 26 t - 6 t + 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation (1620*t^17+8316*t^16-1044*t^15-53274*t^14+52291*t^13-25068*t^12+39517*t^11-\ 13477*t^10-2487*t^9-5401*t^8+1945*t^7+141*t^6-410*t^5-13*t^4+23*t^3+26*t^2-6*t+ 1)/(t-1)/(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t ^8+409*t^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 (1620 t + 2160 t - 59688 t + 66174 t - 39995 t + 41227 t 10 9 8 7 6 5 4 - 18547 t + 10687 t - 3203 t - 219 t - 2867 t - 437 t + 284 t 3 2 / + 118 t - 3 t - 19 t + 8) t / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation (1620*t^16+2160*t^15-59688*t^14+66174*t^13-39995*t^12+41227*t^11-18547*t^10+ 10687*t^9-3203*t^8-219*t^7-2867*t^6-437*t^5+284*t^4+118*t^3-3*t^2-19*t+8)*t/(t-\ 1)/(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t^8+409 *t^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) This theorem took, 0.585, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 21, :consider the sequence, (x y + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 13, 4, 20, 70, 5, 25, 20, 25, 125, 58, 13, 65, 256, 4, 20, 97, 20, 100, 347, 70, 350, 419, 5, 25, 20, 25, 125, 65, 20, 100, 350, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 35, 9, 45, 45, 0, 0, 45, 0, 0, 168, 35, 175, 143, 9, 45, 72, 45 , 225, 234, 45, 225, 526, 0, 0, 45, 0, 0, 175, 45, 225, 225, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267896524511317242722702136297257202930175362507071549601832518100467000060234\ 56808179616928100585937500000, 267896524511317243420256499444604614847475565661\ 72061890155641714073341721996257547289133071899414062500000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 70, 419, 4089, 29964, 254810, 1994140, 16312963, 130200467, 1053041769, 8454480642, 68148008738, 548106635614, 4413612943480, 35517131370215, 285914797108143, 2301176444858019, 18522931032077429, 149088421286304895, 1200029882993312707, 9659008455747134855, 77745848735926487139, 625776973122396328941, 5036898499829037784091, 40542090017523583991989, 326324320951145176414549, 2626591573435280454033953, 21141498307393145731709844 , 170168400675891567386020059, 1369689433209436619579814392, 11024661783793074775730027692, 88737758592232342740280410043, 714252269520383160124513756598, 5749033062799057816752594533214, 46274100120634991807830744306257, 372461303152855925853960097938263, 2997949648427038991013899409910615, 24130566112606310911001586682555006, 194227485115114895481448982103860894, 1563341523160241100076175055900336708] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 45, 526, 3590, 32063, 245344, 2035011, 16131234, 130995094, 1049527331, 8469919742, 68079882400, 548406442929, 4412291323632, 35522951332834, 285889151640917, 2301289406557217, 18522433347575473, 149090613660887226, 1200020224434273258, 9659051004601910053, 77745661289893631906, 625777798888770010901, 5036894861999550621247, 40542106043524364034663, 326324250350410412032395, 2626591884458387301564568, 21141496937217068921860226 , 170168406712041161477229740, 1369689406617888001670541496, 11024661900939026515155881862, 88737758076159618777682921536, 714252271793881090069601821174, 5749033052783429777004848956949, 46274100164757667296961306425737, 372461302958478659979995467240933, 2997949649283345244641617795843832, 24130566108833954223783048973451097, 194227485131733570422619952438571134, 1563341523087029477893471616820464186] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 6.084, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 22, :consider the sequence, (x y + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 24, 6, 30, 86, 5, 25, 30, 25, 125, 116, 24, 120, 366, 6, 30, 100, 30, 150, 436, 86, 430, 698, 5, 25, 30, 25, 125, 120, 30, 150, 430, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 40, 8, 40, 70, 0, 0, 40, 0, 0, 196, 40, 200, 280, 8, 40, 96, 40 , 200, 404, 70, 350, 766, 0, 0, 40, 0, 0, 200, 40, 200, 350, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [221074292181328352337481411059315988494392125439801022315531214096695296000000\ 0000000000000000000000000000000000, 2210742921813283523374814110593182009075969\ 726148126128035763279846440960000000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 86, 698, 6854, 59996, 546800, 4893716, 44113334, 396663728, 3570746276, 32132948468, 289206985922, 2602829013758, 23425609838726, 210830182948616, 1897473521721596, 17077257939071726, 153695341104621086, 1383258009883223744, 12449322266527371560, 112043899486013477528, 1008395097093088522304, 9075555862427839871786, 81680002785491168796212, 735120024954756891341354, 6616080224983201946144222, 59544722023854977700374462, 535902498220454388053652014, 4823122483974206892784913582, 43408102355836154749438068458, 390672921202377979369401930140, 3516056290822046182871417655458, 31644506617395928634275740070172, 284800559556568708440672858462542, 2563205036009082710856195670838972, 23068845324081801088343776131861896, 207619607916735808199149054907088680, 1868576471250623206680729843672741350, 16817188241255605303809346462741948178, 151354694171300463655493939933951869112] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 70, 766, 6574, 61060, 542986, 4907140, 44067814, 396817972, 3570232852, 32134671220, 289201229152, 2602848478612, 23425543844608, 210830409126766, 1897472743671178, 17077260634346776, 153695331750326290, 1383258042448248274, 12449322153171081220, 112043899880796840694, 1008395095719748482148, 9075555867202437300616, 81680002768912779532558, 735120025012277223690466, 6616080224783814941698402, 59544722024545791322650064, 535902498218062073827323832, 4823122483982490221311860910, 43408102355807477495816233336, 390672921202477266765234508798, 3516056290821702404920662828712, 31644506617397119114030612810024, 284800559556564585419200136914384, 2563205036009096992057555146678022, 23068845324081751617086001016824568, 207619607916735979585079728248974482, 1868576471250622612913346132028371850, 16817188241255607360978943010660115862, 151354694171300456528170224745960448818] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 6.817, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 23, :consider the sequence, (x y + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 28, 6, 30, 66, 5, 25, 30, 25, 125, 140, 28, 140, 262, 6, 30, 100, 30, 150, 394, 66, 330, 651, 5, 25, 30, 25, 125, 140, 30, 150, 330, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 32, 8, 40, 77, 0, 0, 40, 0, 0, 154, 32, 160, 311, 8, 40, 96, 40 , 200, 404, 77, 385, 665, 0, 0, 40, 0, 0, 160, 40, 200, 385, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267472790943687754713045839179456699788223350084558470644928980995853850468750\ 000000000000000000000000000000000, 26747279094368775471304583917945669978822334\ 9911969458421259195445479157343750000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 66, 651, 5973, 52866, 467838, 4153845, 36917445, 328087866, 2915204382, 25901506647, 230135669685, 2044782871338, 18168187221804, 161426859510441, 1434299236695075, 12743938962713700, 113231591516223276, 1006077769355418447, 8939135009787587061, 79425405274544254272, 705705304433212164522, 6270285620201477521311, 55712322858527558034285, 495011408817065983739508, 4398242297026386902181258, 39078968602851831534695949, 347221840890572644884915903, 3085112302134291192548948076, 27411633704758373646510075396, 243556016370680989436453180013, 2164027644220976459349399444525, 19227673841672401095642365956188, 170840442980948006608464455697966, 1517940089802767494889392038275745, 13487099869481098744232321728369329, 119834678661785966012895741566618796, 1064747080465268292233048978076378048, 9460419621593504740170051979636841847, 84057088353341425342543117371967154421] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 77, 665, 5858, 52400, 467621, 4158083, 36934214, 328091786, 2915033849, 25900915739, 230135710868, 2044789537190, 18168207997361, 161426850899369, 1434298978727900, 12743938239783212, 113231592099185237, 1006077779243383547, 8939135034650340932, 79425405243245508710, 705705304057358994197, 6270285619358526038363, 55712322860039974615418, 495011408831240837961224, 4398242297054477624721671, 39078968602783076172667931, 347221840890042088053707834, 3085112302133375023315674224, 27411633704761375636255038749, 243556016370700700773997973617, 2164027644221005514548965147368, 19227673841672273776193659617740, 170840442980947279706099940477185, 1517940089802766608341576229876335, 13487099869481104025506971448981640, 119834678661785992618133734419724052, 1064747080465268317766145146712180803, 9460419621593504524955223210634180325, 84057088353341424376285267435024177166] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (1620 t + 8316 t - 1044 t - 53274 t + 52291 t - 25068 t + 39517 t 10 9 8 7 6 5 4 - 13477 t - 2487 t - 5401 t + 1945 t + 141 t - 410 t - 13 t 3 2 / + 23 t + 26 t - 6 t + 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation (1620*t^17+8316*t^16-1044*t^15-53274*t^14+52291*t^13-25068*t^12+39517*t^11-\ 13477*t^10-2487*t^9-5401*t^8+1945*t^7+141*t^6-410*t^5-13*t^4+23*t^3+26*t^2-6*t+ 1)/(t-1)/(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t ^8+409*t^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 t (1620 t + 2160 t - 59688 t + 66174 t - 39995 t + 41227 t 10 9 8 7 6 5 4 - 18547 t + 10687 t - 3203 t - 219 t - 2867 t - 437 t + 284 t 3 2 / + 118 t - 3 t - 19 t + 8) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation t*(1620*t^16+2160*t^15-59688*t^14+66174*t^13-39995*t^12+41227*t^11-18547*t^10+ 10687*t^9-3203*t^8-219*t^7-2867*t^6-437*t^5+284*t^4+118*t^3-3*t^2-19*t+8)/(t-1) /(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t^8+409*t ^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) This theorem took, 0.599, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 24, :consider the sequence, (x y + x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 15, 5, 25, 64, 5, 25, 25, 25, 125, 69, 15, 75, 250, 5, 25, 74, 25, 125, 286, 64, 320, 429, 5, 25, 25, 25, 125, 75, 25, 125, 320, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 31, 7, 35, 48, 0, 0, 35, 0, 0, 155, 31, 155, 158, 7, 35, 70, 35 , 175, 248, 48, 240, 503, 0, 0, 35, 0, 0, 155, 35, 175, 240, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [101836551193649006065079686140632373943739732244404333325412600905728000000000\ 00000000000000000000000000000, 101836551193649006065079691975600565036310342269\ 14140499006773316812800000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 64, 429, 4006, 30798, 266692, 2153646, 18119446, 148777242, 1239214174, 10234774206, 84953304130, 703073716854, 5828783273998, 48273448966206, 400039073259730, 3313910922564594, 27458142093378550, 227482090002795558, 1884756729471575818, 15615089755609261482, 129373352272110257806, 1071861290620359999630, 8880475772390349306226, 73575222340947702672258, 609576602307635586043894, 5050381598001600261155046, 41842783913517064989833914 , 346670327091033676855209738, 2872188464542741235787436510, 23796281606585450560211998638, 197153877668618906087417346178, 1633433705647574361069557016018, 13533113450615314186001200535782, 112122796321976102809788236672694, 928945264730756183024908675705738, 7696376874186098727853955502247578, 63765024237824341130886783476027758, 528297713206739211064083533462846142, 4376983740854466777597514852721001874] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 48, 503, 3660, 32582, 258246, 2195576, 17915400, 149776664, 1234326804, 10258678844, 84836370444, 703645648796, 5825985709296, 48287132620124, 399972142010436, 3314238304285592, 27456540765929064, 227489922602220572, 1884718417754902956, 15615277150331001536, 129372435665311345440, 1071865774033937259812, 8880453842597538150660, 73575329606495486966168, 609576077637938931929640, 5050384164326913590424380, 41842771360808843314865100 , 346670388490297000128410048, 2872188164219538450840512016, 23796283075561155614025417860, 197153870483394447793375324884, 1633433740792778150649698188616, 13533113278709014448212238308248, 112122797162824147580740025067980, 928945260617902637381240831868444, 7696376894303365329243710324597840, 63765024139424431914560015701903584, 528297713688044267344070354224380308, 4376983738500251625920886974311288548] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (2736 t - 6240 t + 4344 t - 3460 t + 3220 t - 4488 t + 6856 t 10 9 8 7 6 5 4 - 5716 t + 3972 t - 2228 t + 1017 t - 626 t + 385 t - 237 t 3 2 / + 107 t - 32 t + 7 t - 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation (2736*t^17-6240*t^16+4344*t^15-3460*t^14+3220*t^13-4488*t^12+6856*t^11-5716*t^ 10+3972*t^9-2228*t^8+1017*t^7-626*t^6+385*t^5-237*t^4+107*t^3-32*t^2+7*t-1)/(t-\ 1)/(48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t ^7-608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 10 - (3312 t - 8304 t + 6144 t - 4700 t + 5876 t - 6784 t + 8420 t 9 8 7 6 5 4 3 - 7868 t + 7094 t - 6268 t + 4345 t - 2408 t + 1051 t - 388 t 2 / + 123 t - 36 t + 7) t / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation -(3312*t^16-8304*t^15+6144*t^14-4700*t^13+5876*t^12-6784*t^11+8420*t^10-7868*t^ 9+7094*t^8-6268*t^7+4345*t^6-2408*t^5+1051*t^4-388*t^3+123*t^2-36*t+7)*t/(t-1)/ (48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t^7-\ 608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) This theorem took, 0.735, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 25, :consider the sequence, (x y + x y + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 12, 4, 20, 69, 5, 25, 20, 25, 125, 52, 12, 60, 281, 4, 20, 97, 20, 100, 353, 69, 345, 448, 5, 25, 20, 25, 125, 60, 20, 100, 345, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 37, 9, 45, 44, 0, 0, 45, 0, 0, 177, 37, 185, 156, 9, 45, 72, 45 , 225, 228, 44, 220, 573, 0, 0, 45, 0, 0, 185, 45, 225, 220, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [732833319837921614402774463453741284334138041426903611089765648122778819121644\ 15407460182905197143554687500, 732833319837921629096453848732335133943344756704\ 97458382308510777387283798104048255481757223606109619140625] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 69, 448, 4905, 39772, 379821, 3309016, 30328017, 270217780, 2445631893, 21942327664, 197822745849, 1778695728268, 16016806476285, 144108533677192, 1297190426141601, 11673645720040036, 105068152056532197, 945586665627930400, 8510413505055670473, 76593053973479549884, 689340823621423370829, 6204050723292273228088, 55836539956133144599665, 502528442372684873662612, 4522758067516731001635381, 40704812176837743321359056, 366343361745603868359028377, 3297089994940113922897271020, 29673811258312629767745361053, 267064294805555645601358640104, 2403578685846290921953975807809, 21632208009635167739877042035908, 194689872901623762447437079495045, 1752208852040077598084215209596032, 15769879688733379702471529415661161, 141928917096737010723675802094466076, 1277360254379950129505922032082616557, 11496242286872966000589099222581439640, 103466180594589619830122888334043503633] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 44, 573, 4280, 42897, 364196, 3387141, 29937392, 272170905, 2435866268, 21991155789, 197578605224, 1779916431393, 16010702960660, 144139051255317, 1297037838250976, 11674408659493161, 105064337359266572, 945605739114258525, 8510318137624029848, 76593530810637753009, 689338439435632355204, 6204062644221228306213, 55836480351488369209040, 502528740395908750615737, 4522756577400611616869756, 40704819627418340245187181, 366343324492700883739887752, 3297090181204628845992974145, 29673810326990055152266845428, 267064299462168518678751218229, 2403578662563226556567012917184, 21632208126050489566811856489033, 194689872319547153312763007229420, 1752208854950460643757585570924157, 15769879674181464474104677609020536, 141928917169496586865510061127669201, 1277360254016152248796750736916600932, 11496242288691955404134955698411517765, 103466180585494672812393605954893113008] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 8 t + t + 1 - --------------------------- (t + 1) (9 t - 1) (5 t + 1) and in Maple notation -(8*t^2+t+1)/(t+1)/(9*t-1)/(5*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is t (17 t + 9) - --------------------------- (t + 1) (9 t - 1) (5 t + 1) and in Maple notation -t*(17*t+9)/(t+1)/(9*t-1)/(5*t+1) This theorem took, 0.194, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 26, :consider the sequence, (x y + x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 29, 6, 30, 67, 5, 25, 30, 25, 125, 143, 29, 145, 278, 6, 30, 100, 30, 150, 395, 67, 335, 611, 5, 25, 30, 25, 125, 145, 30, 150, 335, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 72, 0, 0, 40, 0, 0, 145, 30, 150, 273, 8, 40, 96, 40 , 200, 399, 72, 360, 649, 0, 0, 40, 0, 0, 150, 40, 200, 360, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [129220123879748073779379580591391094634430281282713001766865300651387463073486\ 328125000000000000000000000000000, 12922012387974807377937958059139109463443028\ 1282713001766865159073169429473632812500000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 67, 611, 5494, 48314, 422874, 3689558, 32174370, 280452859, 2444397429, 21303655546, 185664827159, 1618081633231, 14101653642728, 122896329972807, 1071044665871792, 9334179172772070, 81347582127413449, 708945982875028536, 6178479884426814199, 53845588145390805065, 469265484432470598487, 4089659008461396323969, 35641468110821559426351, 310616177589646433390771, 2707026811408276293973830, 23591798129505189614198473, 205603038961941389789381319, 1791835001220079964620582162, 15615881398452881048052303257, 136092749435239089859430623948, 1186051300996155095402806677731, 10336463143202698867240248018027, 90082503363088702122984404598209, 785070995730009914324607464002251, 6841910974124274155772136730948818, 59627404441700699159162100427897389, 519654139596462248393001267980260222, 4528797242277500134540531342085187119, 39468567454475281659687849459075748831] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 72, 649, 5592, 48715, 423917, 3694237, 32188379, 280514385, 2444587103, 21304444473, 185667322645, 1618091694822, 14101686298195, 122896458796310, 1071045093416068, 9334180829484185, 81347587717332819, 708946004204392340, 6178479957329249577, 53845588420266594790, 469265485381983306849, 4089659012008081106053, 35641468123176617388062, 310616177635450543312960, 2707026811568906401488974, 23591798130097137948662265, 205603038964028360348779070, 1791835001227734226162795212, 15615881398479981654561809833, 136092749435338108973064592668, 1186051300996506868907059293769, 10336463143203980285577289191521, 90082503363093266740026826929316, 785070995730026502055853965647386, 6841910974124333370804964966062837, 59627404441700913933464380499431163, 519654139596463016407568462973453801, 4528797242277502915898809751999011405, 39468567454475291619135831790978866159] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 8.475, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 27, :consider the sequence, (x y + x y + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 15, 5, 25, 64, 5, 25, 25, 25, 125, 69, 15, 75, 250, 5, 25, 74, 25, 125, 286, 64, 320, 429, 5, 25, 25, 25, 125, 75, 25, 125, 320, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 31, 7, 35, 48, 0, 0, 35, 0, 0, 155, 31, 155, 158, 7, 35, 70, 35 , 175, 248, 48, 240, 503, 0, 0, 35, 0, 0, 155, 35, 175, 240, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [101836551193649006065079686140632373943739732244404333325412600905728000000000\ 00000000000000000000000000000, 101836551193649006065079691975600565036310342269\ 14140499006773316812800000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 64, 429, 4006, 30798, 266692, 2153646, 18119446, 148777242, 1239214174, 10234774206, 84953304130, 703073716854, 5828783273998, 48273448966206, 400039073259730, 3313910922564594, 27458142093378550, 227482090002795558, 1884756729471575818, 15615089755609261482, 129373352272110257806, 1071861290620359999630, 8880475772390349306226, 73575222340947702672258, 609576602307635586043894, 5050381598001600261155046, 41842783913517064989833914 , 346670327091033676855209738, 2872188464542741235787436510, 23796281606585450560211998638, 197153877668618906087417346178, 1633433705647574361069557016018, 13533113450615314186001200535782, 112122796321976102809788236672694, 928945264730756183024908675705738, 7696376874186098727853955502247578, 63765024237824341130886783476027758, 528297713206739211064083533462846142, 4376983740854466777597514852721001874] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 48, 503, 3660, 32582, 258246, 2195576, 17915400, 149776664, 1234326804, 10258678844, 84836370444, 703645648796, 5825985709296, 48287132620124, 399972142010436, 3314238304285592, 27456540765929064, 227489922602220572, 1884718417754902956, 15615277150331001536, 129372435665311345440, 1071865774033937259812, 8880453842597538150660, 73575329606495486966168, 609576077637938931929640, 5050384164326913590424380, 41842771360808843314865100 , 346670388490297000128410048, 2872188164219538450840512016, 23796283075561155614025417860, 197153870483394447793375324884, 1633433740792778150649698188616, 13533113278709014448212238308248, 112122797162824147580740025067980, 928945260617902637381240831868444, 7696376894303365329243710324597840, 63765024139424431914560015701903584, 528297713688044267344070354224380308, 4376983738500251625920886974311288548] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (2736 t - 6240 t + 4344 t - 3460 t + 3220 t - 4488 t + 6856 t 10 9 8 7 6 5 4 - 5716 t + 3972 t - 2228 t + 1017 t - 626 t + 385 t - 237 t 3 2 / + 107 t - 32 t + 7 t - 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation (2736*t^17-6240*t^16+4344*t^15-3460*t^14+3220*t^13-4488*t^12+6856*t^11-5716*t^ 10+3972*t^9-2228*t^8+1017*t^7-626*t^6+385*t^5-237*t^4+107*t^3-32*t^2+7*t-1)/(t-\ 1)/(48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t ^7-608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 10 - t (3312 t - 8304 t + 6144 t - 4700 t + 5876 t - 6784 t + 8420 t 9 8 7 6 5 4 3 - 7868 t + 7094 t - 6268 t + 4345 t - 2408 t + 1051 t - 388 t 2 / + 123 t - 36 t + 7) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation -t*(3312*t^16-8304*t^15+6144*t^14-4700*t^13+5876*t^12-6784*t^11+8420*t^10-7868* t^9+7094*t^8-6268*t^7+4345*t^6-2408*t^5+1051*t^4-388*t^3+123*t^2-36*t+7)/(t-1)/ (48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t^7-\ 608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) This theorem took, 0.741, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 28, :consider the sequence, (x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 26, 5, 25, 82, 5, 25, 25, 25, 125, 131, 26, 130, 335, 5, 25, 125, 25, 125, 474, 82, 410, 815, 5, 25, 25, 25, 125, 130, 25, 125, 410, 25, 125 , 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 42, 10, 50, 87, 0, 0, 50, 0, 0, 196, 42, 210, 354, 10, 50, 100 , 50, 250, 454, 87, 435, 841, 0, 0, 50, 0, 0, 210, 50, 250, 435, 0, 0, 250, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [207626640204256061661849504295284493000850492381094725098125226973158360217930\ 74905872344970703125000000000000000, 207626640204256061661849504295284493000808\ 79511269677378171321839042758265626616775989532470703125000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 82, 815, 7602, 68102, 610573, 5455979, 48707739, 434544464, 3875631319, 34562074775, 308199440121, 2748232198217, 24505857107734, 218516467310537, 1948486623258966, 17374416916255994, 154925478432043081, 1381450611225161429, 12318216679895859075, 109839942573734730422, 979428522100686750547, 8733437029690378022345, 77874924348574670067627, 694400590672174960238351, 6191879915045693415478450, 55212189310162357695360725, 492319923800252186866328760, 4389952841617646905667958338, 39144639530887331097004578805, 349047668452254899745621406595, 3112412741853660735162042834081, 27752980326672011732577602589002, 247469722332829459164099025315987, 2206655384416411971198345236200547, 19676459567137748067779845997056983, 175452435315156874254324345671899781, 1564486586267293169054331485165058916, 13950323768454355195030038082661324795, 124393225837125259480767261999894392208] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 87, 841, 7565, 68356, 611016, 5459056, 48717506, 434579104, 3875796546, 34562644321, 308201946752, 2748241201942, 24505894541043, 218516610243211, 1948487196556223, 17374419144627724, 154925487234913770, 1381450645764438988, 12318216815733715766, 109839943108165387234, 979428524198161678932, 8733437037948333166909, 77874924380983702309748, 694400590799740502527984, 6191879915546555070931173, 55212189312132634994954077, 492319923807992280243622229, 4389952841648080748856557164, 39144639531006931745458969224, 349047668452725038986470839758, 3112412741855508637531093118132, 27752980326679274891142058453210, 247469722332858008911494872921628, 2206655384416524183216323110377643, 19676459567138189146212167376209606, 175452435315158607889602803134194610, 1564486586267299983448611709053331743, 13950323768454381979080742696756493191, 124393225837125364759198698441246530189] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 10.926, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 29, :consider the sequence, (x y + x y + x + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 28, 6, 30, 66, 5, 25, 30, 25, 125, 140, 28, 140, 262, 6, 30, 100, 30, 150, 394, 66, 330, 651, 5, 25, 30, 25, 125, 140, 30, 150, 330, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 32, 8, 40, 77, 0, 0, 40, 0, 0, 154, 32, 160, 311, 8, 40, 96, 40 , 200, 404, 77, 385, 665, 0, 0, 40, 0, 0, 160, 40, 200, 385, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267472790943687754713045839179456699788223350084558470644928980995853850468750\ 000000000000000000000000000000000, 26747279094368775471304583917945669978822334\ 9911969458421259195445479157343750000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 66, 651, 5973, 52866, 467838, 4153845, 36917445, 328087866, 2915204382, 25901506647, 230135669685, 2044782871338, 18168187221804, 161426859510441, 1434299236695075, 12743938962713700, 113231591516223276, 1006077769355418447, 8939135009787587061, 79425405274544254272, 705705304433212164522, 6270285620201477521311, 55712322858527558034285, 495011408817065983739508, 4398242297026386902181258, 39078968602851831534695949, 347221840890572644884915903, 3085112302134291192548948076, 27411633704758373646510075396, 243556016370680989436453180013, 2164027644220976459349399444525, 19227673841672401095642365956188, 170840442980948006608464455697966, 1517940089802767494889392038275745, 13487099869481098744232321728369329, 119834678661785966012895741566618796, 1064747080465268292233048978076378048, 9460419621593504740170051979636841847, 84057088353341425342543117371967154421] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 77, 665, 5858, 52400, 467621, 4158083, 36934214, 328091786, 2915033849, 25900915739, 230135710868, 2044789537190, 18168207997361, 161426850899369, 1434298978727900, 12743938239783212, 113231592099185237, 1006077779243383547, 8939135034650340932, 79425405243245508710, 705705304057358994197, 6270285619358526038363, 55712322860039974615418, 495011408831240837961224, 4398242297054477624721671, 39078968602783076172667931, 347221840890042088053707834, 3085112302133375023315674224, 27411633704761375636255038749, 243556016370700700773997973617, 2164027644221005514548965147368, 19227673841672273776193659617740, 170840442980947279706099940477185, 1517940089802766608341576229876335, 13487099869481104025506971448981640, 119834678661785992618133734419724052, 1064747080465268317766145146712180803, 9460419621593504524955223210634180325, 84057088353341424376285267435024177166] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (1620 t + 8316 t - 1044 t - 53274 t + 52291 t - 25068 t + 39517 t 10 9 8 7 6 5 4 - 13477 t - 2487 t - 5401 t + 1945 t + 141 t - 410 t - 13 t 3 2 / + 23 t + 26 t - 6 t + 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation (1620*t^17+8316*t^16-1044*t^15-53274*t^14+52291*t^13-25068*t^12+39517*t^11-\ 13477*t^10-2487*t^9-5401*t^8+1945*t^7+141*t^6-410*t^5-13*t^4+23*t^3+26*t^2-6*t+ 1)/(t-1)/(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t ^8+409*t^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 t (1620 t + 2160 t - 59688 t + 66174 t - 39995 t + 41227 t 10 9 8 7 6 5 4 - 18547 t + 10687 t - 3203 t - 219 t - 2867 t - 437 t + 284 t 3 2 / + 118 t - 3 t - 19 t + 8) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation t*(1620*t^16+2160*t^15-59688*t^14+66174*t^13-39995*t^12+41227*t^11-18547*t^10+ 10687*t^9-3203*t^8-219*t^7-2867*t^6-437*t^5+284*t^4+118*t^3-3*t^2-19*t+8)/(t-1) /(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t^8+409*t ^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) This theorem took, 0.923, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 30, :consider the sequence, (x y + x y + x + x y + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 5 t + 1) (3 t - 1) (2 t - 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t^2-5*t+1)*(3*t-1)*(2*t-1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.068, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 31, :consider the sequence, (x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 14, 4, 20, 77, 5, 25, 20, 25, 125, 62, 14, 70, 309, 4, 20, 97, 20, 100, 382, 77, 385, 572, 5, 25, 20, 25, 125, 70, 20, 100, 385, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 39, 9, 45, 55, 0, 0, 45, 0, 0, 187, 39, 195, 190, 9, 45, 72, 45 , 225, 260, 55, 275, 682, 0, 0, 45, 0, 0, 195, 45, 225, 275, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [149705283111816902022749552413030261336338518382644366607174502686306120737075\ 8056640625000000000000000000000, 1497052831118169020996123785262971954827181470\ 694272562053092186112769263820648193359375000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 77, 572, 6095, 51503, 489074, 4303691, 39430124, 351999521, 3187156502, 28600703204, 257936265680, 2319001958492, 20885745717140, 187901874798527, 1691527824959939, 15221747449215083, 137007230138151521, 1233007814224183466, 11097393122576030276, 99874905451533009401, 898883224867843165532, 8089902557481179799065, 72809378580796430316710, 655283086643773913875208, 5897554984486051157304938, 53077957369271298701124143, 477701819613893150965109225, 4299315313019049609289974380, 38693843557176451529855216630, 348244561864609442112896023268, 3134201218936842320723142353762, 28207810116084828704069285306003, 253870295627453966870074707273464, 2284832636446257388227133175275373, 20563493857566817405453136664685973, 185071444032752012720185980804713291, 1665642999957928133145955093713824417, 14990786980216637147546113583065840763, 134917082925546749171586461162039209295] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 55, 682, 5515, 54553, 472897, 4389418, 38974327, 354421024, 3174275635, 28669191040, 257571928768, 2320939692619, 20875437648319, 187956704039146, 1691236158567028, 15223298899237321, 136998977244962458, 1233051714185466535, 11097159599636697145, 99876147647625334303, 898876617110295305122, 8089937706715270110775, 72809191607700213963475, 655284081227601660624991, 5897549693893301023152568, 53077985512040593111590940, 477701669911195247715124744, 4299316109347487900246763109, 38693839321186144163398570549, 348244584397535979767212488235, 3134201099075177910263851940419, 28207810753676987256245861953270, 253870292235845916727855784077774, 2284832654487578214669819344547688, 20563493761597807592700520767757159, 185071444543249595544748987480932055, 1665642997242387086690556558066894946, 14990786994661687156323408908041711471, 134917082848707744582845418333241390786] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 21 20 19 18 17 16 - (9072 t - 34884 t + 94851 t - 92004 t - 436641 t + 807473 t 15 14 13 12 11 + 381103 t - 1220119 t + 202509 t + 948007 t - 256030 t 10 9 8 7 6 5 4 - 363108 t + 100243 t + 68583 t - 21040 t - 5850 t + 2515 t + 61 t 3 2 / - 158 t + 22 t + 4 t - 1) / ((9 t - 1) / 8 7 6 5 4 3 2 (171 t - 246 t + 104 t + 103 t - 131 t + 13 t + 27 t - 10 t + 1) ( 11 10 9 8 7 6 5 4 1008 t - 12 t - 1981 t - 2597 t - 2554 t - 1019 t + 564 t + 576 t 3 2 + 112 t - 25 t - 11 t - 1)) and in Maple notation -(9072*t^21-34884*t^20+94851*t^19-92004*t^18-436641*t^17+807473*t^16+381103*t^ 15-1220119*t^14+202509*t^13+948007*t^12-256030*t^11-363108*t^10+100243*t^9+ 68583*t^8-21040*t^7-5850*t^6+2515*t^5+61*t^4-158*t^3+22*t^2+4*t-1)/(9*t-1)/(171 *t^8-246*t^7+104*t^6+103*t^5-131*t^4+13*t^3+27*t^2-10*t+1)/(1008*t^11-12*t^10-\ 1981*t^9-2597*t^8-2554*t^7-1019*t^6+564*t^5+576*t^4+112*t^3-25*t^2-11*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 20 19 18 17 16 15 - t (9072 t - 34884 t - 292977 t + 448995 t + 851340 t - 868390 t 14 13 12 11 10 - 507251 t + 1049613 t - 179830 t - 616508 t + 417408 t 9 8 7 6 5 4 3 + 232439 t - 211817 t - 57833 t + 49003 t + 8327 t - 5880 t - 604 t 2 / + 361 t + 17 t - 9) / ((9 t - 1) / 8 7 6 5 4 3 2 (171 t - 246 t + 104 t + 103 t - 131 t + 13 t + 27 t - 10 t + 1) ( 11 10 9 8 7 6 5 4 1008 t - 12 t - 1981 t - 2597 t - 2554 t - 1019 t + 564 t + 576 t 3 2 + 112 t - 25 t - 11 t - 1)) and in Maple notation -t*(9072*t^20-34884*t^19-292977*t^18+448995*t^17+851340*t^16-868390*t^15-507251 *t^14+1049613*t^13-179830*t^12-616508*t^11+417408*t^10+232439*t^9-211817*t^8-\ 57833*t^7+49003*t^6+8327*t^5-5880*t^4-604*t^3+361*t^2+17*t-9)/(9*t-1)/(171*t^8-\ 246*t^7+104*t^6+103*t^5-131*t^4+13*t^3+27*t^2-10*t+1)/(1008*t^11-12*t^10-1981*t ^9-2597*t^8-2554*t^7-1019*t^6+564*t^5+576*t^4+112*t^3-25*t^2-11*t-1) This theorem took, 0.721, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 32, :consider the sequence, (x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 15, 5, 25, 64, 5, 25, 25, 25, 125, 69, 15, 75, 250, 5, 25, 74, 25, 125, 286, 64, 320, 429, 5, 25, 25, 25, 125, 75, 25, 125, 320, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 31, 7, 35, 48, 0, 0, 35, 0, 0, 155, 31, 155, 158, 7, 35, 70, 35 , 175, 248, 48, 240, 503, 0, 0, 35, 0, 0, 155, 35, 175, 240, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [101836551193649006065079686140632373943739732244404333325412600905728000000000\ 00000000000000000000000000000, 101836551193649006065079691975600565036310342269\ 14140499006773316812800000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 64, 429, 4006, 30798, 266692, 2153646, 18119446, 148777242, 1239214174, 10234774206, 84953304130, 703073716854, 5828783273998, 48273448966206, 400039073259730, 3313910922564594, 27458142093378550, 227482090002795558, 1884756729471575818, 15615089755609261482, 129373352272110257806, 1071861290620359999630, 8880475772390349306226, 73575222340947702672258, 609576602307635586043894, 5050381598001600261155046, 41842783913517064989833914 , 346670327091033676855209738, 2872188464542741235787436510, 23796281606585450560211998638, 197153877668618906087417346178, 1633433705647574361069557016018, 13533113450615314186001200535782, 112122796321976102809788236672694, 928945264730756183024908675705738, 7696376874186098727853955502247578, 63765024237824341130886783476027758, 528297713206739211064083533462846142, 4376983740854466777597514852721001874] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 48, 503, 3660, 32582, 258246, 2195576, 17915400, 149776664, 1234326804, 10258678844, 84836370444, 703645648796, 5825985709296, 48287132620124, 399972142010436, 3314238304285592, 27456540765929064, 227489922602220572, 1884718417754902956, 15615277150331001536, 129372435665311345440, 1071865774033937259812, 8880453842597538150660, 73575329606495486966168, 609576077637938931929640, 5050384164326913590424380, 41842771360808843314865100 , 346670388490297000128410048, 2872188164219538450840512016, 23796283075561155614025417860, 197153870483394447793375324884, 1633433740792778150649698188616, 13533113278709014448212238308248, 112122797162824147580740025067980, 928945260617902637381240831868444, 7696376894303365329243710324597840, 63765024139424431914560015701903584, 528297713688044267344070354224380308, 4376983738500251625920886974311288548] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (2736 t - 6240 t + 4344 t - 3460 t + 3220 t - 4488 t + 6856 t 10 9 8 7 6 5 4 - 5716 t + 3972 t - 2228 t + 1017 t - 626 t + 385 t - 237 t 3 2 / + 107 t - 32 t + 7 t - 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation (2736*t^17-6240*t^16+4344*t^15-3460*t^14+3220*t^13-4488*t^12+6856*t^11-5716*t^ 10+3972*t^9-2228*t^8+1017*t^7-626*t^6+385*t^5-237*t^4+107*t^3-32*t^2+7*t-1)/(t-\ 1)/(48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t ^7-608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 10 - t (3312 t - 8304 t + 6144 t - 4700 t + 5876 t - 6784 t + 8420 t 9 8 7 6 5 4 3 - 7868 t + 7094 t - 6268 t + 4345 t - 2408 t + 1051 t - 388 t 2 / + 123 t - 36 t + 7) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation -t*(3312*t^16-8304*t^15+6144*t^14-4700*t^13+5876*t^12-6784*t^11+8420*t^10-7868* t^9+7094*t^8-6268*t^7+4345*t^6-2408*t^5+1051*t^4-388*t^3+123*t^2-36*t+7)/(t-1)/ (48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t^7-\ 608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) This theorem took, 1.043, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 33, :consider the sequence, (x y + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 3, 5, 25, 8, 3, 15, 55, 5, 25, 15, 25, 125, 33, 8, 40, 185, 3, 15, 73, 15, 75, 246, 55, 275, 295, 5, 25, 15, 25, 125, 40, 15, 75, 275, 25, 125, 75, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 36, 0, 0, 40, 0, 0, 140, 30, 150, 90, 8, 40, 48, 40, 200, 142, 36, 180, 390, 0, 0, 40, 0, 0, 150, 40, 200, 180, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [524569192196997746788106616626123185942745576537979614773144931748694289126433\ 432102203369140625000000, 52456919219700138958117688024303396302753498057758144\ 1643466733014767014537937939167022705078125000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 55, 295, 2673, 16743, 128731, 859309, 6238833, 42802095, 304010935, 2110214689, 14858636997, 103646493867, 727261783723, 5083451774845, 35618882099625, 249182951147607, 1744976448013279, 12211823754516121, 85496729635183533, 598416407769202275, 4189195324948036723, 29323144664212423381, 205267649402496635409, 1436848931416694748303, 10058055844725589746151, 70405895503727705064625, 492843547389970393038549, 3449894862545730230943963, 24149309870287589242579771, 169044968504025294738106669, 1183315701392208340895080185, 8283205874047572754907206215, 57982459661544859718162586991, 405877136438958994551756961609, 2841140328080995352953019847549, 19887980663162536336836380255187, 139215872145600280436663382805891, 974511072159238107645626501834245, 6821577656057012519383321246582113] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 36, 390, 2294, 18638, 121116, 897384, 6085670, 43567910, 300929856, 2125620084, 14796655634, 103956400682, 726014914956, 5089686118680, 35593799037038, 249308366460542, 1744471855962264, 12214346714771196, 85486578835640234, 598467161766918770, 4188991122897499716, 29324165674465108416, 205263541501707795590, 1436869470920638947398, 10057973206726377576720, 70406308693723765911780, 492841884974234265851138, 3449903174624410866881018, 24149276427729612623096316, 169045135716815177835523944, 1183315028633487574023830558, 8283209237841176589263454350, 57982446127762644724053885960, 405877204107870069522300466764, 2841140055824010212567052202202, 19887982024447462038766218481922, 139215866668648990677444431622708, 974511099543994556441721257750160, 6821577545878037993063671492799030] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 58 t - 140 t - 214 t + 144 t + 23 t - 21 t + 7 t - 1 - ---------------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -(58*t^7-140*t^6-214*t^5+144*t^4+23*t^3-21*t^2+7*t-1)/(7*t-1)/(2*t^2+3*t-1)/(58 *t^4-23*t^2+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 2 t (29 t + 70 t + 55 t - 101 t + t + 22 t - 4) - ---------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -2*t*(29*t^6+70*t^5+55*t^4-101*t^3+t^2+22*t-4)/(7*t-1)/(2*t^2+3*t-1)/(58*t^4-23 *t^2+1) This theorem took, 0.446, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 34, :consider the sequence, (x y + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 19, 4, 20, 63, 5, 25, 20, 25, 125, 92, 19, 95, 252, 4, 20, 97, 20, 100, 348, 63, 315, 560, 5, 25, 20, 25, 125, 95, 20, 100, 315, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 35, 9, 45, 68, 0, 0, 45, 0, 0, 163, 35, 175, 259, 9, 45, 72, 45 , 225, 298, 68, 340, 613, 0, 0, 45, 0, 0, 175, 45, 225, 340, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [221956190468470136620174544998346999366245202257902488545310481507228122711181\ 6406250000000000000000000000000, 2219561904684701366201745449983469984352863726\ 659528815253365020937187538146972656250000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 63, 560, 5141, 44375, 384191, 3306131, 28493651, 245347625, 2112983225, 18193228205, 156647667200, 1348698513428, 11611920684335, 99974279596235, 860739631674608, 7410614344295480, 63802307922148484, 549311017502408027, 4729335896085973187, 40717579338739403726, 350561099676969390011, 3018182380974736902233, 25985269848255786282509, 223722147447079509844889, 1926152754522913296739271, 16583357833535059481404427, 142775673504831946854951140, 1229237958883696536185240243, 10583217169520270948455114730, 91117008577579560459960932747, 784478776034624115614195830034, 6754029347944443607629234268211, 58149326439158853778219502023739, 500641023470063586946040291883320, 4310306752090451345580011162659046, 37109911945103171144074538195795384, 319500593293992663581894682418514594, 2750764519898926791240627489140204393, 23682915158064631416417905788505314397] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 68, 613, 5119, 44482, 382912, 3307108, 28483630, 245388772, 2112905668, 18193758520, 156646107526, 1348703680921, 11611896541468, 99974351650279, 860739346505599, 7410615438436942, 63802304284444897, 549311031889362733, 4729335844438968634, 40717579523743611097, 350561098971864925102, 3018182383493047945126, 25985269838960629804549, 223722147481553991108298, 1926152754398189989565671, 16583357833997413401654538, 142775673503135856937321663, 1229237958889898709634286422, 10583217169497356909662552747, 91117008577663482275472757981, 784478776034315681047061579887, 6754029347945579372027267507335, 58149326439154687661419463968159, 500641023470078919312706553059060, 4310306752090394969151265097854690, 37109911945103378275513381300376938, 319500593293991901313178188396808761, 2750764519898929593525201595472424502, 23682915158064621110924748614088306625] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 12.040, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 35, :consider the sequence, (x y + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 29, 6, 30, 67, 5, 25, 30, 25, 125, 143, 29, 145, 278, 6, 30, 100, 30, 150, 395, 67, 335, 611, 5, 25, 30, 25, 125, 145, 30, 150, 335, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 72, 0, 0, 40, 0, 0, 145, 30, 150, 273, 8, 40, 96, 40 , 200, 399, 72, 360, 649, 0, 0, 40, 0, 0, 150, 40, 200, 360, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [129220123879748073779379580591391094634430281282713001766865300651387463073486\ 328125000000000000000000000000000, 12922012387974807377937958059139109463443028\ 1282713001766865159073169429473632812500000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 67, 611, 5494, 48314, 422874, 3689558, 32174370, 280452859, 2444397429, 21303655546, 185664827159, 1618081633231, 14101653642728, 122896329972807, 1071044665871792, 9334179172772070, 81347582127413449, 708945982875028536, 6178479884426814199, 53845588145390805065, 469265484432470598487, 4089659008461396323969, 35641468110821559426351, 310616177589646433390771, 2707026811408276293973830, 23591798129505189614198473, 205603038961941389789381319, 1791835001220079964620582162, 15615881398452881048052303257, 136092749435239089859430623948, 1186051300996155095402806677731, 10336463143202698867240248018027, 90082503363088702122984404598209, 785070995730009914324607464002251, 6841910974124274155772136730948818, 59627404441700699159162100427897389, 519654139596462248393001267980260222, 4528797242277500134540531342085187119, 39468567454475281659687849459075748831] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 72, 649, 5592, 48715, 423917, 3694237, 32188379, 280514385, 2444587103, 21304444473, 185667322645, 1618091694822, 14101686298195, 122896458796310, 1071045093416068, 9334180829484185, 81347587717332819, 708946004204392340, 6178479957329249577, 53845588420266594790, 469265485381983306849, 4089659012008081106053, 35641468123176617388062, 310616177635450543312960, 2707026811568906401488974, 23591798130097137948662265, 205603038964028360348779070, 1791835001227734226162795212, 15615881398479981654561809833, 136092749435338108973064592668, 1186051300996506868907059293769, 10336463143203980285577289191521, 90082503363093266740026826929316, 785070995730026502055853965647386, 6841910974124333370804964966062837, 59627404441700913933464380499431163, 519654139596463016407568462973453801, 4528797242277502915898809751999011405, 39468567454475291619135831790978866159] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.524, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 36, :consider the sequence, (x y + x + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 25, 5, 25, 50, 5, 25, 25, 25, 125, 125, 25, 125, 202, 5, 25, 74, 25, 125, 238, 50, 250, 440, 5, 25, 25, 25, 125, 125, 25, 125, 250, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 23, 7, 35, 58, 0, 0, 35, 0, 0, 91, 23, 115, 194, 7, 35, 70, 35, 175, 266, 58, 290, 472, 0, 0, 35, 0, 0, 115, 35, 175, 290, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [136162413819432278698506963161122032529321126599757500889853390225408000000000\ 0000000000000000000000000000, 1361624138194322786985069631611220325302089628744\ 064983416503711825920000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 50, 440, 3680, 30080, 243200, 1955840, 15687680, 125665280, 1005977600, 8050442240, 64414023680, 515354132480, 4123000832000, 32984677744640, 263880106311680, 2111051587911680, 16888455652966400, 135107817022423040, 1080863223374151680, 8646908535772282880, 69175279281294540800, 553402278230821437440, 4427218401768431943680, 35417747917834897326080, 283341986157428945715200, 2266735900518430634147840, 18133887249183441346887680 , 145071098173611515869921280, 1160568786109468067338649600, 9284550291758048300226314240, 74276402345593601447878983680, 594211218810865671767305748480, 4753689750671392814875541504000, 38029518006109012281952714096640, 304236144051823577307415241031680, 2433889152426394534666496041287680, 19471113219458379942160664782438400, 155768905755855934196600104068055040, 1246151246047603052210059975778631680] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 58, 472, 3808, 30592, 245248, 1964032, 15720448, 125796352, 1006501888, 8052539392, 64422412288, 515387686912, 4123135049728, 32985214615552, 263882253795328, 2111060177846272, 16888490012704768, 135107954461376512, 1080863773129965568, 8646910734795538432, 69175288077387563008, 553402313415193526272, 4427218542505920299008, 35417748480784850747392, 283341988409228759400448, 2266735909525629888888832, 18133887285212238365851648 , 145071098317726703945777152, 1160568786685928819642073088, 9284550294063891309440008192, 74276402354816973484733759488, 594211218847759159914724851712, 4753689750818966767465217916928, 38029518006699308092311419748352, 304236144054184760548850063638528, 2433889152435839267632235331715072, 19471113219496158874023621944147968, 155768905756007049924051932714893312, 1246151246048207515119867290365984768] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 22 t - 7 t + 1 ------------------- (8 t - 1) (4 t - 1) and in Maple notation (22*t^2-7*t+1)/(8*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is t (26 t - 7) - ------------------- (8 t - 1) (4 t - 1) and in Maple notation -t*(26*t-7)/(8*t-1)/(4*t-1) This theorem took, 0.064, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 37, :consider the sequence, (x y + x + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 26, 5, 25, 82, 5, 25, 25, 25, 125, 131, 26, 130, 335, 5, 25, 125, 25, 125, 474, 82, 410, 815, 5, 25, 25, 25, 125, 130, 25, 125, 410, 25, 125 , 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 42, 10, 50, 87, 0, 0, 50, 0, 0, 196, 42, 210, 354, 10, 50, 100 , 50, 250, 454, 87, 435, 841, 0, 0, 50, 0, 0, 210, 50, 250, 435, 0, 0, 250, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [207626640204256061661849504295284493000850492381094725098125226973158360217930\ 74905872344970703125000000000000000, 207626640204256061661849504295284493000808\ 79511269677378171321839042758265626616775989532470703125000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 82, 815, 7602, 68102, 610573, 5455979, 48707739, 434544464, 3875631319, 34562074775, 308199440121, 2748232198217, 24505857107734, 218516467310537, 1948486623258966, 17374416916255994, 154925478432043081, 1381450611225161429, 12318216679895859075, 109839942573734730422, 979428522100686750547, 8733437029690378022345, 77874924348574670067627, 694400590672174960238351, 6191879915045693415478450, 55212189310162357695360725, 492319923800252186866328760, 4389952841617646905667958338, 39144639530887331097004578805, 349047668452254899745621406595, 3112412741853660735162042834081, 27752980326672011732577602589002, 247469722332829459164099025315987, 2206655384416411971198345236200547, 19676459567137748067779845997056983, 175452435315156874254324345671899781, 1564486586267293169054331485165058916, 13950323768454355195030038082661324795, 124393225837125259480767261999894392208] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 87, 841, 7565, 68356, 611016, 5459056, 48717506, 434579104, 3875796546, 34562644321, 308201946752, 2748241201942, 24505894541043, 218516610243211, 1948487196556223, 17374419144627724, 154925487234913770, 1381450645764438988, 12318216815733715766, 109839943108165387234, 979428524198161678932, 8733437037948333166909, 77874924380983702309748, 694400590799740502527984, 6191879915546555070931173, 55212189312132634994954077, 492319923807992280243622229, 4389952841648080748856557164, 39144639531006931745458969224, 349047668452725038986470839758, 3112412741855508637531093118132, 27752980326679274891142058453210, 247469722332858008911494872921628, 2206655384416524183216323110377643, 19676459567138189146212167376209606, 175452435315158607889602803134194610, 1564486586267299983448611709053331743, 13950323768454381979080742696756493191, 124393225837125364759198698441246530189] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.427, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 38, :consider the sequence, (x y + x + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 12, 4, 20, 69, 5, 25, 20, 25, 125, 52, 12, 60, 281, 4, 20, 97, 20, 100, 353, 69, 345, 448, 5, 25, 20, 25, 125, 60, 20, 100, 345, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 37, 9, 45, 44, 0, 0, 45, 0, 0, 177, 37, 185, 156, 9, 45, 72, 45 , 225, 228, 44, 220, 573, 0, 0, 45, 0, 0, 185, 45, 225, 220, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [732833319837921614402774463453741284334138041426903611089765648122778819121644\ 15407460182905197143554687500, 732833319837921629096453848732335133943344756704\ 97458382308510777387283798104048255481757223606109619140625] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 69, 448, 4905, 39772, 379821, 3309016, 30328017, 270217780, 2445631893, 21942327664, 197822745849, 1778695728268, 16016806476285, 144108533677192, 1297190426141601, 11673645720040036, 105068152056532197, 945586665627930400, 8510413505055670473, 76593053973479549884, 689340823621423370829, 6204050723292273228088, 55836539956133144599665, 502528442372684873662612, 4522758067516731001635381, 40704812176837743321359056, 366343361745603868359028377, 3297089994940113922897271020, 29673811258312629767745361053, 267064294805555645601358640104, 2403578685846290921953975807809, 21632208009635167739877042035908, 194689872901623762447437079495045, 1752208852040077598084215209596032, 15769879688733379702471529415661161, 141928917096737010723675802094466076, 1277360254379950129505922032082616557, 11496242286872966000589099222581439640, 103466180594589619830122888334043503633] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 44, 573, 4280, 42897, 364196, 3387141, 29937392, 272170905, 2435866268, 21991155789, 197578605224, 1779916431393, 16010702960660, 144139051255317, 1297037838250976, 11674408659493161, 105064337359266572, 945605739114258525, 8510318137624029848, 76593530810637753009, 689338439435632355204, 6204062644221228306213, 55836480351488369209040, 502528740395908750615737, 4522756577400611616869756, 40704819627418340245187181, 366343324492700883739887752, 3297090181204628845992974145, 29673810326990055152266845428, 267064299462168518678751218229, 2403578662563226556567012917184, 21632208126050489566811856489033, 194689872319547153312763007229420, 1752208854950460643757585570924157, 15769879674181464474104677609020536, 141928917169496586865510061127669201, 1277360254016152248796750736916600932, 11496242288691955404134955698411517765, 103466180585494672812393605954893113008] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 8 t + t + 1 - --------------------------- (t + 1) (9 t - 1) (5 t + 1) and in Maple notation -(8*t^2+t+1)/(t+1)/(9*t-1)/(5*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is t (17 t + 9) - --------------------------- (t + 1) (9 t - 1) (5 t + 1) and in Maple notation -t*(17*t+9)/(t+1)/(9*t-1)/(5*t+1) This theorem took, 0.290, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 39, :consider the sequence, (x y + x + x y + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 19, 4, 20, 63, 5, 25, 20, 25, 125, 92, 19, 95, 252, 4, 20, 97, 20, 100, 348, 63, 315, 560, 5, 25, 20, 25, 125, 95, 20, 100, 315, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 35, 9, 45, 68, 0, 0, 45, 0, 0, 163, 35, 175, 259, 9, 45, 72, 45 , 225, 298, 68, 340, 613, 0, 0, 45, 0, 0, 175, 45, 225, 340, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [221956190468470136620174544998346999366245202257902488545310481507228122711181\ 6406250000000000000000000000000, 2219561904684701366201745449983469984352863726\ 659528815253365020937187538146972656250000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 63, 560, 5141, 44375, 384191, 3306131, 28493651, 245347625, 2112983225, 18193228205, 156647667200, 1348698513428, 11611920684335, 99974279596235, 860739631674608, 7410614344295480, 63802307922148484, 549311017502408027, 4729335896085973187, 40717579338739403726, 350561099676969390011, 3018182380974736902233, 25985269848255786282509, 223722147447079509844889, 1926152754522913296739271, 16583357833535059481404427, 142775673504831946854951140, 1229237958883696536185240243, 10583217169520270948455114730, 91117008577579560459960932747, 784478776034624115614195830034, 6754029347944443607629234268211, 58149326439158853778219502023739, 500641023470063586946040291883320, 4310306752090451345580011162659046, 37109911945103171144074538195795384, 319500593293992663581894682418514594, 2750764519898926791240627489140204393, 23682915158064631416417905788505314397] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 68, 613, 5119, 44482, 382912, 3307108, 28483630, 245388772, 2112905668, 18193758520, 156646107526, 1348703680921, 11611896541468, 99974351650279, 860739346505599, 7410615438436942, 63802304284444897, 549311031889362733, 4729335844438968634, 40717579523743611097, 350561098971864925102, 3018182383493047945126, 25985269838960629804549, 223722147481553991108298, 1926152754398189989565671, 16583357833997413401654538, 142775673503135856937321663, 1229237958889898709634286422, 10583217169497356909662552747, 91117008577663482275472757981, 784478776034315681047061579887, 6754029347945579372027267507335, 58149326439154687661419463968159, 500641023470078919312706553059060, 4310306752090394969151265097854690, 37109911945103378275513381300376938, 319500593293991901313178188396808761, 2750764519898929593525201595472424502, 23682915158064621110924748614088306625] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.636, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 40, :consider the sequence, (x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 28, 6, 30, 66, 5, 25, 30, 25, 125, 140, 28, 140, 262, 6, 30, 100, 30, 150, 394, 66, 330, 651, 5, 25, 30, 25, 125, 140, 30, 150, 330, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 32, 8, 40, 77, 0, 0, 40, 0, 0, 154, 32, 160, 311, 8, 40, 96, 40 , 200, 404, 77, 385, 665, 0, 0, 40, 0, 0, 160, 40, 200, 385, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267472790943687754713045839179456699788223350084558470644928980995853850468750\ 000000000000000000000000000000000, 26747279094368775471304583917945669978822334\ 9911969458421259195445479157343750000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 66, 651, 5973, 52866, 467838, 4153845, 36917445, 328087866, 2915204382, 25901506647, 230135669685, 2044782871338, 18168187221804, 161426859510441, 1434299236695075, 12743938962713700, 113231591516223276, 1006077769355418447, 8939135009787587061, 79425405274544254272, 705705304433212164522, 6270285620201477521311, 55712322858527558034285, 495011408817065983739508, 4398242297026386902181258, 39078968602851831534695949, 347221840890572644884915903, 3085112302134291192548948076, 27411633704758373646510075396, 243556016370680989436453180013, 2164027644220976459349399444525, 19227673841672401095642365956188, 170840442980948006608464455697966, 1517940089802767494889392038275745, 13487099869481098744232321728369329, 119834678661785966012895741566618796, 1064747080465268292233048978076378048, 9460419621593504740170051979636841847, 84057088353341425342543117371967154421] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 77, 665, 5858, 52400, 467621, 4158083, 36934214, 328091786, 2915033849, 25900915739, 230135710868, 2044789537190, 18168207997361, 161426850899369, 1434298978727900, 12743938239783212, 113231592099185237, 1006077779243383547, 8939135034650340932, 79425405243245508710, 705705304057358994197, 6270285619358526038363, 55712322860039974615418, 495011408831240837961224, 4398242297054477624721671, 39078968602783076172667931, 347221840890042088053707834, 3085112302133375023315674224, 27411633704761375636255038749, 243556016370700700773997973617, 2164027644221005514548965147368, 19227673841672273776193659617740, 170840442980947279706099940477185, 1517940089802766608341576229876335, 13487099869481104025506971448981640, 119834678661785992618133734419724052, 1064747080465268317766145146712180803, 9460419621593504524955223210634180325, 84057088353341424376285267435024177166] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (1620 t + 8316 t - 1044 t - 53274 t + 52291 t - 25068 t + 39517 t 10 9 8 7 6 5 4 - 13477 t - 2487 t - 5401 t + 1945 t + 141 t - 410 t - 13 t 3 2 / + 23 t + 26 t - 6 t + 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation (1620*t^17+8316*t^16-1044*t^15-53274*t^14+52291*t^13-25068*t^12+39517*t^11-\ 13477*t^10-2487*t^9-5401*t^8+1945*t^7+141*t^6-410*t^5-13*t^4+23*t^3+26*t^2-6*t+ 1)/(t-1)/(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t ^8+409*t^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 t (1620 t + 2160 t - 59688 t + 66174 t - 39995 t + 41227 t 10 9 8 7 6 5 4 - 18547 t + 10687 t - 3203 t - 219 t - 2867 t - 437 t + 284 t 3 2 / + 118 t - 3 t - 19 t + 8) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation t*(1620*t^16+2160*t^15-59688*t^14+66174*t^13-39995*t^12+41227*t^11-18547*t^10+ 10687*t^9-3203*t^8-219*t^7-2867*t^6-437*t^5+284*t^4+118*t^3-3*t^2-19*t+8)/(t-1) /(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t^8+409*t ^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) This theorem took, 0.481, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 41, :consider the sequence, (x y + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 3, 5, 25, 8, 3, 15, 55, 5, 25, 15, 25, 125, 33, 8, 40, 185, 3, 15, 73, 15, 75, 246, 55, 275, 295, 5, 25, 15, 25, 125, 40, 15, 75, 275, 25, 125, 75, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 36, 0, 0, 40, 0, 0, 140, 30, 150, 90, 8, 40, 48, 40, 200, 142, 36, 180, 390, 0, 0, 40, 0, 0, 150, 40, 200, 180, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [524569192196997746788106616626123185942745576537979614773144931748694289126433\ 432102203369140625000000, 52456919219700138958117688024303396302753498057758144\ 1643466733014767014537937939167022705078125000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 55, 295, 2673, 16743, 128731, 859309, 6238833, 42802095, 304010935, 2110214689, 14858636997, 103646493867, 727261783723, 5083451774845, 35618882099625, 249182951147607, 1744976448013279, 12211823754516121, 85496729635183533, 598416407769202275, 4189195324948036723, 29323144664212423381, 205267649402496635409, 1436848931416694748303, 10058055844725589746151, 70405895503727705064625, 492843547389970393038549, 3449894862545730230943963, 24149309870287589242579771, 169044968504025294738106669, 1183315701392208340895080185, 8283205874047572754907206215, 57982459661544859718162586991, 405877136438958994551756961609, 2841140328080995352953019847549, 19887980663162536336836380255187, 139215872145600280436663382805891, 974511072159238107645626501834245, 6821577656057012519383321246582113] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 36, 390, 2294, 18638, 121116, 897384, 6085670, 43567910, 300929856, 2125620084, 14796655634, 103956400682, 726014914956, 5089686118680, 35593799037038, 249308366460542, 1744471855962264, 12214346714771196, 85486578835640234, 598467161766918770, 4188991122897499716, 29324165674465108416, 205263541501707795590, 1436869470920638947398, 10057973206726377576720, 70406308693723765911780, 492841884974234265851138, 3449903174624410866881018, 24149276427729612623096316, 169045135716815177835523944, 1183315028633487574023830558, 8283209237841176589263454350, 57982446127762644724053885960, 405877204107870069522300466764, 2841140055824010212567052202202, 19887982024447462038766218481922, 139215866668648990677444431622708, 974511099543994556441721257750160, 6821577545878037993063671492799030] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 58 t - 140 t - 214 t + 144 t + 23 t - 21 t + 7 t - 1 - ---------------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -(58*t^7-140*t^6-214*t^5+144*t^4+23*t^3-21*t^2+7*t-1)/(7*t-1)/(2*t^2+3*t-1)/(58 *t^4-23*t^2+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 2 t (29 t + 70 t + 55 t - 101 t + t + 22 t - 4) - ---------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -2*t*(29*t^6+70*t^5+55*t^4-101*t^3+t^2+22*t-4)/(7*t-1)/(2*t^2+3*t-1)/(58*t^4-23 *t^2+1) This theorem took, 0.137, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 42, :consider the sequence, (x y + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 5 t + 1) (3 t - 1) (2 t - 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t^2-5*t+1)*(3*t-1)*(2*t-1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.063, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 43, :consider the sequence, (x y + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 13, 4, 20, 70, 5, 25, 20, 25, 125, 58, 13, 65, 256, 4, 20, 97, 20, 100, 347, 70, 350, 419, 5, 25, 20, 25, 125, 65, 20, 100, 350, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 35, 9, 45, 45, 0, 0, 45, 0, 0, 168, 35, 175, 143, 9, 45, 72, 45 , 225, 234, 45, 225, 526, 0, 0, 45, 0, 0, 175, 45, 225, 225, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267896524511317242722702136297257202930175362507071549601832518100467000060234\ 56808179616928100585937500000, 267896524511317243420256499444604614847475565661\ 72061890155641714073341721996257547289133071899414062500000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 70, 419, 4089, 29964, 254810, 1994140, 16312963, 130200467, 1053041769, 8454480642, 68148008738, 548106635614, 4413612943480, 35517131370215, 285914797108143, 2301176444858019, 18522931032077429, 149088421286304895, 1200029882993312707, 9659008455747134855, 77745848735926487139, 625776973122396328941, 5036898499829037784091, 40542090017523583991989, 326324320951145176414549, 2626591573435280454033953, 21141498307393145731709844 , 170168400675891567386020059, 1369689433209436619579814392, 11024661783793074775730027692, 88737758592232342740280410043, 714252269520383160124513756598, 5749033062799057816752594533214, 46274100120634991807830744306257, 372461303152855925853960097938263, 2997949648427038991013899409910615, 24130566112606310911001586682555006, 194227485115114895481448982103860894, 1563341523160241100076175055900336708] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 45, 526, 3590, 32063, 245344, 2035011, 16131234, 130995094, 1049527331, 8469919742, 68079882400, 548406442929, 4412291323632, 35522951332834, 285889151640917, 2301289406557217, 18522433347575473, 149090613660887226, 1200020224434273258, 9659051004601910053, 77745661289893631906, 625777798888770010901, 5036894861999550621247, 40542106043524364034663, 326324250350410412032395, 2626591884458387301564568, 21141496937217068921860226 , 170168406712041161477229740, 1369689406617888001670541496, 11024661900939026515155881862, 88737758076159618777682921536, 714252271793881090069601821174, 5749033052783429777004848956949, 46274100164757667296961306425737, 372461302958478659979995467240933, 2997949649283345244641617795843832, 24130566108833954223783048973451097, 194227485131733570422619952438571134, 1563341523087029477893471616820464186] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.444, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 44, :consider the sequence, (x y + x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 5 t + 1) (3 t - 1) (2 t - 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t^2-5*t+1)*(3*t-1)*(2*t-1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.070, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 45, :consider the sequence, (x y + x y + x y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 12, 4, 20, 69, 5, 25, 20, 25, 125, 52, 12, 60, 281, 4, 20, 97, 20, 100, 353, 69, 345, 448, 5, 25, 20, 25, 125, 60, 20, 100, 345, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 37, 9, 45, 44, 0, 0, 45, 0, 0, 177, 37, 185, 156, 9, 45, 72, 45 , 225, 228, 44, 220, 573, 0, 0, 45, 0, 0, 185, 45, 225, 220, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [732833319837921614402774463453741284334138041426903611089765648122778819121644\ 15407460182905197143554687500, 732833319837921629096453848732335133943344756704\ 97458382308510777387283798104048255481757223606109619140625] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 69, 448, 4905, 39772, 379821, 3309016, 30328017, 270217780, 2445631893, 21942327664, 197822745849, 1778695728268, 16016806476285, 144108533677192, 1297190426141601, 11673645720040036, 105068152056532197, 945586665627930400, 8510413505055670473, 76593053973479549884, 689340823621423370829, 6204050723292273228088, 55836539956133144599665, 502528442372684873662612, 4522758067516731001635381, 40704812176837743321359056, 366343361745603868359028377, 3297089994940113922897271020, 29673811258312629767745361053, 267064294805555645601358640104, 2403578685846290921953975807809, 21632208009635167739877042035908, 194689872901623762447437079495045, 1752208852040077598084215209596032, 15769879688733379702471529415661161, 141928917096737010723675802094466076, 1277360254379950129505922032082616557, 11496242286872966000589099222581439640, 103466180594589619830122888334043503633] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 44, 573, 4280, 42897, 364196, 3387141, 29937392, 272170905, 2435866268, 21991155789, 197578605224, 1779916431393, 16010702960660, 144139051255317, 1297037838250976, 11674408659493161, 105064337359266572, 945605739114258525, 8510318137624029848, 76593530810637753009, 689338439435632355204, 6204062644221228306213, 55836480351488369209040, 502528740395908750615737, 4522756577400611616869756, 40704819627418340245187181, 366343324492700883739887752, 3297090181204628845992974145, 29673810326990055152266845428, 267064299462168518678751218229, 2403578662563226556567012917184, 21632208126050489566811856489033, 194689872319547153312763007229420, 1752208854950460643757585570924157, 15769879674181464474104677609020536, 141928917169496586865510061127669201, 1277360254016152248796750736916600932, 11496242288691955404134955698411517765, 103466180585494672812393605954893113008] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 8 t + t + 1 - --------------------------- (t + 1) (9 t - 1) (5 t + 1) and in Maple notation -(8*t^2+t+1)/(t+1)/(9*t-1)/(5*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is t (17 t + 9) - --------------------------- (t + 1) (9 t - 1) (5 t + 1) and in Maple notation -t*(17*t+9)/(t+1)/(9*t-1)/(5*t+1) This theorem took, 0.063, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 46, :consider the sequence, (x y + x y + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 15, 5, 25, 64, 5, 25, 25, 25, 125, 69, 15, 75, 250, 5, 25, 74, 25, 125, 286, 64, 320, 429, 5, 25, 25, 25, 125, 75, 25, 125, 320, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 31, 7, 35, 48, 0, 0, 35, 0, 0, 155, 31, 155, 158, 7, 35, 70, 35 , 175, 248, 48, 240, 503, 0, 0, 35, 0, 0, 155, 35, 175, 240, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [101836551193649006065079686140632373943739732244404333325412600905728000000000\ 00000000000000000000000000000, 101836551193649006065079691975600565036310342269\ 14140499006773316812800000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 64, 429, 4006, 30798, 266692, 2153646, 18119446, 148777242, 1239214174, 10234774206, 84953304130, 703073716854, 5828783273998, 48273448966206, 400039073259730, 3313910922564594, 27458142093378550, 227482090002795558, 1884756729471575818, 15615089755609261482, 129373352272110257806, 1071861290620359999630, 8880475772390349306226, 73575222340947702672258, 609576602307635586043894, 5050381598001600261155046, 41842783913517064989833914 , 346670327091033676855209738, 2872188464542741235787436510, 23796281606585450560211998638, 197153877668618906087417346178, 1633433705647574361069557016018, 13533113450615314186001200535782, 112122796321976102809788236672694, 928945264730756183024908675705738, 7696376874186098727853955502247578, 63765024237824341130886783476027758, 528297713206739211064083533462846142, 4376983740854466777597514852721001874] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 48, 503, 3660, 32582, 258246, 2195576, 17915400, 149776664, 1234326804, 10258678844, 84836370444, 703645648796, 5825985709296, 48287132620124, 399972142010436, 3314238304285592, 27456540765929064, 227489922602220572, 1884718417754902956, 15615277150331001536, 129372435665311345440, 1071865774033937259812, 8880453842597538150660, 73575329606495486966168, 609576077637938931929640, 5050384164326913590424380, 41842771360808843314865100 , 346670388490297000128410048, 2872188164219538450840512016, 23796283075561155614025417860, 197153870483394447793375324884, 1633433740792778150649698188616, 13533113278709014448212238308248, 112122797162824147580740025067980, 928945260617902637381240831868444, 7696376894303365329243710324597840, 63765024139424431914560015701903584, 528297713688044267344070354224380308, 4376983738500251625920886974311288548] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (2736 t - 6240 t + 4344 t - 3460 t + 3220 t - 4488 t + 6856 t 10 9 8 7 6 5 4 - 5716 t + 3972 t - 2228 t + 1017 t - 626 t + 385 t - 237 t 3 2 / + 107 t - 32 t + 7 t - 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation (2736*t^17-6240*t^16+4344*t^15-3460*t^14+3220*t^13-4488*t^12+6856*t^11-5716*t^ 10+3972*t^9-2228*t^8+1017*t^7-626*t^6+385*t^5-237*t^4+107*t^3-32*t^2+7*t-1)/(t-\ 1)/(48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t ^7-608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 10 - t (3312 t - 8304 t + 6144 t - 4700 t + 5876 t - 6784 t + 8420 t 9 8 7 6 5 4 3 - 7868 t + 7094 t - 6268 t + 4345 t - 2408 t + 1051 t - 388 t 2 / + 123 t - 36 t + 7) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation -t*(3312*t^16-8304*t^15+6144*t^14-4700*t^13+5876*t^12-6784*t^11+8420*t^10-7868* t^9+7094*t^8-6268*t^7+4345*t^6-2408*t^5+1051*t^4-388*t^3+123*t^2-36*t+7)/(t-1)/ (48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t^7-\ 608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) This theorem took, 0.671, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 47, :consider the sequence, (x y + x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 3, 5, 25, 8, 3, 15, 55, 5, 25, 15, 25, 125, 33, 8, 40, 185, 3, 15, 73, 15, 75, 246, 55, 275, 295, 5, 25, 15, 25, 125, 40, 15, 75, 275, 25, 125, 75, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 36, 0, 0, 40, 0, 0, 140, 30, 150, 90, 8, 40, 48, 40, 200, 142, 36, 180, 390, 0, 0, 40, 0, 0, 150, 40, 200, 180, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [524569192196997746788106616626123185942745576537979614773144931748694289126433\ 432102203369140625000000, 52456919219700138958117688024303396302753498057758144\ 1643466733014767014537937939167022705078125000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 55, 295, 2673, 16743, 128731, 859309, 6238833, 42802095, 304010935, 2110214689, 14858636997, 103646493867, 727261783723, 5083451774845, 35618882099625, 249182951147607, 1744976448013279, 12211823754516121, 85496729635183533, 598416407769202275, 4189195324948036723, 29323144664212423381, 205267649402496635409, 1436848931416694748303, 10058055844725589746151, 70405895503727705064625, 492843547389970393038549, 3449894862545730230943963, 24149309870287589242579771, 169044968504025294738106669, 1183315701392208340895080185, 8283205874047572754907206215, 57982459661544859718162586991, 405877136438958994551756961609, 2841140328080995352953019847549, 19887980663162536336836380255187, 139215872145600280436663382805891, 974511072159238107645626501834245, 6821577656057012519383321246582113] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 36, 390, 2294, 18638, 121116, 897384, 6085670, 43567910, 300929856, 2125620084, 14796655634, 103956400682, 726014914956, 5089686118680, 35593799037038, 249308366460542, 1744471855962264, 12214346714771196, 85486578835640234, 598467161766918770, 4188991122897499716, 29324165674465108416, 205263541501707795590, 1436869470920638947398, 10057973206726377576720, 70406308693723765911780, 492841884974234265851138, 3449903174624410866881018, 24149276427729612623096316, 169045135716815177835523944, 1183315028633487574023830558, 8283209237841176589263454350, 57982446127762644724053885960, 405877204107870069522300466764, 2841140055824010212567052202202, 19887982024447462038766218481922, 139215866668648990677444431622708, 974511099543994556441721257750160, 6821577545878037993063671492799030] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 58 t - 140 t - 214 t + 144 t + 23 t - 21 t + 7 t - 1 - ---------------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -(58*t^7-140*t^6-214*t^5+144*t^4+23*t^3-21*t^2+7*t-1)/(7*t-1)/(2*t^2+3*t-1)/(58 *t^4-23*t^2+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 2 t (29 t + 70 t + 55 t - 101 t + t + 22 t - 4) - ---------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -2*t*(29*t^6+70*t^5+55*t^4-101*t^3+t^2+22*t-4)/(7*t-1)/(2*t^2+3*t-1)/(58*t^4-23 *t^2+1) This theorem took, 0.139, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 48, :consider the sequence, (x y + x y + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 3, 5, 25, 9, 3, 15, 64, 5, 25, 15, 25, 125, 39, 9, 45, 248, 3, 15, 73, 15, 75, 290, 64, 320, 342, 5, 25, 15, 25, 125, 45, 15, 75, 320, 25, 125, 75, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 34, 8, 40, 36, 0, 0, 40, 0, 0, 164, 34, 170, 108, 8, 40, 48, 40 , 200, 162, 36, 180, 500, 0, 0, 40, 0, 0, 170, 40, 200, 180, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [137935242201087480368791871580266240368460514124667269449590593414306640625000\ 000000000000000000000000000, 13793524220109436812498031593450345369172511734358\ 0347757696062164306640625000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 64, 342, 3862, 25695, 238153, 1757157, 15056914, 116552439, 965317699, 7638222204, 62312428795, 497999413212, 4035224920261, 32395336701435, 261696215806606, 2105222124849390, 16983080526892576, 136746668333945013, 1102469189443493719, 8880698698716458313, 71577351363003644746, 576683696298528852057, 4647415430764955152615, 37446415406823070942746, 301758435040983910631329, 2431502552048629236052194, 19593534156560847569625805 , 157883067821238610826170323, 1272238664653485402082115764, 10251672796306872066189437358, 82608646766980963899405252082, 665661092128298747676289323963, 5363927814152791043982966518101, 43222638542569672783522380909813, 348289647972587435439853788312562, 2806527006960345615775054734183939, 22615089433353459319734378500993059, 182233035054771847675378530044690736, 1468439636259198162437202361619648215] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 36, 500, 3033, 30233, 213804, 1889318, 14343231, 120416639, 944416611, 7751337887, 61700361651, 501311680805, 4017300668040, 32492335703936, 261171296038926, 2108062796604320, 16967707847633655, 136829859793228487, 1102018986677129394, 8883135038076890198, 71564166746347903263, 576755046844825610117, 4647029306461169978259, 37448504977180689608465, 301747127013005786149809, 2431563747164154620500667, 19593202989869434128875724 , 157884859980259863937195736, 1272228966111766678778274480, 10251725281443860817983315204, 82608362735660651252692129341, 665662629207091547534491422869, 5363919496016558141922688723128, 43222683557432111691917537653502, 348289404367796154123612820469199, 2806528325264815036792357094694995, 22615082299147826076503513673205803, 182233073662613359730499821756114939, 1468439427326977408762078918936040655] Using the found enumerative automaton with, 18, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 12 11 10 9 8 7 6 5 (6912 t + 1344 t - 1680 t - 2724 t - 1122 t + 2089 t - 549 t + 283 t 4 3 2 / - 182 t + 28 t + 10 t - 6 t + 1) / ((t + 1) / 6 5 4 3 2 (144 t - 72 t + 45 t - 110 t + 69 t - 15 t + 1) 6 5 4 3 2 (144 t + 8 t + 37 t - 40 t - 11 t + 5 t + 1)) and in Maple notation (6912*t^12+1344*t^11-1680*t^10-2724*t^9-1122*t^8+2089*t^7-549*t^6+283*t^5-182*t ^4+28*t^3+10*t^2-6*t+1)/(t+1)/(144*t^6-72*t^5+45*t^4-110*t^3+69*t^2-15*t+1)/( 144*t^6+8*t^5+37*t^4-40*t^3-11*t^2+5*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 10 9 8 7 6 5 4 3 (2496 t + 2352 t + 900 t - 739 t - 2370 t + 1136 t - 352 t + 305 t 2 / - 40 t - 36 t + 8) t / ((t + 1) / 6 5 4 3 2 (144 t - 72 t + 45 t - 110 t + 69 t - 15 t + 1) 6 5 4 3 2 (144 t + 8 t + 37 t - 40 t - 11 t + 5 t + 1)) and in Maple notation (2496*t^10+2352*t^9+900*t^8-739*t^7-2370*t^6+1136*t^5-352*t^4+305*t^3-40*t^2-36 *t+8)*t/(t+1)/(144*t^6-72*t^5+45*t^4-110*t^3+69*t^2-15*t+1)/(144*t^6+8*t^5+37*t ^4-40*t^3-11*t^2+5*t+1) This theorem took, 0.229, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 49, :consider the sequence, (x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 26, 5, 25, 82, 5, 25, 25, 25, 125, 131, 26, 130, 335, 5, 25, 125, 25, 125, 474, 82, 410, 815, 5, 25, 25, 25, 125, 130, 25, 125, 410, 25, 125 , 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 42, 10, 50, 87, 0, 0, 50, 0, 0, 196, 42, 210, 354, 10, 50, 100 , 50, 250, 454, 87, 435, 841, 0, 0, 50, 0, 0, 210, 50, 250, 435, 0, 0, 250, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [207626640204256061661849504295284493000850492381094725098125226973158360217930\ 74905872344970703125000000000000000, 207626640204256061661849504295284493000808\ 79511269677378171321839042758265626616775989532470703125000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 82, 815, 7602, 68102, 610573, 5455979, 48707739, 434544464, 3875631319, 34562074775, 308199440121, 2748232198217, 24505857107734, 218516467310537, 1948486623258966, 17374416916255994, 154925478432043081, 1381450611225161429, 12318216679895859075, 109839942573734730422, 979428522100686750547, 8733437029690378022345, 77874924348574670067627, 694400590672174960238351, 6191879915045693415478450, 55212189310162357695360725, 492319923800252186866328760, 4389952841617646905667958338, 39144639530887331097004578805, 349047668452254899745621406595, 3112412741853660735162042834081, 27752980326672011732577602589002, 247469722332829459164099025315987, 2206655384416411971198345236200547, 19676459567137748067779845997056983, 175452435315156874254324345671899781, 1564486586267293169054331485165058916, 13950323768454355195030038082661324795, 124393225837125259480767261999894392208] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 87, 841, 7565, 68356, 611016, 5459056, 48717506, 434579104, 3875796546, 34562644321, 308201946752, 2748241201942, 24505894541043, 218516610243211, 1948487196556223, 17374419144627724, 154925487234913770, 1381450645764438988, 12318216815733715766, 109839943108165387234, 979428524198161678932, 8733437037948333166909, 77874924380983702309748, 694400590799740502527984, 6191879915546555070931173, 55212189312132634994954077, 492319923807992280243622229, 4389952841648080748856557164, 39144639531006931745458969224, 349047668452725038986470839758, 3112412741855508637531093118132, 27752980326679274891142058453210, 247469722332858008911494872921628, 2206655384416524183216323110377643, 19676459567138189146212167376209606, 175452435315158607889602803134194610, 1564486586267299983448611709053331743, 13950323768454381979080742696756493191, 124393225837125364759198698441246530189] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.991, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 50, :consider the sequence, (x y + x y + x + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 5 t + 1) (3 t - 1) (2 t - 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t^2-5*t+1)*(3*t-1)*(2*t-1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.313, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 51, :consider the sequence, (x y + x y + x + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 24, 6, 30, 86, 5, 25, 30, 25, 125, 116, 24, 120, 366, 6, 30, 100, 30, 150, 436, 86, 430, 698, 5, 25, 30, 25, 125, 120, 30, 150, 430, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 40, 8, 40, 70, 0, 0, 40, 0, 0, 196, 40, 200, 280, 8, 40, 96, 40 , 200, 404, 70, 350, 766, 0, 0, 40, 0, 0, 200, 40, 200, 350, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [221074292181328352337481411059315988494392125439801022315531214096695296000000\ 0000000000000000000000000000000000, 2210742921813283523374814110593182009075969\ 726148126128035763279846440960000000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 86, 698, 6854, 59996, 546800, 4893716, 44113334, 396663728, 3570746276, 32132948468, 289206985922, 2602829013758, 23425609838726, 210830182948616, 1897473521721596, 17077257939071726, 153695341104621086, 1383258009883223744, 12449322266527371560, 112043899486013477528, 1008395097093088522304, 9075555862427839871786, 81680002785491168796212, 735120024954756891341354, 6616080224983201946144222, 59544722023854977700374462, 535902498220454388053652014, 4823122483974206892784913582, 43408102355836154749438068458, 390672921202377979369401930140, 3516056290822046182871417655458, 31644506617395928634275740070172, 284800559556568708440672858462542, 2563205036009082710856195670838972, 23068845324081801088343776131861896, 207619607916735808199149054907088680, 1868576471250623206680729843672741350, 16817188241255605303809346462741948178, 151354694171300463655493939933951869112] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 70, 766, 6574, 61060, 542986, 4907140, 44067814, 396817972, 3570232852, 32134671220, 289201229152, 2602848478612, 23425543844608, 210830409126766, 1897472743671178, 17077260634346776, 153695331750326290, 1383258042448248274, 12449322153171081220, 112043899880796840694, 1008395095719748482148, 9075555867202437300616, 81680002768912779532558, 735120025012277223690466, 6616080224783814941698402, 59544722024545791322650064, 535902498218062073827323832, 4823122483982490221311860910, 43408102355807477495816233336, 390672921202477266765234508798, 3516056290821702404920662828712, 31644506617397119114030612810024, 284800559556564585419200136914384, 2563205036009096992057555146678022, 23068845324081751617086001016824568, 207619607916735979585079728248974482, 1868576471250622612913346132028371850, 16817188241255607360978943010660115862, 151354694171300456528170224745960448818] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.834, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 52, :consider the sequence, (x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 25, 5, 25, 50, 5, 25, 25, 25, 125, 125, 25, 125, 202, 5, 25, 74, 25, 125, 238, 50, 250, 440, 5, 25, 25, 25, 125, 125, 25, 125, 250, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 23, 7, 35, 58, 0, 0, 35, 0, 0, 91, 23, 115, 194, 7, 35, 70, 35, 175, 266, 58, 290, 472, 0, 0, 35, 0, 0, 115, 35, 175, 290, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [136162413819432278698506963161122032529321126599757500889853390225408000000000\ 0000000000000000000000000000, 1361624138194322786985069631611220325302089628744\ 064983416503711825920000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 50, 440, 3680, 30080, 243200, 1955840, 15687680, 125665280, 1005977600, 8050442240, 64414023680, 515354132480, 4123000832000, 32984677744640, 263880106311680, 2111051587911680, 16888455652966400, 135107817022423040, 1080863223374151680, 8646908535772282880, 69175279281294540800, 553402278230821437440, 4427218401768431943680, 35417747917834897326080, 283341986157428945715200, 2266735900518430634147840, 18133887249183441346887680 , 145071098173611515869921280, 1160568786109468067338649600, 9284550291758048300226314240, 74276402345593601447878983680, 594211218810865671767305748480, 4753689750671392814875541504000, 38029518006109012281952714096640, 304236144051823577307415241031680, 2433889152426394534666496041287680, 19471113219458379942160664782438400, 155768905755855934196600104068055040, 1246151246047603052210059975778631680] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 58, 472, 3808, 30592, 245248, 1964032, 15720448, 125796352, 1006501888, 8052539392, 64422412288, 515387686912, 4123135049728, 32985214615552, 263882253795328, 2111060177846272, 16888490012704768, 135107954461376512, 1080863773129965568, 8646910734795538432, 69175288077387563008, 553402313415193526272, 4427218542505920299008, 35417748480784850747392, 283341988409228759400448, 2266735909525629888888832, 18133887285212238365851648 , 145071098317726703945777152, 1160568786685928819642073088, 9284550294063891309440008192, 74276402354816973484733759488, 594211218847759159914724851712, 4753689750818966767465217916928, 38029518006699308092311419748352, 304236144054184760548850063638528, 2433889152435839267632235331715072, 19471113219496158874023621944147968, 155768905756007049924051932714893312, 1246151246048207515119867290365984768] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 22 t - 7 t + 1 ------------------- (8 t - 1) (4 t - 1) and in Maple notation (22*t^2-7*t+1)/(8*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is t (26 t - 7) - ------------------- (8 t - 1) (4 t - 1) and in Maple notation -t*(26*t-7)/(8*t-1)/(4*t-1) This theorem took, 0.064, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 53, :consider the sequence, (x y + x y + x + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 14, 4, 20, 77, 5, 25, 20, 25, 125, 62, 14, 70, 309, 4, 20, 97, 20, 100, 382, 77, 385, 572, 5, 25, 20, 25, 125, 70, 20, 100, 385, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 39, 9, 45, 55, 0, 0, 45, 0, 0, 187, 39, 195, 190, 9, 45, 72, 45 , 225, 260, 55, 275, 682, 0, 0, 45, 0, 0, 195, 45, 225, 275, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [149705283111816902022749552413030261336338518382644366607174502686306120737075\ 8056640625000000000000000000000, 1497052831118169020996123785262971954827181470\ 694272562053092186112769263820648193359375000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 77, 572, 6095, 51503, 489074, 4303691, 39430124, 351999521, 3187156502, 28600703204, 257936265680, 2319001958492, 20885745717140, 187901874798527, 1691527824959939, 15221747449215083, 137007230138151521, 1233007814224183466, 11097393122576030276, 99874905451533009401, 898883224867843165532, 8089902557481179799065, 72809378580796430316710, 655283086643773913875208, 5897554984486051157304938, 53077957369271298701124143, 477701819613893150965109225, 4299315313019049609289974380, 38693843557176451529855216630, 348244561864609442112896023268, 3134201218936842320723142353762, 28207810116084828704069285306003, 253870295627453966870074707273464, 2284832636446257388227133175275373, 20563493857566817405453136664685973, 185071444032752012720185980804713291, 1665642999957928133145955093713824417, 14990786980216637147546113583065840763, 134917082925546749171586461162039209295] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 55, 682, 5515, 54553, 472897, 4389418, 38974327, 354421024, 3174275635, 28669191040, 257571928768, 2320939692619, 20875437648319, 187956704039146, 1691236158567028, 15223298899237321, 136998977244962458, 1233051714185466535, 11097159599636697145, 99876147647625334303, 898876617110295305122, 8089937706715270110775, 72809191607700213963475, 655284081227601660624991, 5897549693893301023152568, 53077985512040593111590940, 477701669911195247715124744, 4299316109347487900246763109, 38693839321186144163398570549, 348244584397535979767212488235, 3134201099075177910263851940419, 28207810753676987256245861953270, 253870292235845916727855784077774, 2284832654487578214669819344547688, 20563493761597807592700520767757159, 185071444543249595544748987480932055, 1665642997242387086690556558066894946, 14990786994661687156323408908041711471, 134917082848707744582845418333241390786] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 21 20 19 18 17 16 - (9072 t - 34884 t + 94851 t - 92004 t - 436641 t + 807473 t 15 14 13 12 11 + 381103 t - 1220119 t + 202509 t + 948007 t - 256030 t 10 9 8 7 6 5 4 - 363108 t + 100243 t + 68583 t - 21040 t - 5850 t + 2515 t + 61 t 3 2 / - 158 t + 22 t + 4 t - 1) / ((9 t - 1) / 8 7 6 5 4 3 2 (171 t - 246 t + 104 t + 103 t - 131 t + 13 t + 27 t - 10 t + 1) ( 11 10 9 8 7 6 5 4 1008 t - 12 t - 1981 t - 2597 t - 2554 t - 1019 t + 564 t + 576 t 3 2 + 112 t - 25 t - 11 t - 1)) and in Maple notation -(9072*t^21-34884*t^20+94851*t^19-92004*t^18-436641*t^17+807473*t^16+381103*t^ 15-1220119*t^14+202509*t^13+948007*t^12-256030*t^11-363108*t^10+100243*t^9+ 68583*t^8-21040*t^7-5850*t^6+2515*t^5+61*t^4-158*t^3+22*t^2+4*t-1)/(9*t-1)/(171 *t^8-246*t^7+104*t^6+103*t^5-131*t^4+13*t^3+27*t^2-10*t+1)/(1008*t^11-12*t^10-\ 1981*t^9-2597*t^8-2554*t^7-1019*t^6+564*t^5+576*t^4+112*t^3-25*t^2-11*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 20 19 18 17 16 15 - t (9072 t - 34884 t - 292977 t + 448995 t + 851340 t - 868390 t 14 13 12 11 10 - 507251 t + 1049613 t - 179830 t - 616508 t + 417408 t 9 8 7 6 5 4 3 + 232439 t - 211817 t - 57833 t + 49003 t + 8327 t - 5880 t - 604 t 2 / + 361 t + 17 t - 9) / ((9 t - 1) / 8 7 6 5 4 3 2 (171 t - 246 t + 104 t + 103 t - 131 t + 13 t + 27 t - 10 t + 1) ( 11 10 9 8 7 6 5 4 1008 t - 12 t - 1981 t - 2597 t - 2554 t - 1019 t + 564 t + 576 t 3 2 + 112 t - 25 t - 11 t - 1)) and in Maple notation -t*(9072*t^20-34884*t^19-292977*t^18+448995*t^17+851340*t^16-868390*t^15-507251 *t^14+1049613*t^13-179830*t^12-616508*t^11+417408*t^10+232439*t^9-211817*t^8-\ 57833*t^7+49003*t^6+8327*t^5-5880*t^4-604*t^3+361*t^2+17*t-9)/(9*t-1)/(171*t^8-\ 246*t^7+104*t^6+103*t^5-131*t^4+13*t^3+27*t^2-10*t+1)/(1008*t^11-12*t^10-1981*t ^9-2597*t^8-2554*t^7-1019*t^6+564*t^5+576*t^4+112*t^3-25*t^2-11*t-1) This theorem took, 0.682, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 54, :consider the sequence, (x y + x y + x + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 27, 6, 30, 48, 5, 25, 30, 25, 125, 132, 27, 135, 177, 6, 30, 61, 30, 150, 214, 48, 240, 379, 5, 25, 30, 25, 125, 135, 30, 150, 240, 25, 125, 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 5, 0, 0, 13, 5, 25, 44, 0, 0, 25, 0, 0, 53, 13, 65, 121, 5, 25, 60, 25, 125, 197, 44, 220, 363, 0, 0, 25, 0, 0, 65, 25, 125, 220, 0, 0, 125, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [366830848846142692962304570645980613361416583355553641715164870938000000000000\ 0000000000000000000000000, 3668308488461426929623045706458981255899353596369488\ 344628976290620000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 48, 379, 2986, 23514, 185160, 1458148, 11483992, 90450744, 712442784, 5611763248, 44203382176, 348189424800, 2742697165440, 21604357739584, 170178828793216, 1340510014140288, 10559291746718208, 83176304277883648, 655185843525933568, 5160947243464038912, 40653163755582375936, 320227988361766577152, 2522459657373943355392, 19869602240458152130560, 156514334359940578320384, 1232875051891459599314944, 9711448480782083401031680, 76497802015142861561831424, 602578876448813819043938304, 4746558630604264739218407424, 37388995392029106173793107968, 294515897780613578028315279360, 2319923633603318660292224876544, 18274211023394088172592641540096, 143947319511546693069368416927744, 1133883742946253568935188212875264, 8931686584242594025124485371789312, 70355559584874706091288411404238848, 554195976069593874335484953474105344] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 5, 44, 363, 2922, 23258, 184136, 1454052, 11467608, 90385208, 712180640, 5610714672, 44199187872, 348172647584, 2742630056576, 21604089304128, 170177755051392, 1340505719172992, 10559274566849024, 83176235558406912, 655185568648026624, 5160946143952411136, 40653159357535864832, 320227970769580532736, 2522459587005199177728, 19869601958983175419904, 156514333234040671477760, 1232875047387859971944448, 9711448462767684891549696, 76497801943085267523903488, 602578876160583442892226560, 4746558629451343234611560448, 37388995387417420155365720064, 294515897762166833954605727744, 2319923633529531683997386670080, 18274211023098940267413288714240, 143947319510366101448651005624320, 1133883742941531202452318567661568, 8931686584223704559193006790934528, 70355559584799148227562497080819712, 554195976069291642880581296180428800] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (3 t - 5 t + 1) (3 t - 1) (2 t - 1) ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation (3*t^2-5*t+1)*(3*t-1)*(2*t-1)/(12*t^3-34*t^2+12*t-1)/(4*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (3 t - 1) (6 t - 21 t + 5) - ------------------------------------ 3 2 (12 t - 34 t + 12 t - 1) (4 t - 1) and in Maple notation -t*(3*t-1)*(6*t^2-21*t+5)/(12*t^3-34*t^2+12*t-1)/(4*t-1) This theorem took, 0.063, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 55, :consider the sequence, (x y + x y + x + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 19, 4, 20, 63, 5, 25, 20, 25, 125, 92, 19, 95, 252, 4, 20, 97, 20, 100, 348, 63, 315, 560, 5, 25, 20, 25, 125, 95, 20, 100, 315, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 35, 9, 45, 68, 0, 0, 45, 0, 0, 163, 35, 175, 259, 9, 45, 72, 45 , 225, 298, 68, 340, 613, 0, 0, 45, 0, 0, 175, 45, 225, 340, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [221956190468470136620174544998346999366245202257902488545310481507228122711181\ 6406250000000000000000000000000, 2219561904684701366201745449983469984352863726\ 659528815253365020937187538146972656250000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 63, 560, 5141, 44375, 384191, 3306131, 28493651, 245347625, 2112983225, 18193228205, 156647667200, 1348698513428, 11611920684335, 99974279596235, 860739631674608, 7410614344295480, 63802307922148484, 549311017502408027, 4729335896085973187, 40717579338739403726, 350561099676969390011, 3018182380974736902233, 25985269848255786282509, 223722147447079509844889, 1926152754522913296739271, 16583357833535059481404427, 142775673504831946854951140, 1229237958883696536185240243, 10583217169520270948455114730, 91117008577579560459960932747, 784478776034624115614195830034, 6754029347944443607629234268211, 58149326439158853778219502023739, 500641023470063586946040291883320, 4310306752090451345580011162659046, 37109911945103171144074538195795384, 319500593293992663581894682418514594, 2750764519898926791240627489140204393, 23682915158064631416417905788505314397] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 68, 613, 5119, 44482, 382912, 3307108, 28483630, 245388772, 2112905668, 18193758520, 156646107526, 1348703680921, 11611896541468, 99974351650279, 860739346505599, 7410615438436942, 63802304284444897, 549311031889362733, 4729335844438968634, 40717579523743611097, 350561098971864925102, 3018182383493047945126, 25985269838960629804549, 223722147481553991108298, 1926152754398189989565671, 16583357833997413401654538, 142775673503135856937321663, 1229237958889898709634286422, 10583217169497356909662552747, 91117008577663482275472757981, 784478776034315681047061579887, 6754029347945579372027267507335, 58149326439154687661419463968159, 500641023470078919312706553059060, 4310306752090394969151265097854690, 37109911945103378275513381300376938, 319500593293991901313178188396808761, 2750764519898929593525201595472424502, 23682915158064621110924748614088306625] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.043, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 56, :consider the sequence, (x y + x y + x + x y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 29, 6, 30, 67, 5, 25, 30, 25, 125, 143, 29, 145, 278, 6, 30, 100, 30, 150, 395, 67, 335, 611, 5, 25, 30, 25, 125, 145, 30, 150, 335, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 72, 0, 0, 40, 0, 0, 145, 30, 150, 273, 8, 40, 96, 40 , 200, 399, 72, 360, 649, 0, 0, 40, 0, 0, 150, 40, 200, 360, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [129220123879748073779379580591391094634430281282713001766865300651387463073486\ 328125000000000000000000000000000, 12922012387974807377937958059139109463443028\ 1282713001766865159073169429473632812500000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 67, 611, 5494, 48314, 422874, 3689558, 32174370, 280452859, 2444397429, 21303655546, 185664827159, 1618081633231, 14101653642728, 122896329972807, 1071044665871792, 9334179172772070, 81347582127413449, 708945982875028536, 6178479884426814199, 53845588145390805065, 469265484432470598487, 4089659008461396323969, 35641468110821559426351, 310616177589646433390771, 2707026811408276293973830, 23591798129505189614198473, 205603038961941389789381319, 1791835001220079964620582162, 15615881398452881048052303257, 136092749435239089859430623948, 1186051300996155095402806677731, 10336463143202698867240248018027, 90082503363088702122984404598209, 785070995730009914324607464002251, 6841910974124274155772136730948818, 59627404441700699159162100427897389, 519654139596462248393001267980260222, 4528797242277500134540531342085187119, 39468567454475281659687849459075748831] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 72, 649, 5592, 48715, 423917, 3694237, 32188379, 280514385, 2444587103, 21304444473, 185667322645, 1618091694822, 14101686298195, 122896458796310, 1071045093416068, 9334180829484185, 81347587717332819, 708946004204392340, 6178479957329249577, 53845588420266594790, 469265485381983306849, 4089659012008081106053, 35641468123176617388062, 310616177635450543312960, 2707026811568906401488974, 23591798130097137948662265, 205603038964028360348779070, 1791835001227734226162795212, 15615881398479981654561809833, 136092749435338108973064592668, 1186051300996506868907059293769, 10336463143203980285577289191521, 90082503363093266740026826929316, 785070995730026502055853965647386, 6841910974124333370804964966062837, 59627404441700913933464380499431163, 519654139596463016407568462973453801, 4528797242277502915898809751999011405, 39468567454475291619135831790978866159] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.052, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 57, :consider the sequence, (x y + x y + x + x y + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 3, 5, 25, 8, 3, 15, 55, 5, 25, 15, 25, 125, 33, 8, 40, 185, 3, 15, 73, 15, 75, 246, 55, 275, 295, 5, 25, 15, 25, 125, 40, 15, 75, 275, 25, 125, 75, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 36, 0, 0, 40, 0, 0, 140, 30, 150, 90, 8, 40, 48, 40, 200, 142, 36, 180, 390, 0, 0, 40, 0, 0, 150, 40, 200, 180, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [524569192196997746788106616626123185942745576537979614773144931748694289126433\ 432102203369140625000000, 52456919219700138958117688024303396302753498057758144\ 1643466733014767014537937939167022705078125000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 55, 295, 2673, 16743, 128731, 859309, 6238833, 42802095, 304010935, 2110214689, 14858636997, 103646493867, 727261783723, 5083451774845, 35618882099625, 249182951147607, 1744976448013279, 12211823754516121, 85496729635183533, 598416407769202275, 4189195324948036723, 29323144664212423381, 205267649402496635409, 1436848931416694748303, 10058055844725589746151, 70405895503727705064625, 492843547389970393038549, 3449894862545730230943963, 24149309870287589242579771, 169044968504025294738106669, 1183315701392208340895080185, 8283205874047572754907206215, 57982459661544859718162586991, 405877136438958994551756961609, 2841140328080995352953019847549, 19887980663162536336836380255187, 139215872145600280436663382805891, 974511072159238107645626501834245, 6821577656057012519383321246582113] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 36, 390, 2294, 18638, 121116, 897384, 6085670, 43567910, 300929856, 2125620084, 14796655634, 103956400682, 726014914956, 5089686118680, 35593799037038, 249308366460542, 1744471855962264, 12214346714771196, 85486578835640234, 598467161766918770, 4188991122897499716, 29324165674465108416, 205263541501707795590, 1436869470920638947398, 10057973206726377576720, 70406308693723765911780, 492841884974234265851138, 3449903174624410866881018, 24149276427729612623096316, 169045135716815177835523944, 1183315028633487574023830558, 8283209237841176589263454350, 57982446127762644724053885960, 405877204107870069522300466764, 2841140055824010212567052202202, 19887982024447462038766218481922, 139215866668648990677444431622708, 974511099543994556441721257750160, 6821577545878037993063671492799030] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 58 t - 140 t - 214 t + 144 t + 23 t - 21 t + 7 t - 1 - ---------------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -(58*t^7-140*t^6-214*t^5+144*t^4+23*t^3-21*t^2+7*t-1)/(7*t-1)/(2*t^2+3*t-1)/(58 *t^4-23*t^2+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 2 t (29 t + 70 t + 55 t - 101 t + t + 22 t - 4) - ---------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -2*t*(29*t^6+70*t^5+55*t^4-101*t^3+t^2+22*t-4)/(7*t-1)/(2*t^2+3*t-1)/(58*t^4-23 *t^2+1) This theorem took, 0.143, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 58, :consider the sequence, (x y + x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 28, 6, 30, 66, 5, 25, 30, 25, 125, 140, 28, 140, 262, 6, 30, 100, 30, 150, 394, 66, 330, 651, 5, 25, 30, 25, 125, 140, 30, 150, 330, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 32, 8, 40, 77, 0, 0, 40, 0, 0, 154, 32, 160, 311, 8, 40, 96, 40 , 200, 404, 77, 385, 665, 0, 0, 40, 0, 0, 160, 40, 200, 385, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267472790943687754713045839179456699788223350084558470644928980995853850468750\ 000000000000000000000000000000000, 26747279094368775471304583917945669978822334\ 9911969458421259195445479157343750000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 66, 651, 5973, 52866, 467838, 4153845, 36917445, 328087866, 2915204382, 25901506647, 230135669685, 2044782871338, 18168187221804, 161426859510441, 1434299236695075, 12743938962713700, 113231591516223276, 1006077769355418447, 8939135009787587061, 79425405274544254272, 705705304433212164522, 6270285620201477521311, 55712322858527558034285, 495011408817065983739508, 4398242297026386902181258, 39078968602851831534695949, 347221840890572644884915903, 3085112302134291192548948076, 27411633704758373646510075396, 243556016370680989436453180013, 2164027644220976459349399444525, 19227673841672401095642365956188, 170840442980948006608464455697966, 1517940089802767494889392038275745, 13487099869481098744232321728369329, 119834678661785966012895741566618796, 1064747080465268292233048978076378048, 9460419621593504740170051979636841847, 84057088353341425342543117371967154421] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 77, 665, 5858, 52400, 467621, 4158083, 36934214, 328091786, 2915033849, 25900915739, 230135710868, 2044789537190, 18168207997361, 161426850899369, 1434298978727900, 12743938239783212, 113231592099185237, 1006077779243383547, 8939135034650340932, 79425405243245508710, 705705304057358994197, 6270285619358526038363, 55712322860039974615418, 495011408831240837961224, 4398242297054477624721671, 39078968602783076172667931, 347221840890042088053707834, 3085112302133375023315674224, 27411633704761375636255038749, 243556016370700700773997973617, 2164027644221005514548965147368, 19227673841672273776193659617740, 170840442980947279706099940477185, 1517940089802766608341576229876335, 13487099869481104025506971448981640, 119834678661785992618133734419724052, 1064747080465268317766145146712180803, 9460419621593504524955223210634180325, 84057088353341424376285267435024177166] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (1620 t + 8316 t - 1044 t - 53274 t + 52291 t - 25068 t + 39517 t 10 9 8 7 6 5 4 - 13477 t - 2487 t - 5401 t + 1945 t + 141 t - 410 t - 13 t 3 2 / + 23 t + 26 t - 6 t + 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation (1620*t^17+8316*t^16-1044*t^15-53274*t^14+52291*t^13-25068*t^12+39517*t^11-\ 13477*t^10-2487*t^9-5401*t^8+1945*t^7+141*t^6-410*t^5-13*t^4+23*t^3+26*t^2-6*t+ 1)/(t-1)/(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t ^8+409*t^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 t (1620 t + 2160 t - 59688 t + 66174 t - 39995 t + 41227 t 10 9 8 7 6 5 4 - 18547 t + 10687 t - 3203 t - 219 t - 2867 t - 437 t + 284 t 3 2 / + 118 t - 3 t - 19 t + 8) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation t*(1620*t^16+2160*t^15-59688*t^14+66174*t^13-39995*t^12+41227*t^11-18547*t^10+ 10687*t^9-3203*t^8-219*t^7-2867*t^6-437*t^5+284*t^4+118*t^3-3*t^2-19*t+8)/(t-1) /(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t^8+409*t ^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) This theorem took, 0.459, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 59, :consider the sequence, (x y + x y + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 6, 5, 25, 28, 6, 30, 66, 5, 25, 30, 25, 125, 140, 28, 140, 262, 6, 30, 100, 30, 150, 394, 66, 330, 651, 5, 25, 30, 25, 125, 140, 30, 150, 330, 25, 125 , 150, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 32, 8, 40, 77, 0, 0, 40, 0, 0, 154, 32, 160, 311, 8, 40, 96, 40 , 200, 404, 77, 385, 665, 0, 0, 40, 0, 0, 160, 40, 200, 385, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267472790943687754713045839179456699788223350084558470644928980995853850468750\ 000000000000000000000000000000000, 26747279094368775471304583917945669978822334\ 9911969458421259195445479157343750000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 66, 651, 5973, 52866, 467838, 4153845, 36917445, 328087866, 2915204382, 25901506647, 230135669685, 2044782871338, 18168187221804, 161426859510441, 1434299236695075, 12743938962713700, 113231591516223276, 1006077769355418447, 8939135009787587061, 79425405274544254272, 705705304433212164522, 6270285620201477521311, 55712322858527558034285, 495011408817065983739508, 4398242297026386902181258, 39078968602851831534695949, 347221840890572644884915903, 3085112302134291192548948076, 27411633704758373646510075396, 243556016370680989436453180013, 2164027644220976459349399444525, 19227673841672401095642365956188, 170840442980948006608464455697966, 1517940089802767494889392038275745, 13487099869481098744232321728369329, 119834678661785966012895741566618796, 1064747080465268292233048978076378048, 9460419621593504740170051979636841847, 84057088353341425342543117371967154421] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 77, 665, 5858, 52400, 467621, 4158083, 36934214, 328091786, 2915033849, 25900915739, 230135710868, 2044789537190, 18168207997361, 161426850899369, 1434298978727900, 12743938239783212, 113231592099185237, 1006077779243383547, 8939135034650340932, 79425405243245508710, 705705304057358994197, 6270285619358526038363, 55712322860039974615418, 495011408831240837961224, 4398242297054477624721671, 39078968602783076172667931, 347221840890042088053707834, 3085112302133375023315674224, 27411633704761375636255038749, 243556016370700700773997973617, 2164027644221005514548965147368, 19227673841672273776193659617740, 170840442980947279706099940477185, 1517940089802766608341576229876335, 13487099869481104025506971448981640, 119834678661785992618133734419724052, 1064747080465268317766145146712180803, 9460419621593504524955223210634180325, 84057088353341424376285267435024177166] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (1620 t + 8316 t - 1044 t - 53274 t + 52291 t - 25068 t + 39517 t 10 9 8 7 6 5 4 - 13477 t - 2487 t - 5401 t + 1945 t + 141 t - 410 t - 13 t 3 2 / + 23 t + 26 t - 6 t + 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation (1620*t^17+8316*t^16-1044*t^15-53274*t^14+52291*t^13-25068*t^12+39517*t^11-\ 13477*t^10-2487*t^9-5401*t^8+1945*t^7+141*t^6-410*t^5-13*t^4+23*t^3+26*t^2-6*t+ 1)/(t-1)/(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t ^8+409*t^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 t (1620 t + 2160 t - 59688 t + 66174 t - 39995 t + 41227 t 10 9 8 7 6 5 4 - 18547 t + 10687 t - 3203 t - 219 t - 2867 t - 437 t + 284 t 3 2 / + 118 t - 3 t - 19 t + 8) / ((t - 1) / 8 7 6 5 4 3 2 9 (54 t - 108 t - 123 t + 10 t + 50 t + 28 t + 3 t + t + 1) (342 t 8 7 6 5 4 3 2 - 504 t + 409 t - 349 t + 264 t - 100 t + 29 t - 30 t + 12 t - 1)) and in Maple notation t*(1620*t^16+2160*t^15-59688*t^14+66174*t^13-39995*t^12+41227*t^11-18547*t^10+ 10687*t^9-3203*t^8-219*t^7-2867*t^6-437*t^5+284*t^4+118*t^3-3*t^2-19*t+8)/(t-1) /(54*t^8-108*t^7-123*t^6+10*t^5+50*t^4+28*t^3+3*t^2+t+1)/(342*t^9-504*t^8+409*t ^7-349*t^6+264*t^5-100*t^4+29*t^3-30*t^2+12*t-1) This theorem took, 0.456, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 60, :consider the sequence, (x y + x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 14, 4, 20, 77, 5, 25, 20, 25, 125, 62, 14, 70, 309, 4, 20, 97, 20, 100, 382, 77, 385, 572, 5, 25, 20, 25, 125, 70, 20, 100, 385, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 39, 9, 45, 55, 0, 0, 45, 0, 0, 187, 39, 195, 190, 9, 45, 72, 45 , 225, 260, 55, 275, 682, 0, 0, 45, 0, 0, 195, 45, 225, 275, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [149705283111816902022749552413030261336338518382644366607174502686306120737075\ 8056640625000000000000000000000, 1497052831118169020996123785262971954827181470\ 694272562053092186112769263820648193359375000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 77, 572, 6095, 51503, 489074, 4303691, 39430124, 351999521, 3187156502, 28600703204, 257936265680, 2319001958492, 20885745717140, 187901874798527, 1691527824959939, 15221747449215083, 137007230138151521, 1233007814224183466, 11097393122576030276, 99874905451533009401, 898883224867843165532, 8089902557481179799065, 72809378580796430316710, 655283086643773913875208, 5897554984486051157304938, 53077957369271298701124143, 477701819613893150965109225, 4299315313019049609289974380, 38693843557176451529855216630, 348244561864609442112896023268, 3134201218936842320723142353762, 28207810116084828704069285306003, 253870295627453966870074707273464, 2284832636446257388227133175275373, 20563493857566817405453136664685973, 185071444032752012720185980804713291, 1665642999957928133145955093713824417, 14990786980216637147546113583065840763, 134917082925546749171586461162039209295] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 55, 682, 5515, 54553, 472897, 4389418, 38974327, 354421024, 3174275635, 28669191040, 257571928768, 2320939692619, 20875437648319, 187956704039146, 1691236158567028, 15223298899237321, 136998977244962458, 1233051714185466535, 11097159599636697145, 99876147647625334303, 898876617110295305122, 8089937706715270110775, 72809191607700213963475, 655284081227601660624991, 5897549693893301023152568, 53077985512040593111590940, 477701669911195247715124744, 4299316109347487900246763109, 38693839321186144163398570549, 348244584397535979767212488235, 3134201099075177910263851940419, 28207810753676987256245861953270, 253870292235845916727855784077774, 2284832654487578214669819344547688, 20563493761597807592700520767757159, 185071444543249595544748987480932055, 1665642997242387086690556558066894946, 14990786994661687156323408908041711471, 134917082848707744582845418333241390786] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 21 20 19 18 17 16 - (9072 t - 34884 t + 94851 t - 92004 t - 436641 t + 807473 t 15 14 13 12 11 + 381103 t - 1220119 t + 202509 t + 948007 t - 256030 t 10 9 8 7 6 5 4 - 363108 t + 100243 t + 68583 t - 21040 t - 5850 t + 2515 t + 61 t 3 2 / - 158 t + 22 t + 4 t - 1) / ((9 t - 1) / 8 7 6 5 4 3 2 (171 t - 246 t + 104 t + 103 t - 131 t + 13 t + 27 t - 10 t + 1) ( 11 10 9 8 7 6 5 4 1008 t - 12 t - 1981 t - 2597 t - 2554 t - 1019 t + 564 t + 576 t 3 2 + 112 t - 25 t - 11 t - 1)) and in Maple notation -(9072*t^21-34884*t^20+94851*t^19-92004*t^18-436641*t^17+807473*t^16+381103*t^ 15-1220119*t^14+202509*t^13+948007*t^12-256030*t^11-363108*t^10+100243*t^9+ 68583*t^8-21040*t^7-5850*t^6+2515*t^5+61*t^4-158*t^3+22*t^2+4*t-1)/(9*t-1)/(171 *t^8-246*t^7+104*t^6+103*t^5-131*t^4+13*t^3+27*t^2-10*t+1)/(1008*t^11-12*t^10-\ 1981*t^9-2597*t^8-2554*t^7-1019*t^6+564*t^5+576*t^4+112*t^3-25*t^2-11*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 20 19 18 17 16 15 - t (9072 t - 34884 t - 292977 t + 448995 t + 851340 t - 868390 t 14 13 12 11 10 - 507251 t + 1049613 t - 179830 t - 616508 t + 417408 t 9 8 7 6 5 4 3 + 232439 t - 211817 t - 57833 t + 49003 t + 8327 t - 5880 t - 604 t 2 / + 361 t + 17 t - 9) / ((9 t - 1) / 8 7 6 5 4 3 2 (171 t - 246 t + 104 t + 103 t - 131 t + 13 t + 27 t - 10 t + 1) ( 11 10 9 8 7 6 5 4 1008 t - 12 t - 1981 t - 2597 t - 2554 t - 1019 t + 564 t + 576 t 3 2 + 112 t - 25 t - 11 t - 1)) and in Maple notation -t*(9072*t^20-34884*t^19-292977*t^18+448995*t^17+851340*t^16-868390*t^15-507251 *t^14+1049613*t^13-179830*t^12-616508*t^11+417408*t^10+232439*t^9-211817*t^8-\ 57833*t^7+49003*t^6+8327*t^5-5880*t^4-604*t^3+361*t^2+17*t-9)/(9*t-1)/(171*t^8-\ 246*t^7+104*t^6+103*t^5-131*t^4+13*t^3+27*t^2-10*t+1)/(1008*t^11-12*t^10-1981*t ^9-2597*t^8-2554*t^7-1019*t^6+564*t^5+576*t^4+112*t^3-25*t^2-11*t-1) This theorem took, 0.713, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 61, :consider the sequence, (x y + x y + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 4, 5, 25, 13, 4, 20, 70, 5, 25, 20, 25, 125, 58, 13, 65, 256, 4, 20, 97, 20, 100, 347, 70, 350, 419, 5, 25, 20, 25, 125, 65, 20, 100, 350, 25, 125, 100, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 35, 9, 45, 45, 0, 0, 45, 0, 0, 168, 35, 175, 143, 9, 45, 72, 45 , 225, 234, 45, 225, 526, 0, 0, 45, 0, 0, 175, 45, 225, 225, 0, 0, 225, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [267896524511317242722702136297257202930175362507071549601832518100467000060234\ 56808179616928100585937500000, 267896524511317243420256499444604614847475565661\ 72061890155641714073341721996257547289133071899414062500000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 70, 419, 4089, 29964, 254810, 1994140, 16312963, 130200467, 1053041769, 8454480642, 68148008738, 548106635614, 4413612943480, 35517131370215, 285914797108143, 2301176444858019, 18522931032077429, 149088421286304895, 1200029882993312707, 9659008455747134855, 77745848735926487139, 625776973122396328941, 5036898499829037784091, 40542090017523583991989, 326324320951145176414549, 2626591573435280454033953, 21141498307393145731709844 , 170168400675891567386020059, 1369689433209436619579814392, 11024661783793074775730027692, 88737758592232342740280410043, 714252269520383160124513756598, 5749033062799057816752594533214, 46274100120634991807830744306257, 372461303152855925853960097938263, 2997949648427038991013899409910615, 24130566112606310911001586682555006, 194227485115114895481448982103860894, 1563341523160241100076175055900336708] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 45, 526, 3590, 32063, 245344, 2035011, 16131234, 130995094, 1049527331, 8469919742, 68079882400, 548406442929, 4412291323632, 35522951332834, 285889151640917, 2301289406557217, 18522433347575473, 149090613660887226, 1200020224434273258, 9659051004601910053, 77745661289893631906, 625777798888770010901, 5036894861999550621247, 40542106043524364034663, 326324250350410412032395, 2626591884458387301564568, 21141496937217068921860226 , 170168406712041161477229740, 1369689406617888001670541496, 11024661900939026515155881862, 88737758076159618777682921536, 714252271793881090069601821174, 5749033052783429777004848956949, 46274100164757667296961306425737, 372461302958478659979995467240933, 2997949649283345244641617795843832, 24130566108833954223783048973451097, 194227485131733570422619952438571134, 1563341523087029477893471616820464186] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.440, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 62, :consider the sequence, (x y + x y + x y + x y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 5, 5, 25, 15, 5, 25, 64, 5, 25, 25, 25, 125, 69, 15, 75, 250, 5, 25, 74, 25, 125, 286, 64, 320, 429, 5, 25, 25, 25, 125, 75, 25, 125, 320, 25, 125, 125, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 31, 7, 35, 48, 0, 0, 35, 0, 0, 155, 31, 155, 158, 7, 35, 70, 35 , 175, 248, 48, 240, 503, 0, 0, 35, 0, 0, 155, 35, 175, 240, 0, 0, 175, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [101836551193649006065079686140632373943739732244404333325412600905728000000000\ 00000000000000000000000000000, 101836551193649006065079691975600565036310342269\ 14140499006773316812800000000000000000000000000000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 64, 429, 4006, 30798, 266692, 2153646, 18119446, 148777242, 1239214174, 10234774206, 84953304130, 703073716854, 5828783273998, 48273448966206, 400039073259730, 3313910922564594, 27458142093378550, 227482090002795558, 1884756729471575818, 15615089755609261482, 129373352272110257806, 1071861290620359999630, 8880475772390349306226, 73575222340947702672258, 609576602307635586043894, 5050381598001600261155046, 41842783913517064989833914 , 346670327091033676855209738, 2872188464542741235787436510, 23796281606585450560211998638, 197153877668618906087417346178, 1633433705647574361069557016018, 13533113450615314186001200535782, 112122796321976102809788236672694, 928945264730756183024908675705738, 7696376874186098727853955502247578, 63765024237824341130886783476027758, 528297713206739211064083533462846142, 4376983740854466777597514852721001874] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 48, 503, 3660, 32582, 258246, 2195576, 17915400, 149776664, 1234326804, 10258678844, 84836370444, 703645648796, 5825985709296, 48287132620124, 399972142010436, 3314238304285592, 27456540765929064, 227489922602220572, 1884718417754902956, 15615277150331001536, 129372435665311345440, 1071865774033937259812, 8880453842597538150660, 73575329606495486966168, 609576077637938931929640, 5050384164326913590424380, 41842771360808843314865100 , 346670388490297000128410048, 2872188164219538450840512016, 23796283075561155614025417860, 197153870483394447793375324884, 1633433740792778150649698188616, 13533113278709014448212238308248, 112122797162824147580740025067980, 928945260617902637381240831868444, 7696376894303365329243710324597840, 63765024139424431914560015701903584, 528297713688044267344070354224380308, 4376983738500251625920886974311288548] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 17 16 15 14 13 12 11 (2736 t - 6240 t + 4344 t - 3460 t + 3220 t - 4488 t + 6856 t 10 9 8 7 6 5 4 - 5716 t + 3972 t - 2228 t + 1017 t - 626 t + 385 t - 237 t 3 2 / + 107 t - 32 t + 7 t - 1) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation (2736*t^17-6240*t^16+4344*t^15-3460*t^14+3220*t^13-4488*t^12+6856*t^11-5716*t^ 10+3972*t^9-2228*t^8+1017*t^7-626*t^6+385*t^5-237*t^4+107*t^3-32*t^2+7*t-1)/(t-\ 1)/(48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t ^7-608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 16 15 14 13 12 11 10 - t (3312 t - 8304 t + 6144 t - 4700 t + 5876 t - 6784 t + 8420 t 9 8 7 6 5 4 3 - 7868 t + 7094 t - 6268 t + 4345 t - 2408 t + 1051 t - 388 t 2 / + 123 t - 36 t + 7) / ((t - 1) / 8 7 6 5 4 3 2 9 (48 t - 12 t - 50 t - 6 t + 32 t - 28 t + 12 t - t - 1) (168 t 8 7 6 5 4 3 2 - 516 t + 654 t - 608 t + 396 t - 200 t + 104 t - 41 t + 12 t - 1)) and in Maple notation -t*(3312*t^16-8304*t^15+6144*t^14-4700*t^13+5876*t^12-6784*t^11+8420*t^10-7868* t^9+7094*t^8-6268*t^7+4345*t^6-2408*t^5+1051*t^4-388*t^3+123*t^2-36*t+7)/(t-1)/ (48*t^8-12*t^7-50*t^6-6*t^5+32*t^4-28*t^3+12*t^2-t-1)/(168*t^9-516*t^8+654*t^7-\ 608*t^6+396*t^5-200*t^4+104*t^3-41*t^2+12*t-1) This theorem took, 0.799, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 2 n Theorem number, 63, :consider the sequence, (x y + x y + x y + x + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 5, 3, 5, 25, 8, 3, 15, 55, 5, 25, 15, 25, 125, 33, 8, 40, 185, 3, 15, 73, 15, 75, 246, 55, 275, 295, 5, 25, 15, 25, 125, 40, 15, 75, 275, 25, 125, 75, 125, 625] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 30, 8, 40, 36, 0, 0, 40, 0, 0, 140, 30, 150, 90, 8, 40, 48, 40, 200, 142, 36, 180, 390, 0, 0, 40, 0, 0, 150, 40, 200, 180, 0, 0, 200, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [524569192196997746788106616626123185942745576537979614773144931748694289126433\ 432102203369140625000000, 52456919219700138958117688024303396302753498057758144\ 1643466733014767014537937939167022705078125000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 3, 55, 295, 2673, 16743, 128731, 859309, 6238833, 42802095, 304010935, 2110214689, 14858636997, 103646493867, 727261783723, 5083451774845, 35618882099625, 249182951147607, 1744976448013279, 12211823754516121, 85496729635183533, 598416407769202275, 4189195324948036723, 29323144664212423381, 205267649402496635409, 1436848931416694748303, 10058055844725589746151, 70405895503727705064625, 492843547389970393038549, 3449894862545730230943963, 24149309870287589242579771, 169044968504025294738106669, 1183315701392208340895080185, 8283205874047572754907206215, 57982459661544859718162586991, 405877136438958994551756961609, 2841140328080995352953019847549, 19887980663162536336836380255187, 139215872145600280436663382805891, 974511072159238107645626501834245, 6821577656057012519383321246582113] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 36, 390, 2294, 18638, 121116, 897384, 6085670, 43567910, 300929856, 2125620084, 14796655634, 103956400682, 726014914956, 5089686118680, 35593799037038, 249308366460542, 1744471855962264, 12214346714771196, 85486578835640234, 598467161766918770, 4188991122897499716, 29324165674465108416, 205263541501707795590, 1436869470920638947398, 10057973206726377576720, 70406308693723765911780, 492841884974234265851138, 3449903174624410866881018, 24149276427729612623096316, 169045135716815177835523944, 1183315028633487574023830558, 8283209237841176589263454350, 57982446127762644724053885960, 405877204107870069522300466764, 2841140055824010212567052202202, 19887982024447462038766218481922, 139215866668648990677444431622708, 974511099543994556441721257750160, 6821577545878037993063671492799030] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 58 t - 140 t - 214 t + 144 t + 23 t - 21 t + 7 t - 1 - ---------------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -(58*t^7-140*t^6-214*t^5+144*t^4+23*t^3-21*t^2+7*t-1)/(7*t-1)/(2*t^2+3*t-1)/(58 *t^4-23*t^2+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 2 t (29 t + 70 t + 55 t - 101 t + t + 22 t - 4) - ---------------------------------------------------- 2 4 2 (7 t - 1) (2 t + 3 t - 1) (58 t - 23 t + 1) and in Maple notation -2*t*(29*t^6+70*t^5+55*t^4-101*t^3+t^2+22*t-4)/(7*t-1)/(2*t^2+3*t-1)/(58*t^4-23 *t^2+1) This theorem took, 0.138, seconds. to state and prove ------------------------------------------------------------------ This concludes this webbook, that took, 131.462, seconds. to generate. k is , 6 Counting the Occurrences of Coefficients that Appear in the Expansion of, n P(x, y) , modolu , 3 For all Polynomials that are Sums of, 6, Monomials taken from, 2 2 2 2 2 2 {1, x, y, x , y , x y, x y , x y, x y } By Shalosh B. Ekhad In this webbook, we will consider the sequences described in the title, that\ after normalization and weeding out obvious symmetry, concerns the following set of, 46, polynomials 2 2 2 2 {x + x y + y + x + y + 1, x y + x y + y + x + y + 1, 2 2 2 2 2 x y + x + y + x + y + 1, x y + x + x y + x + y + 1, 2 2 2 2 2 2 x y + x + x y + y + y + 1, x y + x + x y + y + x + 1, 2 2 2 2 2 2 x y + x + x y + y + x + y, x y + x y + y + x + y + 1, 2 2 2 2 2 x y + x y + x y + x + y + 1, x y + x y + x y + y + y + 1, 2 2 2 2 2 2 x y + x y + x y + y + x + 1, x y + x y + x y + y + x + y, 2 2 2 2 2 2 2 2 x y + x y + x + y + y + 1, x y + x y + x + y + x + y, 2 2 2 2 2 2 2 2 x y + x y + x + x y + y + 1, x y + x y + x + x y + y + y, 2 2 2 2 2 2 2 x y + x y + y + x + y + 1, x y + x + y + x + y + 1, 2 2 2 2 2 2 2 2 x y + x + x y + y + y + 1, x y + x + x y + y + x + y, 2 2 2 2 2 2 2 x y + x y + y + x + y + 1, x y + x y + x y + x + y + 1, 2 2 2 2 2 2 2 2 x y + x y + x y + y + y + 1, x y + x y + x y + y + x + 1, 2 2 2 2 2 2 2 2 x y + x y + x y + y + x + y, x y + x y + x + x + y + 1, 2 2 2 2 2 2 2 2 2 2 x y + x y + x + y + y + 1, x y + x y + x + y + x + 1, 2 2 2 2 2 2 2 2 2 x y + x y + x + y + x + y, x y + x y + x + x y + y + 1, 2 2 2 2 2 2 2 2 x y + x y + x + x y + x + 1, x y + x y + x + x y + x + y, 2 2 2 2 2 2 2 2 2 2 x y + x y + x + x y + y + 1, x y + x y + x + x y + y + y, 2 2 2 2 2 2 2 2 2 x y + x y + x + x y + y + x, x y + x y + x y + x + y + 1, 2 2 2 2 2 2 2 2 2 2 x y + x y + x y + y + y + 1, x y + x y + x y + y + x + 1, 2 2 2 2 2 2 2 2 2 x y + x y + x y + y + x + y, x y + x y + x y + x y + y + 1, 2 2 2 2 2 2 2 2 2 x y + x y + x y + x y + x + y, x y + x y + x y + x y + y + 1, 2 2 2 2 2 2 2 2 2 2 2 x y + x y + x y + x y + y + x, x y + x y + x y + x + y + 1, 2 2 2 2 2 2 2 2 2 2 2 2 x y + x y + x y + x + y + y, x y + x y + x y + x + x y + y } by finding enumerative automata with at most, 500, states . ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 1, :consider the sequence, (x + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 27, 6, 36, 42, 6, 36, 36, 36, 216, 153, 27, 162, 144, 6, 36, 72, 36, 216, 186, 42, 252, 264, 6, 36, 36, 36, 216, 162, 36, 216, 252, 36, 216, 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 9, 6, 36, 42, 0, 0, 36, 0, 0, 27, 9, 54, 72, 6, 36, 72, 36, 216 , 186, 42, 252, 264, 0, 0, 36, 0, 0, 54, 36, 216, 252, 0, 0, 216, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [881081690996846456728566157801394786708421517753374882404256386202655813702315\ 18305558709567695618048000, 881081690996846456728566157801394786708421517753374\ 88240425638620265581370231518305558709567695618048000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 42, 264, 1608, 9696, 58272, 349824, 2099328, 12596736, 75581952, 453494784, 2720974848, 16325861376, 97955192832, 587731206144, 3526387335168, 21158324207616, 126949945638912, 761699674619904, 4570198049292288, 27421188298899456, 164527129799688192, 987162778810712064, 5922976672889438208, 35537860037386960896, 213227160224422428672, 1279362961346735898624, 7676177768080818044928, 46057066608485713575936, 276342399650915892068352, 1658054397905498573635584, 9948326387432997884264448, 59689958324598000190488576, 358139749947588026912735232, 2148838499685528213016018944, 12893030998113169381175328768, 77358185988679016493210402816, 464149115932074099371579277312, 2784894695592444597054109384704, 16709368173554667583973923749888] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 264, 1608, 9696, 58272, 349824, 2099328, 12596736, 75581952, 453494784, 2720974848, 16325861376, 97955192832, 587731206144, 3526387335168, 21158324207616, 126949945638912, 761699674619904, 4570198049292288, 27421188298899456, 164527129799688192, 987162778810712064, 5922976672889438208, 35537860037386960896, 213227160224422428672, 1279362961346735898624, 7676177768080818044928, 46057066608485713575936, 276342399650915892068352, 1658054397905498573635584, 9948326387432997884264448, 59689958324598000190488576, 358139749947588026912735232, 2148838499685528213016018944, 12893030998113169381175328768, 77358185988679016493210402816, 464149115932074099371579277312, 2784894695592444597054109384704, 16709368173554667583973923749888] Using the found enumerative automaton with, 8, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 2 t + 1 ------------------- (2 t - 1) (6 t - 1) and in Maple notation (6*t^2-2*t+1)/(2*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 (t - 1) t - ------------------- (2 t - 1) (6 t - 1) and in Maple notation -6*(t-1)*t/(2*t-1)/(6*t-1) This theorem took, 0.084, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 2, :consider the sequence, (x y + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 12, 6, 36, 54, 6, 36, 36, 36, 216, 24, 12, 72, 108, 6, 36, 72, 36, 216, 108, 54, 324, 486, 6, 36, 36, 36, 216, 72, 36, 216, 324, 36, 216, 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 12, 6, 36, 54, 0, 0, 36, 0, 0, 24, 12, 72, 108, 6, 36, 72, 36, 216, 108, 54, 324, 486, 0, 0, 36, 0, 0, 72, 36, 216, 324, 0, 0, 216, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [966306636225087453637048766703947960402700428515963135931050949587815641956743\ 510217308480995328, 96630663622508745363704876670394796040270042851596313593105\ 0949587815641956743510217308480995328] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 54, 486, 4374, 39366, 354294, 3188646, 28697814, 258280326, 2324522934, 20920706406, 188286357654, 1694577218886, 15251194969974, 137260754729766, 1235346792567894, 11118121133111046, 100063090197999414, 900567811781994726, 8105110306037952534, 72945992754341572806, 656513934789074155254, 5908625413101667397286, 53177628717915006575574, 478598658461235059180166, 4307387926151115532621494, 38766491335360039793593446, 348898422018240358142341014, 3140085798164163223281069126, 28260772183477469009529622134, 254346949651297221085766599206, 2289122546861674989771899392854, 20602102921755074907947094535686, 185418926295795674171523850821174, 1668770336662161067543714657390566, 15018933029959449607893431916515094, 135170397269635046471040887248635846, 1216533575426715418239367985237722614, 10948802178840438764154311867139503526, 98539219609563948877388806804255531734] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 54, 486, 4374, 39366, 354294, 3188646, 28697814, 258280326, 2324522934, 20920706406, 188286357654, 1694577218886, 15251194969974, 137260754729766, 1235346792567894, 11118121133111046, 100063090197999414, 900567811781994726, 8105110306037952534, 72945992754341572806, 656513934789074155254, 5908625413101667397286, 53177628717915006575574, 478598658461235059180166, 4307387926151115532621494, 38766491335360039793593446, 348898422018240358142341014, 3140085798164163223281069126, 28260772183477469009529622134, 254346949651297221085766599206, 2289122546861674989771899392854, 20602102921755074907947094535686, 185418926295795674171523850821174, 1668770336662161067543714657390566, 15018933029959449607893431916515094, 135170397269635046471040887248635846, 1216533575426715418239367985237722614, 10948802178840438764154311867139503526, 98539219609563948877388806804255531734] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 9 t - 1 and in Maple notation (3*t-1)/(9*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 t - ------- 9 t - 1 and in Maple notation -6*t/(9*t-1) This theorem took, 0.033, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 3, :consider the sequence, (x y + x + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 28, 6, 36, 77, 6, 36, 36, 36, 216, 158, 28, 168, 278, 6, 36, 117, 36, 216, 392, 77, 462, 675, 6, 36, 36, 36, 216, 168, 36, 216, 462, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 31, 9, 54, 74, 0, 0, 54, 0, 0, 155, 31, 186, 239, 9, 54, 108, 54, 324, 389, 74, 444, 693, 0, 0, 54, 0, 0, 186, 54, 324, 444, 0, 0, 324, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [299613328556701823530503346293847151895738346180461719251586341044783223611642\ 420477973383027920171690927063040000, 29961332855670182353050334629384715189573\ 8346246482098268362802025778019661713375629549029608511802709072936960000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 77, 675, 6008, 51123, 437885, 3728367, 31781705, 270657096, 2305411847, 19634024127, 167218901039, 1424131240374, 12128787405905, 103295842120815, 879728753140760, 7492287071972223, 63808732900927568, 543432711515831550, 4628192874635482940, 39416414154644556363, 335693381308138687670, 2858962395611823954759, 24348606332265308720441, 207367061161355482791072, 1766059932873189011879561, 15040805751380445405161403, 128096353611566235670259540, 1090943934753636280122575421, 9291120591881330820609418535, 79128650980679560753914685515, 673906160630705513821943193431, 5739381471910396393893973079913, 48879950361766715601051600129659, 416290424161906335326880215484042, 3545374247852355649686925912433057, 30194493622187648132933992767879603, 257154077782512045316397790049647290, 2190075466991156096083962694928877438, 18651971582473286269213914227306180150] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 74, 693, 5921, 51279, 436925, 3730485, 31770341, 270683154, 2305273112, 19634338788, 167217144833, 1424134982826, 12128764736933, 103295885970105, 879728454664877, 7492287573941589, 63808728905198318, 543432717060143445, 4628192820335702552, 39416414212277856309, 335693380560116346638, 2858962396144448887455, 24348606321829202115566, 207367061164981153316613, 1766059932725842337550884, 15040805751376776885944916, 128096353609462281691896881, 1090943934752799827143364103, 9291120591850967197707422582, 79128650980656373193044368723, 673906160630262911633673251414, 5739381471909902023915030051311, 48879950361760203482766704832593, 416290424161896878540062986260340, 3545374247852259006814684014888206, 30194493622187477538101216138157414, 257154077782510599763184489971512887, 2190075466991153128092643602187551090, 18651971582473264492742934266050318820] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 33 32 31 30 29 (122688 t + 2102544 t + 617992 t - 12420845 t + 9805007 t 28 27 26 25 + 17375873 t - 18181150 t - 17054084 t - 33480173 t 24 23 22 21 + 155723414 t - 116864259 t - 92129126 t + 164836827 t 20 19 18 17 16 - 18331059 t - 83845136 t + 32768709 t + 24262132 t - 16929582 t 15 14 13 12 11 - 375168 t + 1902016 t - 775410 t + 838228 t - 36292 t 10 9 8 7 6 5 - 385443 t + 83768 t + 91321 t - 38580 t - 5409 t + 5963 t 4 3 2 / 17 16 15 - 1051 t - 111 t + 77 t - 14 t + 1) / ((1008 t - 932 t + 4237 t / 14 13 12 11 10 9 + 1830 t - 18173 t + 20725 t + 487 t - 14044 t + 9005 t 8 7 6 5 4 3 2 - 1360 t - 1135 t + 1079 t - 530 t + 7 t + 95 t - 19 t - 4 t + 1) ( 17 16 15 14 13 12 11 1080 t + 126 t - 1219 t + 1728 t + 183 t + 2221 t - 9585 t 10 9 8 7 6 5 4 + 3806 t + 4425 t - 1606 t - 2715 t + 1531 t + 382 t - 325 t 3 2 - 65 t + 75 t - 16 t + 1)) and in Maple notation (122688*t^33+2102544*t^32+617992*t^31-12420845*t^30+9805007*t^29+17375873*t^28-\ 18181150*t^27-17054084*t^26-33480173*t^25+155723414*t^24-116864259*t^23-\ 92129126*t^22+164836827*t^21-18331059*t^20-83845136*t^19+32768709*t^18+24262132 *t^17-16929582*t^16-375168*t^15+1902016*t^14-775410*t^13+838228*t^12-36292*t^11 -385443*t^10+83768*t^9+91321*t^8-38580*t^7-5409*t^6+5963*t^5-1051*t^4-111*t^3+ 77*t^2-14*t+1)/(1008*t^17-932*t^16+4237*t^15+1830*t^14-18173*t^13+20725*t^12+ 487*t^11-14044*t^10+9005*t^9-1360*t^8-1135*t^7+1079*t^6-530*t^5+7*t^4+95*t^3-19 *t^2-4*t+1)/(1080*t^17+126*t^16-1219*t^15+1728*t^14+183*t^13+2221*t^12-9585*t^ 11+3806*t^10+4425*t^9-1606*t^8-2715*t^7+1531*t^6+382*t^5-325*t^4-65*t^3+75*t^2-\ 16*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 32 31 30 29 28 - t (412992 t + 2033712 t + 2700016 t - 12129143 t + 6729889 t 27 26 25 24 23 + 24953665 t - 45709397 t + 3245682 t + 15033849 t + 64438613 t 22 21 20 19 - 76383606 t - 63073078 t + 116848389 t + 10207855 t 18 17 16 15 14 - 81835973 t + 14465720 t + 35670075 t - 15204248 t - 6610764 t 13 12 11 10 9 + 5594997 t - 1289457 t + 81262 t + 560897 t - 540268 t 8 7 6 5 4 3 2 + 48591 t + 135182 t - 52083 t - 7805 t + 8492 t - 1247 t - 293 t / 17 16 15 14 13 + 106 t - 9) / ((1008 t - 932 t + 4237 t + 1830 t - 18173 t / 12 11 10 9 8 7 6 + 20725 t + 487 t - 14044 t + 9005 t - 1360 t - 1135 t + 1079 t 5 4 3 2 17 16 15 - 530 t + 7 t + 95 t - 19 t - 4 t + 1) (1080 t + 126 t - 1219 t 14 13 12 11 10 9 8 + 1728 t + 183 t + 2221 t - 9585 t + 3806 t + 4425 t - 1606 t 7 6 5 4 3 2 - 2715 t + 1531 t + 382 t - 325 t - 65 t + 75 t - 16 t + 1)) and in Maple notation -t*(412992*t^32+2033712*t^31+2700016*t^30-12129143*t^29+6729889*t^28+24953665*t ^27-45709397*t^26+3245682*t^25+15033849*t^24+64438613*t^23-76383606*t^22-\ 63073078*t^21+116848389*t^20+10207855*t^19-81835973*t^18+14465720*t^17+35670075 *t^16-15204248*t^15-6610764*t^14+5594997*t^13-1289457*t^12+81262*t^11+560897*t^ 10-540268*t^9+48591*t^8+135182*t^7-52083*t^6-7805*t^5+8492*t^4-1247*t^3-293*t^2 +106*t-9)/(1008*t^17-932*t^16+4237*t^15+1830*t^14-18173*t^13+20725*t^12+487*t^ 11-14044*t^10+9005*t^9-1360*t^8-1135*t^7+1079*t^6-530*t^5+7*t^4+95*t^3-19*t^2-4 *t+1)/(1080*t^17+126*t^16-1219*t^15+1728*t^14+183*t^13+2221*t^12-9585*t^11+3806 *t^10+4425*t^9-1606*t^8-2715*t^7+1531*t^6+382*t^5-325*t^4-65*t^3+75*t^2-16*t+1) This theorem took, 0.585, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 4, :consider the sequence, (x y + x + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 31, 7, 42, 64, 6, 36, 42, 36, 216, 174, 31, 186, 241, 7, 42, 98, 42, 252, 368, 64, 384, 543, 6, 36, 42, 36, 216, 186, 42, 252, 384, 36, 216, 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 25, 7, 42, 61, 0, 0, 42, 0, 0, 138, 25, 150, 220, 7, 42, 98, 42 , 252, 362, 61, 366, 555, 0, 0, 42, 0, 0, 150, 42, 252, 366, 0, 0, 252, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [279735559493400782277913806361690352230628998324154502650854005715091248998327\ 02793355970939788984320000000000000, 279735559493400782277913806361690352230628\ 99832415450265085400571509124899832702793355970939788984320000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 64, 543, 4671, 39270, 330420, 2773815, 23285985, 195439821, 1640315118, 13766732514, 115540222512, 969692984508, 8138325714381, 68302362288702, 573239809645785, 4811017556242698, 40377324152220621, 338873902716074970, 2844059731845134271, 23869279062671259360, 200327185962789626034, 1681281672780125288100, 14110456598822827605621, 118424526149899166888862, 993898978074120766320783, 8341474614503945976901599, 70007314907563792947609963 , 587548888783168414005023559, 4931108944345248862814044041, 41385212167380798288338171235, 347332781625756745824371541078, 2915052379191889965478137403620, 24465097517308823923113930478494, 205327698673242011113872448832031, 1723249368314278247570095208031210, 14462677975665397192876765417003314, 121380606935844035801115112614282369, 1018708413815458897284388233510859368, 8549692233182867385837819107982774828] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 61, 555, 4662, 39363, 330372, 2774358, 23285604, 195443238, 1640313600, 13766754654, 115540209798, 969693084597, 8138325497826, 68302362467127, 573239807229756, 4811017554550527, 40377324130160721, 338873902682564370, 2844059731633707138, 23869279062206859654, 200327185960680725988, 1681281672774814106538, 14110456598802858240909, 118424526149846525841027, 993898978073941483900452, 8341474614503457444379107, 70007314907562207359916969 , 587548888783163974030741407, 4931108944345234909949085924, 41385212167380758767709518539, 347332781625756625162392978927, 2915052379191889623344458211697, 24465097517308822901285880728247, 205327698673242008215909708556073, 1723249368314278239035366253197142, 14462677975665397168655304106789866, 121380606935844035730524771496389115, 1018708413815458897084379239507175121, 8549692233182867385260851027182627231] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 3.990, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 5, :consider the sequence, (x y + x + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 29, 6, 36, 76, 6, 36, 36, 36, 216, 158, 29, 174, 285, 6, 36, 117, 36, 216, 412, 76, 456, 637, 6, 36, 36, 36, 216, 174, 36, 216, 456, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 32, 9, 54, 70, 0, 0, 54, 0, 0, 170, 32, 192, 252, 9, 54, 108, 54, 324, 400, 70, 420, 682, 0, 0, 54, 0, 0, 192, 54, 324, 420, 0, 0, 324, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [812700186500290082064287764591979046704967466797565197329519805077453539737539\ 471257710117776052308603711992627200, 81270018650029008206428776459197904670496\ 7468883797798707420054392833583935463611049961814844475344751808007372800] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 76, 637, 5818, 49828, 432421, 3720358, 32076625, 276218320, 2379381748, 20492498077, 176501517919, 1520161153012, 13092851232571, 112765673914825, 971225572762222, 8364943217572018, 72045350412047827, 620510122975955122, 5344311875503772908, 46029335963126912752, 396440145223143329365, 3414448307985299847079, 29407862431014726238672, 253283193755816887525030, 2181470223923728190494315, 18788504151964148248479706, 161821089464810376032845504, 1393728036221068268462646841, 12003860840226898522560583972, 103386508218576032813383835371, 890444351521448550628600794949, 7669193561309457420376930456258, 66053010253064298514255150860745, 568899471451885197053549659165165, 4899801044316797202490777787237692, 42200865844745664089299914011951497, 363466406480294133323095469217355426, 3130453036810063657817307637696704568, 26961876093505367179344835176052082245] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 70, 682, 5725, 50203, 431128, 3723817, 32060392, 276253753, 2379189907, 20492881981, 176499320434, 1520165469847, 13092826532797, 112765723861030, 971225299033783, 8364943806756745, 72045347409829444, 620510129994039925, 5344311842783450986, 46029336046863561232, 396440144867371548076, 3414448308980631752065, 29407862427141868402432, 253283193767566451304412, 2181470223881399892403969, 18788504152101701968261564, 161821089464344953789633022, 1393728036222664924884819325, 12003860840221744212089379277, 103386508218594416976111484804, 890444351521391044905411256822, 7669193561309667586851060599065, 66053010253063652484402366033673, 568899471451887585249169550331748, 4899801044316789902148509982786481, 42200865844745691098407632430954258, 363466406480294050452647967439779421, 3130453036810063962211184464352661655, 26961876093505366235709745899695033539] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 33 32 31 30 29 - (946161 t - 8152704 t + 39790476 t - 12187101 t - 85244409 t 28 27 26 25 24 + 91806720 t + 33954279 t - 86786249 t + 63910540 t - 2599941 t 23 22 21 20 19 - 40634567 t + 43512646 t - 35716423 t - 26368695 t + 49861891 t 18 17 16 15 14 + 3187633 t - 27939949 t + 7981717 t + 7152140 t - 5952856 t 13 12 11 10 9 8 + 452136 t + 1513311 t - 609999 t - 77139 t + 60603 t + 113 t 7 6 5 4 3 2 + 12019 t - 12116 t + 2952 t + 571 t - 511 t + 136 t - 18 t + 1) / 17 16 15 14 13 12 / ((459 t - 1689 t + 4936 t - 3369 t - 5367 t + 8678 t / 11 10 9 8 7 6 5 - 2730 t - 3996 t + 3494 t + 1144 t - 1742 t - 248 t + 598 t 4 3 2 17 16 15 - 50 t - 87 t + 17 t + 4 t - 1) (9585 t - 2655 t - 11562 t 14 13 12 11 10 9 + 7065 t + 8961 t + 11766 t + 8958 t - 7466 t - 3968 t 8 7 6 5 4 3 2 + 2522 t - 54 t - 1454 t + 922 t + 268 t - 411 t + 141 t - 20 t + 1 )) and in Maple notation -(946161*t^33-8152704*t^32+39790476*t^31-12187101*t^30-85244409*t^29+91806720*t ^28+33954279*t^27-86786249*t^26+63910540*t^25-2599941*t^24-40634567*t^23+ 43512646*t^22-35716423*t^21-26368695*t^20+49861891*t^19+3187633*t^18-27939949*t ^17+7981717*t^16+7152140*t^15-5952856*t^14+452136*t^13+1513311*t^12-609999*t^11 -77139*t^10+60603*t^9+113*t^8+12019*t^7-12116*t^6+2952*t^5+571*t^4-511*t^3+136* t^2-18*t+1)/(459*t^17-1689*t^16+4936*t^15-3369*t^14-5367*t^13+8678*t^12-2730*t^ 11-3996*t^10+3494*t^9+1144*t^8-1742*t^7-248*t^6+598*t^5-50*t^4-87*t^3+17*t^2+4* t-1)/(9585*t^17-2655*t^16-11562*t^15+7065*t^14+8961*t^13+11766*t^12+8958*t^11-\ 7466*t^10-3968*t^9+2522*t^8-54*t^7-1454*t^6+922*t^5+268*t^4-411*t^3+141*t^2-20* t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 32 31 30 29 28 - t (1003671 t + 2509056 t - 25031061 t + 17310141 t + 41754051 t 27 26 25 24 23 - 59920872 t - 13952226 t + 29784109 t - 32141563 t + 23652700 t 22 21 20 19 18 + 8119383 t - 30056505 t + 42076643 t + 14079002 t - 45055040 t 17 16 15 14 13 - 956559 t + 23329735 t - 5317102 t - 6484386 t + 4641306 t 12 11 10 9 8 7 - 56193 t - 1353762 t + 343053 t + 159216 t + 33588 t - 71059 t 6 5 4 3 2 / - 12846 t + 25724 t - 6004 t - 1295 t + 838 t - 146 t + 9) / (( / 17 16 15 14 13 12 11 459 t - 1689 t + 4936 t - 3369 t - 5367 t + 8678 t - 2730 t 10 9 8 7 6 5 4 3 - 3996 t + 3494 t + 1144 t - 1742 t - 248 t + 598 t - 50 t - 87 t 2 17 16 15 14 13 + 17 t + 4 t - 1) (9585 t - 2655 t - 11562 t + 7065 t + 8961 t 12 11 10 9 8 7 6 + 11766 t + 8958 t - 7466 t - 3968 t + 2522 t - 54 t - 1454 t 5 4 3 2 + 922 t + 268 t - 411 t + 141 t - 20 t + 1)) and in Maple notation -t*(1003671*t^32+2509056*t^31-25031061*t^30+17310141*t^29+41754051*t^28-\ 59920872*t^27-13952226*t^26+29784109*t^25-32141563*t^24+23652700*t^23+8119383*t ^22-30056505*t^21+42076643*t^20+14079002*t^19-45055040*t^18-956559*t^17+ 23329735*t^16-5317102*t^15-6484386*t^14+4641306*t^13-56193*t^12-1353762*t^11+ 343053*t^10+159216*t^9+33588*t^8-71059*t^7-12846*t^6+25724*t^5-6004*t^4-1295*t^ 3+838*t^2-146*t+9)/(459*t^17-1689*t^16+4936*t^15-3369*t^14-5367*t^13+8678*t^12-\ 2730*t^11-3996*t^10+3494*t^9+1144*t^8-1742*t^7-248*t^6+598*t^5-50*t^4-87*t^3+17 *t^2+4*t-1)/(9585*t^17-2655*t^16-11562*t^15+7065*t^14+8961*t^13+11766*t^12+8958 *t^11-7466*t^10-3968*t^9+2522*t^8-54*t^7-1454*t^6+922*t^5+268*t^4-411*t^3+141*t ^2-20*t+1) This theorem took, 0.595, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 6, :consider the sequence, (x y + x + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 5, 6, 36, 25, 5, 30, 82, 6, 36, 30, 36, 216, 143, 25, 150, 323, 5, 30, 146, 30, 180, 507, 82, 492, 724, 6, 36, 30, 36, 216, 150, 30, 180, 492, 36, 216 , 180, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 11, 0, 0, 43, 11, 66, 76, 0, 0, 66, 0, 0, 227, 43, 258, 290, 11, 66, 110 , 66, 396, 441, 76, 456, 736, 0, 0, 66, 0, 0, 258, 66, 396, 456, 0, 0, 396, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [503086733270672346209323293985685434235625055532136716009910498657345821060646\ 15508035631146245067660855591441530880, 503086733270672346209323293985690139937\ 76069663093003014872006434176589255583108619524517775106090115687989410529280] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 82, 724, 6534, 57098, 501416, 4375167, 38212114, 333385033, 2909086452, 25379304377, 221418974480, 1931675361528, 16852151839507, 147019096024795, 1282603597149657, 11189499073077515, 97617771493054865, 851622330339807585, 7429596034714861882, 64816166917535579962, 565459479576591520188, 4933096726210082546729, 43036582109025219820817, 375453290318868494817561, 3275473244237423718808546, 28575392060617970595614698, 249293146528610898110494401, 2174845852396421058987505658, 18973463761663239335977735829, 165525444810669031600099342923, 1444052241801668575917296042881, 12597983829222938082712601602945, 109905439683642301901625681181947, 958820541111698714038582565042585, 8364798254790607904545502454123080, 72974917456627524779443798832385399, 636636822023975655528879162700754482, 5554051409481583763661702287021414365, 48453821695539060226632889120210778448] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 11, 76, 736, 6468, 57338, 500429, 4378467, 38200405, 333432838, 2908916019, 25379969978, 221416687664, 1931684236644, 16852120470121, 147019217593036, 1282603164458514, 11189500729727546, 97617765557495858, 851622352934064306, 7429595953253171707, 64816167225871255798, 565459478459741760339, 4933096730419338240263, 43036582093715821023701, 375453290376365309852613, 3275473244027642030717713, 28575392061403538416713424, 249293146525737261870116952, 2174845852407156935726328554, 18973463761623884248838947628, 165525444810815783464137451845, 1444052241801129702922515738964, 12597983829224944414948944700054, 109905439683634924460044834178262, 958820541111726147589574946406427, 8364798254790506916301452041092184, 72974917456627899933088587787488234, 636636822023974273261917108673164388, 5554051409481588894349204600255753504, 48453821695539041308528422349131698946] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 7.041, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 7, :consider the sequence, (x y + x + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 8, 6, 36, 33, 8, 48, 73, 6, 36, 48, 36, 216, 186, 33, 198, 300, 8, 48, 128, 48, 288, 478, 73, 438, 659, 6, 36, 48, 36, 216, 198, 48, 288, 438, 36, 216 , 288, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 33, 8, 48, 70, 0, 0, 48, 0, 0, 186, 33, 198, 276, 8, 48, 128, 48, 288, 478, 70, 420, 671, 0, 0, 48, 0, 0, 198, 48, 288, 420, 0, 0, 288, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [282546496429954267733407208961192302243875569498352220724859563129772361065402\ 73485577767126453613384842541531136000, 282546496429954267733407208961192302243\ 87556949835222072485956312977236106540273485577767126453613384842541531136000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 73, 659, 5977, 53882, 485275, 4368551, 39319189, 353883314, 3184963543, 28664770679, 257983013137, 2321848020722, 20896632549835, 188069701137311, 1692627311185669, 15233645874760874, 137102812865138263, 1233925316455595999, 11105327847917178097, 99947950637296115162, 899531555733220041595, 8095784001653474550071, 72862056014853693550549, 655758504134174532853634, 5901826537207283601308983, 53116438834869979939038119, 478047949513826961761351857, 4302431545624482545010967202, 38721883910620315275498652555, 348496955195583196772085206831, 3136472596760248508910469917829, 28228253370842239815860106689594, 254054280337580155890796522371703, 2286488523038221432152395095001039, 20578396707343992866648872113550417, 185205570366095936062156547055936842, 1666850133294863424350318542999134715, 15001651199653770821514414902988467591, 135014860796883937391715817358290271509] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 70, 671, 5974, 53984, 485308, 4369319, 39319582, 353888852, 3184966780, 28664809931, 257983037254, 2321848296944, 20896632723028, 188069703075239, 1692627312411142, 15233645888339492, 137102812873755940, 1233925316550685691, 11105327847977619934, 99947950637961861104, 899531555733643488748, 8095784001658135125959, 72862056014856658743502, 655758504134207157947732, 5901826537207304360848300, 53116438834870208317885451, 478047949513827107087693014, 4302431545624484143672464464, 38721883910620316292811738468, 348496955195583207962744385479, 3136472596760248516031747612662, 28228253370842239894194807033572, 254054280337580155940645724515860, 2286488523038221432700738255689211, 20578396707343992866997817303400494, 185205570366095936065994949955595024, 1666850133294863424352761161652608188, 15001651199653770821541283725610597799, 135014860796883937391732915695838154622] Using the found enumerative automaton with, 28, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (1458 t - 432 t - 2844 t + 1647 t + 1362 t - 591 t - 393 t + 112 t 3 2 / + 60 t - 17 t - 3 t + 1) / ((t - 1) (3 t - 1) (9 t - 1) (2 t - 1) / 2 2 (3 t + 1) (t + 1) (3 t - 1) (7 t - 1)) and in Maple notation (1458*t^11-432*t^10-2844*t^9+1647*t^8+1362*t^7-591*t^6-393*t^5+112*t^4+60*t^3-\ 17*t^2-3*t+1)/(t-1)/(3*t-1)/(9*t-1)/(2*t-1)/(3*t+1)/(t+1)/(3*t^2-1)/(7*t^2-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 10 9 8 7 6 5 4 3 - t (1458 t - 2322 t - 1584 t + 1413 t + 867 t + 384 t - 488 t - 213 t 2 / + 115 t + 18 t - 8) / ((t - 1) (3 t - 1) (9 t - 1) (2 t - 1) (3 t + 1) / 2 2 (t + 1) (3 t - 1) (7 t - 1)) and in Maple notation -t*(1458*t^10-2322*t^9-1584*t^8+1413*t^7+867*t^6+384*t^5-488*t^4-213*t^3+115*t^ 2+18*t-8)/(t-1)/(3*t-1)/(9*t-1)/(2*t-1)/(3*t+1)/(t+1)/(3*t^2-1)/(7*t^2-1) This theorem took, 0.806, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 8, :consider the sequence, (x y + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 34, 7, 42, 77, 6, 36, 42, 36, 216, 188, 34, 204, 254, 7, 42, 149, 42, 252, 482, 77, 462, 658, 6, 36, 42, 36, 216, 204, 42, 252, 462, 36, 216 , 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 34, 10, 60, 77, 0, 0, 60, 0, 0, 182, 34, 204, 278, 10, 60, 140 , 60, 360, 479, 77, 462, 616, 0, 0, 60, 0, 0, 204, 60, 360, 462, 0, 0, 360, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [229781516295022559458128548631867534118914955817804486710471405971837227876448\ 60087653467753218993940040553332736000, 229781516295022559458128548631867534118\ 91495581780448671047140597183722787644860087653467753218993940040553332736000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 77, 658, 5453, 45577, 383042, 3221155, 27055826, 227216059, 1908180023, 16024507120, 134567698754, 1130038007608, 9489505568165, 79688082972061, 669179749589021, 5619426749810446, 47189041619211893, 396269149112027083, 3327663091978623812, 27943990770473022151, 234659156243507250128, 1970546005494631142419, 16547624285577279039755, 138958374228701036732140, 1166900421932692988161220, 9799025082320178502342846, 82287135004282145960537189, 691004720375385955250467555, 5802699578105778252044016869, 48728064223920716891427911914, 409193033525454114007153634849, 3436191060577574803939338323809, 28855351966909247449613030297858, 242312293599288378604299065189575, 2034813080660802541733194209593114, 17087305855290546797668698292345039, 143490340300652094053767061372543627, 1204957524256039523416056980422676812, 10118608905791517807017831176908672890] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 77, 616, 5354, 45529, 383015, 3219214, 27048365, 227203681, 1908160439, 16024394875, 134567198066, 1130036740981, 9489502927574, 79688074280584, 669179714989853, 5619426643986976, 47189041353138638, 396269148353501695, 3327663089370753905, 27943990762067497834, 234659156219802665027, 1970546005427624341369, 16547624285367017909411, 138958374228030030246139, 1166900421930690703058972, 9799025082314378840824303, 82287135004264632564771782, 691004720375331406017812674, 5802699578105612181936634217, 48728064223920224802143886916, 409193033525452641977255098346, 3436191060577570302807817348849, 28855351966909233712933569859043, 242312293599288337344369848717110, 2034813080660802418113077976626309, 17087305855290546423237020752907281, 143490340300652092914778423888316519, 1204957524256039519974641037087677719, 10118608905791517796667976326828715542] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.395, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 n Theorem number, 9, :consider the sequence, (x y + x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 8, 6, 36, 33, 8, 48, 73, 6, 36, 48, 36, 216, 186, 33, 198, 300, 8, 48, 128, 48, 288, 478, 73, 438, 659, 6, 36, 48, 36, 216, 198, 48, 288, 438, 36, 216 , 288, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 33, 8, 48, 70, 0, 0, 48, 0, 0, 186, 33, 198, 276, 8, 48, 128, 48, 288, 478, 70, 420, 671, 0, 0, 48, 0, 0, 198, 48, 288, 420, 0, 0, 288, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [282546496429954267733407208961192302243875569498352220724859563129772361065402\ 73485577767126453613384842541531136000, 282546496429954267733407208961192302243\ 87556949835222072485956312977236106540273485577767126453613384842541531136000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 73, 659, 5977, 53882, 485275, 4368551, 39319189, 353883314, 3184963543, 28664770679, 257983013137, 2321848020722, 20896632549835, 188069701137311, 1692627311185669, 15233645874760874, 137102812865138263, 1233925316455595999, 11105327847917178097, 99947950637296115162, 899531555733220041595, 8095784001653474550071, 72862056014853693550549, 655758504134174532853634, 5901826537207283601308983, 53116438834869979939038119, 478047949513826961761351857, 4302431545624482545010967202, 38721883910620315275498652555, 348496955195583196772085206831, 3136472596760248508910469917829, 28228253370842239815860106689594, 254054280337580155890796522371703, 2286488523038221432152395095001039, 20578396707343992866648872113550417, 185205570366095936062156547055936842, 1666850133294863424350318542999134715, 15001651199653770821514414902988467591, 135014860796883937391715817358290271509] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 70, 671, 5974, 53984, 485308, 4369319, 39319582, 353888852, 3184966780, 28664809931, 257983037254, 2321848296944, 20896632723028, 188069703075239, 1692627312411142, 15233645888339492, 137102812873755940, 1233925316550685691, 11105327847977619934, 99947950637961861104, 899531555733643488748, 8095784001658135125959, 72862056014856658743502, 655758504134207157947732, 5901826537207304360848300, 53116438834870208317885451, 478047949513827107087693014, 4302431545624484143672464464, 38721883910620316292811738468, 348496955195583207962744385479, 3136472596760248516031747612662, 28228253370842239894194807033572, 254054280337580155940645724515860, 2286488523038221432700738255689211, 20578396707343992866997817303400494, 185205570366095936065994949955595024, 1666850133294863424352761161652608188, 15001651199653770821541283725610597799, 135014860796883937391732915695838154622] Using the found enumerative automaton with, 28, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (1458 t - 432 t - 2844 t + 1647 t + 1362 t - 591 t - 393 t + 112 t 3 2 / + 60 t - 17 t - 3 t + 1) / ((t - 1) (2 t - 1) (t + 1) (3 t - 1) / 2 2 (3 t + 1) (9 t - 1) (7 t - 1) (3 t - 1)) and in Maple notation (1458*t^11-432*t^10-2844*t^9+1647*t^8+1362*t^7-591*t^6-393*t^5+112*t^4+60*t^3-\ 17*t^2-3*t+1)/(t-1)/(2*t-1)/(t+1)/(3*t-1)/(3*t+1)/(9*t-1)/(7*t^2-1)/(3*t^2-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 10 9 8 7 6 5 4 3 - t (1458 t - 2322 t - 1584 t + 1413 t + 867 t + 384 t - 488 t - 213 t 2 / + 115 t + 18 t - 8) / ((t - 1) (2 t - 1) (t + 1) (3 t - 1) (3 t + 1) / 2 2 (9 t - 1) (7 t - 1) (3 t - 1)) and in Maple notation -t*(1458*t^10-2322*t^9-1584*t^8+1413*t^7+867*t^6+384*t^5-488*t^4-213*t^3+115*t^ 2+18*t-8)/(t-1)/(2*t-1)/(t+1)/(3*t-1)/(3*t+1)/(9*t-1)/(7*t^2-1)/(3*t^2-1) This theorem took, 0.426, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 10, :consider the sequence, (x y + x y + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 31, 7, 42, 64, 6, 36, 42, 36, 216, 174, 31, 186, 241, 7, 42, 98, 42, 252, 368, 64, 384, 543, 6, 36, 42, 36, 216, 186, 42, 252, 384, 36, 216, 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 25, 7, 42, 61, 0, 0, 42, 0, 0, 138, 25, 150, 220, 7, 42, 98, 42 , 252, 362, 61, 366, 555, 0, 0, 42, 0, 0, 150, 42, 252, 366, 0, 0, 252, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [279735559493400782277913806361690352230628998324154502650854005715091248998327\ 02793355970939788984320000000000000, 279735559493400782277913806361690352230628\ 99832415450265085400571509124899832702793355970939788984320000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 64, 543, 4671, 39270, 330420, 2773815, 23285985, 195439821, 1640315118, 13766732514, 115540222512, 969692984508, 8138325714381, 68302362288702, 573239809645785, 4811017556242698, 40377324152220621, 338873902716074970, 2844059731845134271, 23869279062671259360, 200327185962789626034, 1681281672780125288100, 14110456598822827605621, 118424526149899166888862, 993898978074120766320783, 8341474614503945976901599, 70007314907563792947609963 , 587548888783168414005023559, 4931108944345248862814044041, 41385212167380798288338171235, 347332781625756745824371541078, 2915052379191889965478137403620, 24465097517308823923113930478494, 205327698673242011113872448832031, 1723249368314278247570095208031210, 14462677975665397192876765417003314, 121380606935844035801115112614282369, 1018708413815458897284388233510859368, 8549692233182867385837819107982774828] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 61, 555, 4662, 39363, 330372, 2774358, 23285604, 195443238, 1640313600, 13766754654, 115540209798, 969693084597, 8138325497826, 68302362467127, 573239807229756, 4811017554550527, 40377324130160721, 338873902682564370, 2844059731633707138, 23869279062206859654, 200327185960680725988, 1681281672774814106538, 14110456598802858240909, 118424526149846525841027, 993898978073941483900452, 8341474614503457444379107, 70007314907562207359916969 , 587548888783163974030741407, 4931108944345234909949085924, 41385212167380758767709518539, 347332781625756625162392978927, 2915052379191889623344458211697, 24465097517308822901285880728247, 205327698673242008215909708556073, 1723249368314278239035366253197142, 14462677975665397168655304106789866, 121380606935844035730524771496389115, 1018708413815458897084379239507175121, 8549692233182867385260851027182627231] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.138, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 11, :consider the sequence, (x y + x y + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 8, 6, 36, 37, 8, 48, 78, 6, 36, 48, 36, 216, 204, 37, 222, 306, 8, 48, 128, 48, 288, 522, 78, 468, 666, 6, 36, 48, 36, 216, 222, 48, 288, 468, 36, 216 , 288, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 37, 8, 48, 78, 0, 0, 48, 0, 0, 204, 37, 222, 306, 8, 48, 128, 48, 288, 534, 78, 468, 666, 0, 0, 48, 0, 0, 222, 48, 288, 468, 0, 0, 288, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [225543730303375062291505574400322835060982140646906869238277291440957520920349\ 742515280223237650424187197126082560000, 22554373030337506229150557440032283506\ 0982140646906869238277291440957520920349742515280223237650424187197126082560000 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 78, 666, 5778, 50706, 448578, 3990546, 35634978, 319035186, 2861238978, 25690684626, 230853364578, 2075503498866, 18666470795778, 167919872997906, 1510808671996578, 13594456938061746, 122333185783110978, 1100897112091330386, 9907464649081962978, 89163525683297603826, 802449794199038056578, 7221916526087500242066, 64996459004564448575778, 584963392659741715565106, 5264642103649645510384578, 47381608351118630015252946, 426433451669698592668026978, 3837894924085072869196745586, 34541017471112369033877726978, 310868936166091600571541643026, 2797819099051306080743727388578, 25180363932800644780292662105266, 226623227643239143344228652595778, 2039608762277352332027626035252306, 18356477141425371239826043288612578, 165208283958403542667898843425564146, 1486874493739083093067876313798382978, 13381870072332455091951607161995280786, 120436828423076339353608786484816530978] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 78, 666, 5778, 50706, 448578, 3990546, 35634978, 319035186, 2861238978, 25690684626, 230853364578, 2075503498866, 18666470795778, 167919872997906, 1510808671996578, 13594456938061746, 122333185783110978, 1100897112091330386, 9907464649081962978, 89163525683297603826, 802449794199038056578, 7221916526087500242066, 64996459004564448575778, 584963392659741715565106, 5264642103649645510384578, 47381608351118630015252946, 426433451669698592668026978, 3837894924085072869196745586, 34541017471112369033877726978, 310868936166091600571541643026, 2797819099051306080743727388578, 25180363932800644780292662105266, 226623227643239143344228652595778, 2039608762277352332027626035252306, 18356477141425371239826043288612578, 165208283958403542667898843425564146, 1486874493739083093067876313798382978, 13381870072332455091951607161995280786, 120436828423076339353608786484816530978] Using the found enumerative automaton with, 28, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 2 72 t - 12 t + 7 t - 1 - ----------------------- (6 t - 1) (9 t - 1) and in Maple notation -(72*t^3-12*t^2+7*t-1)/(6*t-1)/(9*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 2 (36 t + 21 t - 4) t - ---------------------- (6 t - 1) (9 t - 1) and in Maple notation -2*(36*t^2+21*t-4)*t/(6*t-1)/(9*t-1) This theorem took, 0.308, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 12, :consider the sequence, (x y + x y + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 8, 6, 36, 33, 8, 48, 73, 6, 36, 48, 36, 216, 186, 33, 198, 300, 8, 48, 128, 48, 288, 478, 73, 438, 659, 6, 36, 48, 36, 216, 198, 48, 288, 438, 36, 216 , 288, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 33, 8, 48, 70, 0, 0, 48, 0, 0, 186, 33, 198, 276, 8, 48, 128, 48, 288, 478, 70, 420, 671, 0, 0, 48, 0, 0, 198, 48, 288, 420, 0, 0, 288, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [282546496429954267733407208961192302243875569498352220724859563129772361065402\ 73485577767126453613384842541531136000, 282546496429954267733407208961192302243\ 87556949835222072485956312977236106540273485577767126453613384842541531136000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 73, 659, 5977, 53882, 485275, 4368551, 39319189, 353883314, 3184963543, 28664770679, 257983013137, 2321848020722, 20896632549835, 188069701137311, 1692627311185669, 15233645874760874, 137102812865138263, 1233925316455595999, 11105327847917178097, 99947950637296115162, 899531555733220041595, 8095784001653474550071, 72862056014853693550549, 655758504134174532853634, 5901826537207283601308983, 53116438834869979939038119, 478047949513826961761351857, 4302431545624482545010967202, 38721883910620315275498652555, 348496955195583196772085206831, 3136472596760248508910469917829, 28228253370842239815860106689594, 254054280337580155890796522371703, 2286488523038221432152395095001039, 20578396707343992866648872113550417, 185205570366095936062156547055936842, 1666850133294863424350318542999134715, 15001651199653770821514414902988467591, 135014860796883937391715817358290271509] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 70, 671, 5974, 53984, 485308, 4369319, 39319582, 353888852, 3184966780, 28664809931, 257983037254, 2321848296944, 20896632723028, 188069703075239, 1692627312411142, 15233645888339492, 137102812873755940, 1233925316550685691, 11105327847977619934, 99947950637961861104, 899531555733643488748, 8095784001658135125959, 72862056014856658743502, 655758504134207157947732, 5901826537207304360848300, 53116438834870208317885451, 478047949513827107087693014, 4302431545624484143672464464, 38721883910620316292811738468, 348496955195583207962744385479, 3136472596760248516031747612662, 28228253370842239894194807033572, 254054280337580155940645724515860, 2286488523038221432700738255689211, 20578396707343992866997817303400494, 185205570366095936065994949955595024, 1666850133294863424352761161652608188, 15001651199653770821541283725610597799, 135014860796883937391732915695838154622] Using the found enumerative automaton with, 28, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (1458 t - 432 t - 2844 t + 1647 t + 1362 t - 591 t - 393 t + 112 t 3 2 / + 60 t - 17 t - 3 t + 1) / ((9 t - 1) (3 t + 1) (t + 1) (t - 1) / 2 2 (3 t - 1) (2 t - 1) (3 t - 1) (7 t - 1)) and in Maple notation (1458*t^11-432*t^10-2844*t^9+1647*t^8+1362*t^7-591*t^6-393*t^5+112*t^4+60*t^3-\ 17*t^2-3*t+1)/(9*t-1)/(3*t+1)/(t+1)/(t-1)/(3*t-1)/(2*t-1)/(3*t^2-1)/(7*t^2-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 10 9 8 7 6 5 4 3 - t (1458 t - 2322 t - 1584 t + 1413 t + 867 t + 384 t - 488 t - 213 t 2 / + 115 t + 18 t - 8) / ((9 t - 1) (3 t + 1) (t + 1) (t - 1) (3 t - 1) / 2 2 (2 t - 1) (3 t - 1) (7 t - 1)) and in Maple notation -t*(1458*t^10-2322*t^9-1584*t^8+1413*t^7+867*t^6+384*t^5-488*t^4-213*t^3+115*t^ 2+18*t-8)/(9*t-1)/(3*t+1)/(t+1)/(t-1)/(3*t-1)/(2*t-1)/(3*t^2-1)/(7*t^2-1) This theorem took, 0.834, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 13, :consider the sequence, (x y + x y + x + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 32, 7, 42, 86, 6, 36, 42, 36, 216, 178, 32, 192, 351, 7, 42, 149, 42, 252, 560, 86, 516, 824, 6, 36, 42, 36, 216, 192, 42, 252, 516, 36, 216 , 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 44, 10, 60, 86, 0, 0, 60, 0, 0, 235, 44, 264, 330, 10, 60, 140 , 60, 360, 545, 86, 516, 836, 0, 0, 60, 0, 0, 264, 60, 360, 516, 0, 0, 360, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [161882451752427645777103597992252045002160385605011820035879989175970403922774\ 2029729454523064232894701383681310720000, 1618824517524276457771035979922520450\ 0216038560501182003587998917597040392277420297294545230642328947013836813107200\ 00] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 86, 824, 7510, 67807, 603353, 5358482, 47524774, 421270888, 3733269764, 33080131676, 293101909486, 2596916891080, 23008674385319, 203855416752713, 1806140335046164, 16002212793116926, 141777804813824537, 1256134903974001094, 11129207356568800075, 98603457171311361544, 873614891958444024932, 7740123745471728248909, 68576572330468198436350, 607580239852574707970251, 5383088335752972843706541, 47693519439367862246442743, 422558883043642601574711097, 3743821207900250925650001394, 33169808506693792419079754936, 293880539463710306860247206286, 2603746459749225516426139296676, 23068882474484522326144943834620, 204387541889692203035453489024048, 1810849195903926431344483323451967, 16043907470934275583948774031707445, 142147102871673593492123144612000533, 1259406344208142235719905770541008768, 11158189704805234281446781881565969392, 98860227329296622258406325288539781603] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 86, 836, 7576, 67729, 603368, 5358596, 47523922, 421270345, 3733279121, 33080109917, 293101886872, 2596917269227, 23008674822740, 203855413532165, 1806140347207120, 16002212818030630, 141777804726360233, 1256134904102715470, 11129207357856841360, 98603457169267953877, 873614891958428544338, 7740123745512976547030, 68576572330459892468779, 607580239852427950932973, 5383088335754231466726302, 47693519439369052448926529, 422558883043636545290840443, 3743821207900279395738687547, 33169808506693880663418542069, 293880539463710119793893536695, 2603746459749226018713436042060, 23068882474484526025524120678775, 204387541889692200031184912229410, 1810849195903926433387493152938536, 16043907470934275712580587509441668, 142147102871673593522200278929860159, 1259406344208142235493464681737275074, 11158189704805234285128163254169418825, 98860227329296622263960413976059331324] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.658, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 14, :consider the sequence, (x y + x y + x + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 36, 6, 36, 72, 6, 36, 36, 36, 216, 216, 36, 216, 288, 6, 36, 180, 36, 216, 486, 72, 432, 594, 6, 36, 36, 36, 216, 216, 36, 216, 432, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 12, 0, 0, 36, 12, 72, 72, 0, 0, 72, 0, 0, 180, 36, 216, 288, 12, 72, 144 , 72, 432, 594, 72, 432, 594, 0, 0, 72, 0, 0, 216, 72, 432, 432, 0, 0, 432, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [240547113170926548891412063901604051286575230271157810934411248487020110838051\ 861868500776290154664395770796441600000, 24054711317092654889141206390160405128\ 6575230271157810934411248487020110838051861868500776290154664395770796441600000 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 72, 594, 5022, 43254, 377622, 3328614, 29537622, 263319174, 2354756022, 21102104934, 189374748822, 1701107565894, 15290377052022, 137495847222054, 1236757347521622, 11126584462833414, 100113870176333622, 900872491651999974, 8106938385257984022, 72956961229661761734, 656579745640995288822, 5909020278213194198694, 53179997908584167384022, 478612873605250024030854, 4307473217015205321725622, 38767003080544578528218214, 348901492489347590550089622, 3140104220990806617727560774, 28260882720437329376208572022, 254347612873056383285840298534, 2289126526192229962972341588822, 20602126797738404747149747711494, 185419069551695653206739769876022, 1668771196197560941755010171719654, 15018938187171848853161205002489622, 135170428212909441942647525764483014, 1216533761086361791069007816332805622, 10948803292798317001132150853710001574, 98539226293311218299255840723678520022] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 12, 72, 594, 5022, 43254, 377622, 3328614, 29537622, 263319174, 2354756022, 21102104934, 189374748822, 1701107565894, 15290377052022, 137495847222054, 1236757347521622, 11126584462833414, 100113870176333622, 900872491651999974, 8106938385257984022, 72956961229661761734, 656579745640995288822, 5909020278213194198694, 53179997908584167384022, 478612873605250024030854, 4307473217015205321725622, 38767003080544578528218214, 348901492489347590550089622, 3140104220990806617727560774, 28260882720437329376208572022, 254347612873056383285840298534, 2289126526192229962972341588822, 20602126797738404747149747711494, 185419069551695653206739769876022, 1668771196197560941755010171719654, 15018938187171848853161205002489622, 135170428212909441942647525764483014, 1216533761086361791069007816332805622, 10948803292798317001132150853710001574, 98539226293311218299255840723678520022] Using the found enumerative automaton with, 18, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 2 162 t - 36 t + 9 t - 1 - ------------------------ (9 t - 1) (6 t - 1) and in Maple notation -(162*t^3-36*t^2+9*t-1)/(9*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 6 t (27 t - 18 t + 2) ---------------------- (9 t - 1) (6 t - 1) and in Maple notation 6*t*(27*t^2-18*t+2)/(9*t-1)/(6*t-1) This theorem took, 0.175, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 15, :consider the sequence, (x y + x y + x + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 28, 7, 42, 77, 6, 36, 42, 36, 216, 147, 28, 168, 311, 7, 42, 149, 42, 252, 439, 77, 462, 724, 6, 36, 42, 36, 216, 168, 42, 252, 462, 36, 216 , 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 37, 10, 60, 92, 0, 0, 60, 0, 0, 180, 37, 222, 308, 10, 60, 140 , 60, 360, 445, 92, 552, 751, 0, 0, 60, 0, 0, 222, 60, 360, 552, 0, 0, 360, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [145275112114807399460676455560526626512925588704855610996528224210641548197272\ 35118751502088923778088374894592000000, 145275112114807399460676455560526626512\ 92558917415095024960381065523698591607418681608967497077190793197780992000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 77, 724, 6383, 56017, 488591, 4263307, 37187624, 324452392, 2830920443, 24702480682, 215561562704, 1881117258016, 16416048165728, 143260649829919, 1250226524661245, 10910704162226932, 95217836659887839, 830968923324633301, 7251900887307475499, 63287702159447822299, 552315264876608748212, 4820087551317588156352, 42065195448707741450579, 367105560238444852175530, 3203753228593799831582816, 27959356972481461239259084, 244003085966488991263843124, 2129430501397075687469084899, 18583676184069047636574636725, 162180932448388453071873640540, 1415363396075082143143608656387, 12351967162885212594737960059477, 107796410209679843216874794192387, 940746192305092199976547383607663, 8209952425012250828806577176767776, 71648782025216053898957064561260176, 625283522062025643801573608697844919, 5456889454882760219635030520485819678, 47622624742778064322271945899776851276] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 92, 751, 6500, 56188, 489392, 4264180, 37193015, 324456019, 2830958450, 24702490195, 215561848571, 1881117219685, 16416050426645, 143260648757866, 1250226543063896, 10910704147718491, 95217836811965276, 830968923158770636, 7251900888577170632, 63287702157694022188, 552315264887312522471, 4820087551299901060135, 42065195448798885123122, 367105560238272092127655, 3203753228594583566355839, 27959356972479811108306801, 244003085966495790304252589, 2129430501397060173005514850, 18583676184069107062716227768, 162180932448388308833858811991, 1415363396075082665732269787852, 12351967162885211264538880794820, 107796410209679847835000730550956, 940746192305092187781220744314436, 8209952425012250869775944110856259, 71648782025216053787637701934381403, 625283522062025644166152400044690226, 5456889454882760218622240981124395275, 47622624742778064325524196135169135471] Using the found enumerative automaton with, 36, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 13 12 11 10 9 8 - (1539 t - 31617 t - 37170 t + 124542 t - 56910 t - 31980 t 7 6 5 4 3 2 / + 29114 t - 2521 t - 3355 t + 1081 t - 31 t - 47 t + 12 t - 1) / ( / 5 4 3 2 (9 t + 21 t - 24 t + 2 t + 4 t - 1) (3 t + 1) 7 6 5 4 3 2 (927 t + 219 t - 1199 t + 445 t + 122 t - 99 t + 18 t - 1)) and in Maple notation -(1539*t^13-31617*t^12-37170*t^11+124542*t^10-56910*t^9-31980*t^8+29114*t^7-\ 2521*t^6-3355*t^5+1081*t^4-31*t^3-47*t^2+12*t-1)/(9*t^5+21*t^4-24*t^3+2*t^2+4*t -1)/(3*t+1)/(927*t^7+219*t^6-1199*t^5+445*t^4+122*t^3-99*t^2+18*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 12 11 10 9 8 7 - (51597 t + 138726 t - 172926 t - 126576 t + 179646 t - 1575 t 6 5 4 3 2 / - 53702 t + 12785 t + 5271 t - 2197 t - 33 t + 98 t - 10) t / ( / 5 4 3 2 (9 t + 21 t - 24 t + 2 t + 4 t - 1) (3 t + 1) 7 6 5 4 3 2 (927 t + 219 t - 1199 t + 445 t + 122 t - 99 t + 18 t - 1)) and in Maple notation -(51597*t^12+138726*t^11-172926*t^10-126576*t^9+179646*t^8-1575*t^7-53702*t^6+ 12785*t^5+5271*t^4-2197*t^3-33*t^2+98*t-10)*t/(9*t^5+21*t^4-24*t^3+2*t^2+4*t-1) /(3*t+1)/(927*t^7+219*t^6-1199*t^5+445*t^4+122*t^3-99*t^2+18*t-1) This theorem took, 0.825, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 16, :consider the sequence, (x y + x y + x + x y + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 8, 6, 36, 33, 8, 48, 73, 6, 36, 48, 36, 216, 186, 33, 198, 300, 8, 48, 128, 48, 288, 478, 73, 438, 659, 6, 36, 48, 36, 216, 198, 48, 288, 438, 36, 216 , 288, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 33, 8, 48, 70, 0, 0, 48, 0, 0, 186, 33, 198, 276, 8, 48, 128, 48, 288, 478, 70, 420, 671, 0, 0, 48, 0, 0, 198, 48, 288, 420, 0, 0, 288, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [282546496429954267733407208961192302243875569498352220724859563129772361065402\ 73485577767126453613384842541531136000, 282546496429954267733407208961192302243\ 87556949835222072485956312977236106540273485577767126453613384842541531136000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 73, 659, 5977, 53882, 485275, 4368551, 39319189, 353883314, 3184963543, 28664770679, 257983013137, 2321848020722, 20896632549835, 188069701137311, 1692627311185669, 15233645874760874, 137102812865138263, 1233925316455595999, 11105327847917178097, 99947950637296115162, 899531555733220041595, 8095784001653474550071, 72862056014853693550549, 655758504134174532853634, 5901826537207283601308983, 53116438834869979939038119, 478047949513826961761351857, 4302431545624482545010967202, 38721883910620315275498652555, 348496955195583196772085206831, 3136472596760248508910469917829, 28228253370842239815860106689594, 254054280337580155890796522371703, 2286488523038221432152395095001039, 20578396707343992866648872113550417, 185205570366095936062156547055936842, 1666850133294863424350318542999134715, 15001651199653770821514414902988467591, 135014860796883937391715817358290271509] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 70, 671, 5974, 53984, 485308, 4369319, 39319582, 353888852, 3184966780, 28664809931, 257983037254, 2321848296944, 20896632723028, 188069703075239, 1692627312411142, 15233645888339492, 137102812873755940, 1233925316550685691, 11105327847977619934, 99947950637961861104, 899531555733643488748, 8095784001658135125959, 72862056014856658743502, 655758504134207157947732, 5901826537207304360848300, 53116438834870208317885451, 478047949513827107087693014, 4302431545624484143672464464, 38721883910620316292811738468, 348496955195583207962744385479, 3136472596760248516031747612662, 28228253370842239894194807033572, 254054280337580155940645724515860, 2286488523038221432700738255689211, 20578396707343992866997817303400494, 185205570366095936065994949955595024, 1666850133294863424352761161652608188, 15001651199653770821541283725610597799, 135014860796883937391732915695838154622] Using the found enumerative automaton with, 28, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (1458 t - 432 t - 2844 t + 1647 t + 1362 t - 591 t - 393 t + 112 t 3 2 / + 60 t - 17 t - 3 t + 1) / ((t - 1) (3 t - 1) (t + 1) (2 t - 1) / 2 2 (3 t + 1) (9 t - 1) (7 t - 1) (3 t - 1)) and in Maple notation (1458*t^11-432*t^10-2844*t^9+1647*t^8+1362*t^7-591*t^6-393*t^5+112*t^4+60*t^3-\ 17*t^2-3*t+1)/(t-1)/(3*t-1)/(t+1)/(2*t-1)/(3*t+1)/(9*t-1)/(7*t^2-1)/(3*t^2-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 10 9 8 7 6 5 4 3 - t (1458 t - 2322 t - 1584 t + 1413 t + 867 t + 384 t - 488 t - 213 t 2 / + 115 t + 18 t - 8) / ((t - 1) (3 t - 1) (t + 1) (2 t - 1) (3 t + 1) / 2 2 (9 t - 1) (7 t - 1) (3 t - 1)) and in Maple notation -t*(1458*t^10-2322*t^9-1584*t^8+1413*t^7+867*t^6+384*t^5-488*t^4-213*t^3+115*t^ 2+18*t-8)/(t-1)/(3*t-1)/(t+1)/(2*t-1)/(3*t+1)/(9*t-1)/(7*t^2-1)/(3*t^2-1) This theorem took, 0.423, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 17, :consider the sequence, (x y + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 29, 6, 36, 76, 6, 36, 36, 36, 216, 158, 29, 174, 285, 6, 36, 117, 36, 216, 412, 76, 456, 637, 6, 36, 36, 36, 216, 174, 36, 216, 456, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 32, 9, 54, 70, 0, 0, 54, 0, 0, 170, 32, 192, 252, 9, 54, 108, 54, 324, 400, 70, 420, 682, 0, 0, 54, 0, 0, 192, 54, 324, 420, 0, 0, 324, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [812700186500290082064287764591979046704967466797565197329519805077453539737539\ 471257710117776052308603711992627200, 81270018650029008206428776459197904670496\ 7468883797798707420054392833583935463611049961814844475344751808007372800] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 76, 637, 5818, 49828, 432421, 3720358, 32076625, 276218320, 2379381748, 20492498077, 176501517919, 1520161153012, 13092851232571, 112765673914825, 971225572762222, 8364943217572018, 72045350412047827, 620510122975955122, 5344311875503772908, 46029335963126912752, 396440145223143329365, 3414448307985299847079, 29407862431014726238672, 253283193755816887525030, 2181470223923728190494315, 18788504151964148248479706, 161821089464810376032845504, 1393728036221068268462646841, 12003860840226898522560583972, 103386508218576032813383835371, 890444351521448550628600794949, 7669193561309457420376930456258, 66053010253064298514255150860745, 568899471451885197053549659165165, 4899801044316797202490777787237692, 42200865844745664089299914011951497, 363466406480294133323095469217355426, 3130453036810063657817307637696704568, 26961876093505367179344835176052082245] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 70, 682, 5725, 50203, 431128, 3723817, 32060392, 276253753, 2379189907, 20492881981, 176499320434, 1520165469847, 13092826532797, 112765723861030, 971225299033783, 8364943806756745, 72045347409829444, 620510129994039925, 5344311842783450986, 46029336046863561232, 396440144867371548076, 3414448308980631752065, 29407862427141868402432, 253283193767566451304412, 2181470223881399892403969, 18788504152101701968261564, 161821089464344953789633022, 1393728036222664924884819325, 12003860840221744212089379277, 103386508218594416976111484804, 890444351521391044905411256822, 7669193561309667586851060599065, 66053010253063652484402366033673, 568899471451887585249169550331748, 4899801044316789902148509982786481, 42200865844745691098407632430954258, 363466406480294050452647967439779421, 3130453036810063962211184464352661655, 26961876093505366235709745899695033539] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 33 32 31 30 29 - (946161 t - 8152704 t + 39790476 t - 12187101 t - 85244409 t 28 27 26 25 24 + 91806720 t + 33954279 t - 86786249 t + 63910540 t - 2599941 t 23 22 21 20 19 - 40634567 t + 43512646 t - 35716423 t - 26368695 t + 49861891 t 18 17 16 15 14 + 3187633 t - 27939949 t + 7981717 t + 7152140 t - 5952856 t 13 12 11 10 9 8 + 452136 t + 1513311 t - 609999 t - 77139 t + 60603 t + 113 t 7 6 5 4 3 2 + 12019 t - 12116 t + 2952 t + 571 t - 511 t + 136 t - 18 t + 1) / 17 16 15 14 13 12 / ((459 t - 1689 t + 4936 t - 3369 t - 5367 t + 8678 t / 11 10 9 8 7 6 5 - 2730 t - 3996 t + 3494 t + 1144 t - 1742 t - 248 t + 598 t 4 3 2 17 16 15 - 50 t - 87 t + 17 t + 4 t - 1) (9585 t - 2655 t - 11562 t 14 13 12 11 10 9 + 7065 t + 8961 t + 11766 t + 8958 t - 7466 t - 3968 t 8 7 6 5 4 3 2 + 2522 t - 54 t - 1454 t + 922 t + 268 t - 411 t + 141 t - 20 t + 1 )) and in Maple notation -(946161*t^33-8152704*t^32+39790476*t^31-12187101*t^30-85244409*t^29+91806720*t ^28+33954279*t^27-86786249*t^26+63910540*t^25-2599941*t^24-40634567*t^23+ 43512646*t^22-35716423*t^21-26368695*t^20+49861891*t^19+3187633*t^18-27939949*t ^17+7981717*t^16+7152140*t^15-5952856*t^14+452136*t^13+1513311*t^12-609999*t^11 -77139*t^10+60603*t^9+113*t^8+12019*t^7-12116*t^6+2952*t^5+571*t^4-511*t^3+136* t^2-18*t+1)/(459*t^17-1689*t^16+4936*t^15-3369*t^14-5367*t^13+8678*t^12-2730*t^ 11-3996*t^10+3494*t^9+1144*t^8-1742*t^7-248*t^6+598*t^5-50*t^4-87*t^3+17*t^2+4* t-1)/(9585*t^17-2655*t^16-11562*t^15+7065*t^14+8961*t^13+11766*t^12+8958*t^11-\ 7466*t^10-3968*t^9+2522*t^8-54*t^7-1454*t^6+922*t^5+268*t^4-411*t^3+141*t^2-20* t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 32 31 30 29 28 - t (1003671 t + 2509056 t - 25031061 t + 17310141 t + 41754051 t 27 26 25 24 23 - 59920872 t - 13952226 t + 29784109 t - 32141563 t + 23652700 t 22 21 20 19 18 + 8119383 t - 30056505 t + 42076643 t + 14079002 t - 45055040 t 17 16 15 14 13 - 956559 t + 23329735 t - 5317102 t - 6484386 t + 4641306 t 12 11 10 9 8 7 - 56193 t - 1353762 t + 343053 t + 159216 t + 33588 t - 71059 t 6 5 4 3 2 / - 12846 t + 25724 t - 6004 t - 1295 t + 838 t - 146 t + 9) / (( / 17 16 15 14 13 12 11 459 t - 1689 t + 4936 t - 3369 t - 5367 t + 8678 t - 2730 t 10 9 8 7 6 5 4 3 - 3996 t + 3494 t + 1144 t - 1742 t - 248 t + 598 t - 50 t - 87 t 2 17 16 15 14 13 + 17 t + 4 t - 1) (9585 t - 2655 t - 11562 t + 7065 t + 8961 t 12 11 10 9 8 7 6 + 11766 t + 8958 t - 7466 t - 3968 t + 2522 t - 54 t - 1454 t 5 4 3 2 + 922 t + 268 t - 411 t + 141 t - 20 t + 1)) and in Maple notation -t*(1003671*t^32+2509056*t^31-25031061*t^30+17310141*t^29+41754051*t^28-\ 59920872*t^27-13952226*t^26+29784109*t^25-32141563*t^24+23652700*t^23+8119383*t ^22-30056505*t^21+42076643*t^20+14079002*t^19-45055040*t^18-956559*t^17+ 23329735*t^16-5317102*t^15-6484386*t^14+4641306*t^13-56193*t^12-1353762*t^11+ 343053*t^10+159216*t^9+33588*t^8-71059*t^7-12846*t^6+25724*t^5-6004*t^4-1295*t^ 3+838*t^2-146*t+9)/(459*t^17-1689*t^16+4936*t^15-3369*t^14-5367*t^13+8678*t^12-\ 2730*t^11-3996*t^10+3494*t^9+1144*t^8-1742*t^7-248*t^6+598*t^5-50*t^4-87*t^3+17 *t^2+4*t-1)/(9585*t^17-2655*t^16-11562*t^15+7065*t^14+8961*t^13+11766*t^12+8958 *t^11-7466*t^10-3968*t^9+2522*t^8-54*t^7-1454*t^6+922*t^5+268*t^4-411*t^3+141*t ^2-20*t+1) This theorem took, 1.019, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 18, :consider the sequence, (x y + x + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 5, 6, 36, 27, 5, 30, 97, 6, 36, 30, 36, 216, 155, 27, 162, 387, 5, 30, 146, 30, 180, 574, 97, 582, 887, 6, 36, 30, 36, 216, 162, 30, 180, 582, 36, 216 , 180, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 11, 0, 0, 48, 11, 66, 85, 0, 0, 66, 0, 0, 254, 48, 288, 330, 11, 66, 110 , 66, 396, 490, 85, 510, 893, 0, 0, 66, 0, 0, 288, 66, 396, 510, 0, 0, 396, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [130379139191452975111121482686061764938153519792692781573006129910641636654328\ 5259529012474710578340369955417384550400, 1303791391914529751111214826861262156\ 1669447133892196186354012978397682147731155713603986051188993246438045826154496\ 00] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 97, 887, 8263, 74891, 670936, 5991884, 53381032, 474926597, 4224204247, 37558277354, 333908004007, 2968379264507, 26387540188447, 234569819804780, 2085174528625132, 18535786251383531, 164770310647709080, 1464692659709630006, 13020086945990863795, 115739385486233062511, 1028841368945122419928, 9145672452475779710879, 81298560925011837580108, 722686709218150461044660, 6424173698961090170125399, 57106360292307572004036920, 507635150350645691966428618, 4512517421455396761370122506, 40113088028679707782968516730, 356576979245023064845681413773, 3169717126412637485410150812244, 28176543203523716056087758516386, 250469538833579668231382637776389, 2226497034422318469407376956815328, 19791983757207527934072179402346133, 175936736042554151394250166003458976, 1563953137239324315611506454073775468, 13902437151544103152070720801869287800, 123582832599291603126143616160899272506] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 11, 85, 893, 8254, 74693, 671470, 5990759, 53372281, 474963335, 4224054931, 37558375406, 333908699305, 2968373782892, 26387554078978, 234569814284912, 2085174378845353, 18535787038264727, 164770308818456026, 1464692659791222362, 13020086969832260581, 115739385384351774872, 1028841369179068520656, 9145672452699694851836, 81298560921583341914806, 722686709232767803484972, 6424173698936035948578910, 57106360292262176781162176, 507635150351176321257028747, 4512517421453492551785488183, 40113088028682512325935696704, 356576979245032444073227187882, 3169717126412560315120082777197, 28176543203523965981594936620556, 250469538833579391536287153605907, 2226497034422316780900984117037253, 19791983757207538942952842885128091, 175936736042554118953100563190786675, 1563953137239324336401582742829595494, 13902437151544103429837672170142060645, 123582832599291601558243741556643874894] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.780, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 19, :consider the sequence, (x y + x + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 4, 6, 36, 19, 4, 24, 93, 6, 36, 24, 36, 216, 108, 19, 114, 371, 4, 24, 185, 24, 144, 590, 93, 558, 818, 6, 36, 24, 36, 216, 114, 24, 144, 558, 36, 216 , 144, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 13, 0, 0, 52, 13, 78, 87, 0, 0, 78, 0, 0, 270, 52, 312, 323, 13, 78, 104 , 78, 468, 425, 87, 522, 878, 0, 0, 78, 0, 0, 312, 78, 468, 522, 0, 0, 468, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [444424515964846311530976905161491524088067166121010065389622414375724781253322\ 786587299751749537847434871098572800000, 44442451596484631153097695848625278223\ 1642616245215352420979552837094747664239281592981411720070808923733701427200000 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 93, 818, 7709, 68049, 605008, 5332498, 47050902, 414392915, 3650435417, 32144960127, 283067993860, 2492489757103, 21947035078269, 193246047674354, 1701550799306063, 14982266539253220, 131919783409561771, 1161560844597982699, 10227604802565935526, 90054584978018082404, 792935209636557719126, 6981834622745269705209, 61475406301841307954691, 541294050462587302997893, 4766121378922965548342724, 41965938712356211000159703, 369512203055274680931925907, 3253573548561529614990953490, 28647878869670571201408206905, 252246015460080530812026259594, 2221038863019818577546956430453, 19556358985118539813482958073687, 172194725233471792767691051975466, 1516183223093526119326034188227414, 13350069596307880739747672741083666, 117548034770060671029659657764447279, 1035016362919598976768518059621378077, 9113371172954389280156693756686314731, 80243691898496211808647667262174553716] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 13, 87, 878, 7622, 68583, 604117, 5340031, 47032089, 414496385, 3650133719, 32146452990, 283063356532, 2492511273652, 21946963254828, 193246362293705, 1701549701418377, 14982271183129578, 131919766723846276, 1161560913341843221, 10227604550013470667, 90054585999417483857, 792935205826515114893, 6981834637958563591539, 61475406244476183900097, 541294050689550064777576, 4766121378060555291505320, 41965938715746349562546240, 369512203042323167516494250, 3253573548612210951113319465, 28647878869476208831473600631, 252246015460838655654977124640, 2221038863016903308298604253322, 19556358985129885218813426014381, 172194725233428082161841587993683, 1516183223093695955655814677016755, 13350069596307225525295681901307961, 117548034770063213959122741547838389, 1035016362919589156999703480521323995, 9113371172954427360768867118922406098, 80243691898496064657566415308317540028] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.826, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 20, :consider the sequence, (x y + x + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 28, 7, 42, 77, 6, 36, 42, 36, 216, 147, 28, 168, 311, 7, 42, 149, 42, 252, 439, 77, 462, 724, 6, 36, 42, 36, 216, 168, 42, 252, 462, 36, 216 , 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 37, 10, 60, 92, 0, 0, 60, 0, 0, 180, 37, 222, 308, 10, 60, 140 , 60, 360, 445, 92, 552, 751, 0, 0, 60, 0, 0, 222, 60, 360, 552, 0, 0, 360, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [145275112114807399460676455560526626512925588704855610996528224210641548197272\ 35118751502088923778088374894592000000, 145275112114807399460676455560526626512\ 92558917415095024960381065523698591607418681608967497077190793197780992000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 77, 724, 6383, 56017, 488591, 4263307, 37187624, 324452392, 2830920443, 24702480682, 215561562704, 1881117258016, 16416048165728, 143260649829919, 1250226524661245, 10910704162226932, 95217836659887839, 830968923324633301, 7251900887307475499, 63287702159447822299, 552315264876608748212, 4820087551317588156352, 42065195448707741450579, 367105560238444852175530, 3203753228593799831582816, 27959356972481461239259084, 244003085966488991263843124, 2129430501397075687469084899, 18583676184069047636574636725, 162180932448388453071873640540, 1415363396075082143143608656387, 12351967162885212594737960059477, 107796410209679843216874794192387, 940746192305092199976547383607663, 8209952425012250828806577176767776, 71648782025216053898957064561260176, 625283522062025643801573608697844919, 5456889454882760219635030520485819678, 47622624742778064322271945899776851276] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 92, 751, 6500, 56188, 489392, 4264180, 37193015, 324456019, 2830958450, 24702490195, 215561848571, 1881117219685, 16416050426645, 143260648757866, 1250226543063896, 10910704147718491, 95217836811965276, 830968923158770636, 7251900888577170632, 63287702157694022188, 552315264887312522471, 4820087551299901060135, 42065195448798885123122, 367105560238272092127655, 3203753228594583566355839, 27959356972479811108306801, 244003085966495790304252589, 2129430501397060173005514850, 18583676184069107062716227768, 162180932448388308833858811991, 1415363396075082665732269787852, 12351967162885211264538880794820, 107796410209679847835000730550956, 940746192305092187781220744314436, 8209952425012250869775944110856259, 71648782025216053787637701934381403, 625283522062025644166152400044690226, 5456889454882760218622240981124395275, 47622624742778064325524196135169135471] Using the found enumerative automaton with, 36, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 13 12 11 10 9 8 - (1539 t - 31617 t - 37170 t + 124542 t - 56910 t - 31980 t 7 6 5 4 3 2 / + 29114 t - 2521 t - 3355 t + 1081 t - 31 t - 47 t + 12 t - 1) / ( / 5 4 3 2 (9 t + 21 t - 24 t + 2 t + 4 t - 1) 7 6 5 4 3 2 (927 t + 219 t - 1199 t + 445 t + 122 t - 99 t + 18 t - 1) (3 t + 1)) and in Maple notation -(1539*t^13-31617*t^12-37170*t^11+124542*t^10-56910*t^9-31980*t^8+29114*t^7-\ 2521*t^6-3355*t^5+1081*t^4-31*t^3-47*t^2+12*t-1)/(9*t^5+21*t^4-24*t^3+2*t^2+4*t -1)/(927*t^7+219*t^6-1199*t^5+445*t^4+122*t^3-99*t^2+18*t-1)/(3*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 12 11 10 9 8 7 - t (51597 t + 138726 t - 172926 t - 126576 t + 179646 t - 1575 t 6 5 4 3 2 / - 53702 t + 12785 t + 5271 t - 2197 t - 33 t + 98 t - 10) / ( / 5 4 3 2 (9 t + 21 t - 24 t + 2 t + 4 t - 1) 7 6 5 4 3 2 (927 t + 219 t - 1199 t + 445 t + 122 t - 99 t + 18 t - 1) (3 t + 1)) and in Maple notation -t*(51597*t^12+138726*t^11-172926*t^10-126576*t^9+179646*t^8-1575*t^7-53702*t^6 +12785*t^5+5271*t^4-2197*t^3-33*t^2+98*t-10)/(9*t^5+21*t^4-24*t^3+2*t^2+4*t-1)/ (927*t^7+219*t^6-1199*t^5+445*t^4+122*t^3-99*t^2+18*t-1)/(3*t+1) This theorem took, 0.479, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 21, :consider the sequence, (x y + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 28, 6, 36, 77, 6, 36, 36, 36, 216, 158, 28, 168, 278, 6, 36, 117, 36, 216, 392, 77, 462, 675, 6, 36, 36, 36, 216, 168, 36, 216, 462, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 31, 9, 54, 74, 0, 0, 54, 0, 0, 155, 31, 186, 239, 9, 54, 108, 54, 324, 389, 74, 444, 693, 0, 0, 54, 0, 0, 186, 54, 324, 444, 0, 0, 324, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [299613328556701823530503346293847151895738346180461719251586341044783223611642\ 420477973383027920171690927063040000, 29961332855670182353050334629384715189573\ 8346246482098268362802025778019661713375629549029608511802709072936960000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 77, 675, 6008, 51123, 437885, 3728367, 31781705, 270657096, 2305411847, 19634024127, 167218901039, 1424131240374, 12128787405905, 103295842120815, 879728753140760, 7492287071972223, 63808732900927568, 543432711515831550, 4628192874635482940, 39416414154644556363, 335693381308138687670, 2858962395611823954759, 24348606332265308720441, 207367061161355482791072, 1766059932873189011879561, 15040805751380445405161403, 128096353611566235670259540, 1090943934753636280122575421, 9291120591881330820609418535, 79128650980679560753914685515, 673906160630705513821943193431, 5739381471910396393893973079913, 48879950361766715601051600129659, 416290424161906335326880215484042, 3545374247852355649686925912433057, 30194493622187648132933992767879603, 257154077782512045316397790049647290, 2190075466991156096083962694928877438, 18651971582473286269213914227306180150] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 74, 693, 5921, 51279, 436925, 3730485, 31770341, 270683154, 2305273112, 19634338788, 167217144833, 1424134982826, 12128764736933, 103295885970105, 879728454664877, 7492287573941589, 63808728905198318, 543432717060143445, 4628192820335702552, 39416414212277856309, 335693380560116346638, 2858962396144448887455, 24348606321829202115566, 207367061164981153316613, 1766059932725842337550884, 15040805751376776885944916, 128096353609462281691896881, 1090943934752799827143364103, 9291120591850967197707422582, 79128650980656373193044368723, 673906160630262911633673251414, 5739381471909902023915030051311, 48879950361760203482766704832593, 416290424161896878540062986260340, 3545374247852259006814684014888206, 30194493622187477538101216138157414, 257154077782510599763184489971512887, 2190075466991153128092643602187551090, 18651971582473264492742934266050318820] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 33 32 31 30 29 (122688 t + 2102544 t + 617992 t - 12420845 t + 9805007 t 28 27 26 25 + 17375873 t - 18181150 t - 17054084 t - 33480173 t 24 23 22 21 + 155723414 t - 116864259 t - 92129126 t + 164836827 t 20 19 18 17 16 - 18331059 t - 83845136 t + 32768709 t + 24262132 t - 16929582 t 15 14 13 12 11 - 375168 t + 1902016 t - 775410 t + 838228 t - 36292 t 10 9 8 7 6 5 - 385443 t + 83768 t + 91321 t - 38580 t - 5409 t + 5963 t 4 3 2 / 17 16 15 - 1051 t - 111 t + 77 t - 14 t + 1) / ((1008 t - 932 t + 4237 t / 14 13 12 11 10 9 + 1830 t - 18173 t + 20725 t + 487 t - 14044 t + 9005 t 8 7 6 5 4 3 2 - 1360 t - 1135 t + 1079 t - 530 t + 7 t + 95 t - 19 t - 4 t + 1) ( 17 16 15 14 13 12 11 1080 t + 126 t - 1219 t + 1728 t + 183 t + 2221 t - 9585 t 10 9 8 7 6 5 4 + 3806 t + 4425 t - 1606 t - 2715 t + 1531 t + 382 t - 325 t 3 2 - 65 t + 75 t - 16 t + 1)) and in Maple notation (122688*t^33+2102544*t^32+617992*t^31-12420845*t^30+9805007*t^29+17375873*t^28-\ 18181150*t^27-17054084*t^26-33480173*t^25+155723414*t^24-116864259*t^23-\ 92129126*t^22+164836827*t^21-18331059*t^20-83845136*t^19+32768709*t^18+24262132 *t^17-16929582*t^16-375168*t^15+1902016*t^14-775410*t^13+838228*t^12-36292*t^11 -385443*t^10+83768*t^9+91321*t^8-38580*t^7-5409*t^6+5963*t^5-1051*t^4-111*t^3+ 77*t^2-14*t+1)/(1008*t^17-932*t^16+4237*t^15+1830*t^14-18173*t^13+20725*t^12+ 487*t^11-14044*t^10+9005*t^9-1360*t^8-1135*t^7+1079*t^6-530*t^5+7*t^4+95*t^3-19 *t^2-4*t+1)/(1080*t^17+126*t^16-1219*t^15+1728*t^14+183*t^13+2221*t^12-9585*t^ 11+3806*t^10+4425*t^9-1606*t^8-2715*t^7+1531*t^6+382*t^5-325*t^4-65*t^3+75*t^2-\ 16*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 32 31 30 29 28 - t (412992 t + 2033712 t + 2700016 t - 12129143 t + 6729889 t 27 26 25 24 23 + 24953665 t - 45709397 t + 3245682 t + 15033849 t + 64438613 t 22 21 20 19 - 76383606 t - 63073078 t + 116848389 t + 10207855 t 18 17 16 15 14 - 81835973 t + 14465720 t + 35670075 t - 15204248 t - 6610764 t 13 12 11 10 9 + 5594997 t - 1289457 t + 81262 t + 560897 t - 540268 t 8 7 6 5 4 3 2 + 48591 t + 135182 t - 52083 t - 7805 t + 8492 t - 1247 t - 293 t / 17 16 15 14 13 + 106 t - 9) / ((1008 t - 932 t + 4237 t + 1830 t - 18173 t / 12 11 10 9 8 7 6 + 20725 t + 487 t - 14044 t + 9005 t - 1360 t - 1135 t + 1079 t 5 4 3 2 17 16 15 - 530 t + 7 t + 95 t - 19 t - 4 t + 1) (1080 t + 126 t - 1219 t 14 13 12 11 10 9 8 + 1728 t + 183 t + 2221 t - 9585 t + 3806 t + 4425 t - 1606 t 7 6 5 4 3 2 - 2715 t + 1531 t + 382 t - 325 t - 65 t + 75 t - 16 t + 1)) and in Maple notation -t*(412992*t^32+2033712*t^31+2700016*t^30-12129143*t^29+6729889*t^28+24953665*t ^27-45709397*t^26+3245682*t^25+15033849*t^24+64438613*t^23-76383606*t^22-\ 63073078*t^21+116848389*t^20+10207855*t^19-81835973*t^18+14465720*t^17+35670075 *t^16-15204248*t^15-6610764*t^14+5594997*t^13-1289457*t^12+81262*t^11+560897*t^ 10-540268*t^9+48591*t^8+135182*t^7-52083*t^6-7805*t^5+8492*t^4-1247*t^3-293*t^2 +106*t-9)/(1008*t^17-932*t^16+4237*t^15+1830*t^14-18173*t^13+20725*t^12+487*t^ 11-14044*t^10+9005*t^9-1360*t^8-1135*t^7+1079*t^6-530*t^5+7*t^4+95*t^3-19*t^2-4 *t+1)/(1080*t^17+126*t^16-1219*t^15+1728*t^14+183*t^13+2221*t^12-9585*t^11+3806 *t^10+4425*t^9-1606*t^8-2715*t^7+1531*t^6+382*t^5-325*t^4-65*t^3+75*t^2-16*t+1) This theorem took, 0.547, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 22, :consider the sequence, (x y + x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 8, 6, 36, 33, 8, 48, 73, 6, 36, 48, 36, 216, 186, 33, 198, 300, 8, 48, 128, 48, 288, 478, 73, 438, 659, 6, 36, 48, 36, 216, 198, 48, 288, 438, 36, 216 , 288, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 33, 8, 48, 70, 0, 0, 48, 0, 0, 186, 33, 198, 276, 8, 48, 128, 48, 288, 478, 70, 420, 671, 0, 0, 48, 0, 0, 198, 48, 288, 420, 0, 0, 288, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [282546496429954267733407208961192302243875569498352220724859563129772361065402\ 73485577767126453613384842541531136000, 282546496429954267733407208961192302243\ 87556949835222072485956312977236106540273485577767126453613384842541531136000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 73, 659, 5977, 53882, 485275, 4368551, 39319189, 353883314, 3184963543, 28664770679, 257983013137, 2321848020722, 20896632549835, 188069701137311, 1692627311185669, 15233645874760874, 137102812865138263, 1233925316455595999, 11105327847917178097, 99947950637296115162, 899531555733220041595, 8095784001653474550071, 72862056014853693550549, 655758504134174532853634, 5901826537207283601308983, 53116438834869979939038119, 478047949513826961761351857, 4302431545624482545010967202, 38721883910620315275498652555, 348496955195583196772085206831, 3136472596760248508910469917829, 28228253370842239815860106689594, 254054280337580155890796522371703, 2286488523038221432152395095001039, 20578396707343992866648872113550417, 185205570366095936062156547055936842, 1666850133294863424350318542999134715, 15001651199653770821514414902988467591, 135014860796883937391715817358290271509] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 70, 671, 5974, 53984, 485308, 4369319, 39319582, 353888852, 3184966780, 28664809931, 257983037254, 2321848296944, 20896632723028, 188069703075239, 1692627312411142, 15233645888339492, 137102812873755940, 1233925316550685691, 11105327847977619934, 99947950637961861104, 899531555733643488748, 8095784001658135125959, 72862056014856658743502, 655758504134207157947732, 5901826537207304360848300, 53116438834870208317885451, 478047949513827107087693014, 4302431545624484143672464464, 38721883910620316292811738468, 348496955195583207962744385479, 3136472596760248516031747612662, 28228253370842239894194807033572, 254054280337580155940645724515860, 2286488523038221432700738255689211, 20578396707343992866997817303400494, 185205570366095936065994949955595024, 1666850133294863424352761161652608188, 15001651199653770821541283725610597799, 135014860796883937391732915695838154622] Using the found enumerative automaton with, 28, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (1458 t - 432 t - 2844 t + 1647 t + 1362 t - 591 t - 393 t + 112 t 3 2 / 2 + 60 t - 17 t - 3 t + 1) / ((3 t + 1) (9 t - 1) (3 t - 1) (3 t - 1) / 2 (t - 1) (t + 1) (2 t - 1) (7 t - 1)) and in Maple notation (1458*t^11-432*t^10-2844*t^9+1647*t^8+1362*t^7-591*t^6-393*t^5+112*t^4+60*t^3-\ 17*t^2-3*t+1)/(3*t+1)/(9*t-1)/(3*t^2-1)/(3*t-1)/(t-1)/(t+1)/(2*t-1)/(7*t^2-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 10 9 8 7 6 5 4 3 - t (1458 t - 2322 t - 1584 t + 1413 t + 867 t + 384 t - 488 t - 213 t 2 / 2 + 115 t + 18 t - 8) / ((3 t + 1) (9 t - 1) (3 t - 1) (3 t - 1) (t - 1) / 2 (t + 1) (2 t - 1) (7 t - 1)) and in Maple notation -t*(1458*t^10-2322*t^9-1584*t^8+1413*t^7+867*t^6+384*t^5-488*t^4-213*t^3+115*t^ 2+18*t-8)/(3*t+1)/(9*t-1)/(3*t^2-1)/(3*t-1)/(t-1)/(t+1)/(2*t-1)/(7*t^2-1) This theorem took, 0.873, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 23, :consider the sequence, (x y + x y + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 27, 6, 36, 42, 6, 36, 36, 36, 216, 153, 27, 162, 144, 6, 36, 72, 36, 216, 186, 42, 252, 264, 6, 36, 36, 36, 216, 162, 36, 216, 252, 36, 216, 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 9, 6, 36, 42, 0, 0, 36, 0, 0, 27, 9, 54, 72, 6, 36, 72, 36, 216 , 186, 42, 252, 264, 0, 0, 36, 0, 0, 54, 36, 216, 252, 0, 0, 216, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [881081690996846456728566157801394786708421517753374882404256386202655813702315\ 18305558709567695618048000, 881081690996846456728566157801394786708421517753374\ 88240425638620265581370231518305558709567695618048000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 42, 264, 1608, 9696, 58272, 349824, 2099328, 12596736, 75581952, 453494784, 2720974848, 16325861376, 97955192832, 587731206144, 3526387335168, 21158324207616, 126949945638912, 761699674619904, 4570198049292288, 27421188298899456, 164527129799688192, 987162778810712064, 5922976672889438208, 35537860037386960896, 213227160224422428672, 1279362961346735898624, 7676177768080818044928, 46057066608485713575936, 276342399650915892068352, 1658054397905498573635584, 9948326387432997884264448, 59689958324598000190488576, 358139749947588026912735232, 2148838499685528213016018944, 12893030998113169381175328768, 77358185988679016493210402816, 464149115932074099371579277312, 2784894695592444597054109384704, 16709368173554667583973923749888] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 264, 1608, 9696, 58272, 349824, 2099328, 12596736, 75581952, 453494784, 2720974848, 16325861376, 97955192832, 587731206144, 3526387335168, 21158324207616, 126949945638912, 761699674619904, 4570198049292288, 27421188298899456, 164527129799688192, 987162778810712064, 5922976672889438208, 35537860037386960896, 213227160224422428672, 1279362961346735898624, 7676177768080818044928, 46057066608485713575936, 276342399650915892068352, 1658054397905498573635584, 9948326387432997884264448, 59689958324598000190488576, 358139749947588026912735232, 2148838499685528213016018944, 12893030998113169381175328768, 77358185988679016493210402816, 464149115932074099371579277312, 2784894695592444597054109384704, 16709368173554667583973923749888] Using the found enumerative automaton with, 8, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 2 t + 1 ------------------- (2 t - 1) (6 t - 1) and in Maple notation (6*t^2-2*t+1)/(2*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 (t - 1) t - ------------------- (2 t - 1) (6 t - 1) and in Maple notation -6*(t-1)*t/(2*t-1)/(6*t-1) This theorem took, 0.086, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 24, :consider the sequence, (x y + x y + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 5, 6, 36, 25, 5, 30, 82, 6, 36, 30, 36, 216, 143, 25, 150, 323, 5, 30, 146, 30, 180, 507, 82, 492, 724, 6, 36, 30, 36, 216, 150, 30, 180, 492, 36, 216 , 180, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 11, 0, 0, 43, 11, 66, 76, 0, 0, 66, 0, 0, 227, 43, 258, 290, 11, 66, 110 , 66, 396, 441, 76, 456, 736, 0, 0, 66, 0, 0, 258, 66, 396, 456, 0, 0, 396, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [503086733270672346209323293985685434235625055532136716009910498657345821060646\ 15508035631146245067660855591441530880, 503086733270672346209323293985690139937\ 76069663093003014872006434176589255583108619524517775106090115687989410529280] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 82, 724, 6534, 57098, 501416, 4375167, 38212114, 333385033, 2909086452, 25379304377, 221418974480, 1931675361528, 16852151839507, 147019096024795, 1282603597149657, 11189499073077515, 97617771493054865, 851622330339807585, 7429596034714861882, 64816166917535579962, 565459479576591520188, 4933096726210082546729, 43036582109025219820817, 375453290318868494817561, 3275473244237423718808546, 28575392060617970595614698, 249293146528610898110494401, 2174845852396421058987505658, 18973463761663239335977735829, 165525444810669031600099342923, 1444052241801668575917296042881, 12597983829222938082712601602945, 109905439683642301901625681181947, 958820541111698714038582565042585, 8364798254790607904545502454123080, 72974917456627524779443798832385399, 636636822023975655528879162700754482, 5554051409481583763661702287021414365, 48453821695539060226632889120210778448] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 11, 76, 736, 6468, 57338, 500429, 4378467, 38200405, 333432838, 2908916019, 25379969978, 221416687664, 1931684236644, 16852120470121, 147019217593036, 1282603164458514, 11189500729727546, 97617765557495858, 851622352934064306, 7429595953253171707, 64816167225871255798, 565459478459741760339, 4933096730419338240263, 43036582093715821023701, 375453290376365309852613, 3275473244027642030717713, 28575392061403538416713424, 249293146525737261870116952, 2174845852407156935726328554, 18973463761623884248838947628, 165525444810815783464137451845, 1444052241801129702922515738964, 12597983829224944414948944700054, 109905439683634924460044834178262, 958820541111726147589574946406427, 8364798254790506916301452041092184, 72974917456627899933088587787488234, 636636822023974273261917108673164388, 5554051409481588894349204600255753504, 48453821695539041308528422349131698946] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 7.154, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 25, :consider the sequence, (x y + x y + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 31, 7, 42, 64, 6, 36, 42, 36, 216, 174, 31, 186, 241, 7, 42, 98, 42, 252, 368, 64, 384, 543, 6, 36, 42, 36, 216, 186, 42, 252, 384, 36, 216, 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 25, 7, 42, 61, 0, 0, 42, 0, 0, 138, 25, 150, 220, 7, 42, 98, 42 , 252, 362, 61, 366, 555, 0, 0, 42, 0, 0, 150, 42, 252, 366, 0, 0, 252, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [279735559493400782277913806361690352230628998324154502650854005715091248998327\ 02793355970939788984320000000000000, 279735559493400782277913806361690352230628\ 99832415450265085400571509124899832702793355970939788984320000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 64, 543, 4671, 39270, 330420, 2773815, 23285985, 195439821, 1640315118, 13766732514, 115540222512, 969692984508, 8138325714381, 68302362288702, 573239809645785, 4811017556242698, 40377324152220621, 338873902716074970, 2844059731845134271, 23869279062671259360, 200327185962789626034, 1681281672780125288100, 14110456598822827605621, 118424526149899166888862, 993898978074120766320783, 8341474614503945976901599, 70007314907563792947609963 , 587548888783168414005023559, 4931108944345248862814044041, 41385212167380798288338171235, 347332781625756745824371541078, 2915052379191889965478137403620, 24465097517308823923113930478494, 205327698673242011113872448832031, 1723249368314278247570095208031210, 14462677975665397192876765417003314, 121380606935844035801115112614282369, 1018708413815458897284388233510859368, 8549692233182867385837819107982774828] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 61, 555, 4662, 39363, 330372, 2774358, 23285604, 195443238, 1640313600, 13766754654, 115540209798, 969693084597, 8138325497826, 68302362467127, 573239807229756, 4811017554550527, 40377324130160721, 338873902682564370, 2844059731633707138, 23869279062206859654, 200327185960680725988, 1681281672774814106538, 14110456598802858240909, 118424526149846525841027, 993898978073941483900452, 8341474614503457444379107, 70007314907562207359916969 , 587548888783163974030741407, 4931108944345234909949085924, 41385212167380758767709518539, 347332781625756625162392978927, 2915052379191889623344458211697, 24465097517308822901285880728247, 205327698673242008215909708556073, 1723249368314278239035366253197142, 14462677975665397168655304106789866, 121380606935844035730524771496389115, 1018708413815458897084379239507175121, 8549692233182867385260851027182627231] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.364, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 26, :consider the sequence, (x y + x y + x + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 32, 7, 42, 86, 6, 36, 42, 36, 216, 178, 32, 192, 351, 7, 42, 149, 42, 252, 560, 86, 516, 824, 6, 36, 42, 36, 216, 192, 42, 252, 516, 36, 216 , 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 44, 10, 60, 86, 0, 0, 60, 0, 0, 235, 44, 264, 330, 10, 60, 140 , 60, 360, 545, 86, 516, 836, 0, 0, 60, 0, 0, 264, 60, 360, 516, 0, 0, 360, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [161882451752427645777103597992252045002160385605011820035879989175970403922774\ 2029729454523064232894701383681310720000, 1618824517524276457771035979922520450\ 0216038560501182003587998917597040392277420297294545230642328947013836813107200\ 00] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 86, 824, 7510, 67807, 603353, 5358482, 47524774, 421270888, 3733269764, 33080131676, 293101909486, 2596916891080, 23008674385319, 203855416752713, 1806140335046164, 16002212793116926, 141777804813824537, 1256134903974001094, 11129207356568800075, 98603457171311361544, 873614891958444024932, 7740123745471728248909, 68576572330468198436350, 607580239852574707970251, 5383088335752972843706541, 47693519439367862246442743, 422558883043642601574711097, 3743821207900250925650001394, 33169808506693792419079754936, 293880539463710306860247206286, 2603746459749225516426139296676, 23068882474484522326144943834620, 204387541889692203035453489024048, 1810849195903926431344483323451967, 16043907470934275583948774031707445, 142147102871673593492123144612000533, 1259406344208142235719905770541008768, 11158189704805234281446781881565969392, 98860227329296622258406325288539781603] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 86, 836, 7576, 67729, 603368, 5358596, 47523922, 421270345, 3733279121, 33080109917, 293101886872, 2596917269227, 23008674822740, 203855413532165, 1806140347207120, 16002212818030630, 141777804726360233, 1256134904102715470, 11129207357856841360, 98603457169267953877, 873614891958428544338, 7740123745512976547030, 68576572330459892468779, 607580239852427950932973, 5383088335754231466726302, 47693519439369052448926529, 422558883043636545290840443, 3743821207900279395738687547, 33169808506693880663418542069, 293880539463710119793893536695, 2603746459749226018713436042060, 23068882474484526025524120678775, 204387541889692200031184912229410, 1810849195903926433387493152938536, 16043907470934275712580587509441668, 142147102871673593522200278929860159, 1259406344208142235493464681737275074, 11158189704805234285128163254169418825, 98860227329296622263960413976059331324] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.856, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 27, :consider the sequence, (x y + x y + x + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 5, 6, 36, 27, 5, 30, 97, 6, 36, 30, 36, 216, 155, 27, 162, 387, 5, 30, 146, 30, 180, 574, 97, 582, 887, 6, 36, 30, 36, 216, 162, 30, 180, 582, 36, 216 , 180, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 11, 0, 0, 48, 11, 66, 85, 0, 0, 66, 0, 0, 254, 48, 288, 330, 11, 66, 110 , 66, 396, 490, 85, 510, 893, 0, 0, 66, 0, 0, 288, 66, 396, 510, 0, 0, 396, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [130379139191452975111121482686061764938153519792692781573006129910641636654328\ 5259529012474710578340369955417384550400, 1303791391914529751111214826861262156\ 1669447133892196186354012978397682147731155713603986051188993246438045826154496\ 00] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 97, 887, 8263, 74891, 670936, 5991884, 53381032, 474926597, 4224204247, 37558277354, 333908004007, 2968379264507, 26387540188447, 234569819804780, 2085174528625132, 18535786251383531, 164770310647709080, 1464692659709630006, 13020086945990863795, 115739385486233062511, 1028841368945122419928, 9145672452475779710879, 81298560925011837580108, 722686709218150461044660, 6424173698961090170125399, 57106360292307572004036920, 507635150350645691966428618, 4512517421455396761370122506, 40113088028679707782968516730, 356576979245023064845681413773, 3169717126412637485410150812244, 28176543203523716056087758516386, 250469538833579668231382637776389, 2226497034422318469407376956815328, 19791983757207527934072179402346133, 175936736042554151394250166003458976, 1563953137239324315611506454073775468, 13902437151544103152070720801869287800, 123582832599291603126143616160899272506] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 11, 85, 893, 8254, 74693, 671470, 5990759, 53372281, 474963335, 4224054931, 37558375406, 333908699305, 2968373782892, 26387554078978, 234569814284912, 2085174378845353, 18535787038264727, 164770308818456026, 1464692659791222362, 13020086969832260581, 115739385384351774872, 1028841369179068520656, 9145672452699694851836, 81298560921583341914806, 722686709232767803484972, 6424173698936035948578910, 57106360292262176781162176, 507635150351176321257028747, 4512517421453492551785488183, 40113088028682512325935696704, 356576979245032444073227187882, 3169717126412560315120082777197, 28176543203523965981594936620556, 250469538833579391536287153605907, 2226497034422316780900984117037253, 19791983757207538942952842885128091, 175936736042554118953100563190786675, 1563953137239324336401582742829595494, 13902437151544103429837672170142060645, 123582832599291601558243741556643874894] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.887, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 28, :consider the sequence, (x y + x y + x + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 12, 6, 36, 54, 6, 36, 36, 36, 216, 24, 12, 72, 108, 6, 36, 72, 36, 216, 108, 54, 324, 486, 6, 36, 36, 36, 216, 72, 36, 216, 324, 36, 216, 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 12, 6, 36, 54, 0, 0, 36, 0, 0, 24, 12, 72, 108, 6, 36, 72, 36, 216, 108, 54, 324, 486, 0, 0, 36, 0, 0, 72, 36, 216, 324, 0, 0, 216, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [966306636225087453637048766703947960402700428515963135931050949587815641956743\ 510217308480995328, 96630663622508745363704876670394796040270042851596313593105\ 0949587815641956743510217308480995328] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 54, 486, 4374, 39366, 354294, 3188646, 28697814, 258280326, 2324522934, 20920706406, 188286357654, 1694577218886, 15251194969974, 137260754729766, 1235346792567894, 11118121133111046, 100063090197999414, 900567811781994726, 8105110306037952534, 72945992754341572806, 656513934789074155254, 5908625413101667397286, 53177628717915006575574, 478598658461235059180166, 4307387926151115532621494, 38766491335360039793593446, 348898422018240358142341014, 3140085798164163223281069126, 28260772183477469009529622134, 254346949651297221085766599206, 2289122546861674989771899392854, 20602102921755074907947094535686, 185418926295795674171523850821174, 1668770336662161067543714657390566, 15018933029959449607893431916515094, 135170397269635046471040887248635846, 1216533575426715418239367985237722614, 10948802178840438764154311867139503526, 98539219609563948877388806804255531734] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 54, 486, 4374, 39366, 354294, 3188646, 28697814, 258280326, 2324522934, 20920706406, 188286357654, 1694577218886, 15251194969974, 137260754729766, 1235346792567894, 11118121133111046, 100063090197999414, 900567811781994726, 8105110306037952534, 72945992754341572806, 656513934789074155254, 5908625413101667397286, 53177628717915006575574, 478598658461235059180166, 4307387926151115532621494, 38766491335360039793593446, 348898422018240358142341014, 3140085798164163223281069126, 28260772183477469009529622134, 254346949651297221085766599206, 2289122546861674989771899392854, 20602102921755074907947094535686, 185418926295795674171523850821174, 1668770336662161067543714657390566, 15018933029959449607893431916515094, 135170397269635046471040887248635846, 1216533575426715418239367985237722614, 10948802178840438764154311867139503526, 98539219609563948877388806804255531734] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 t - 1 ------- 9 t - 1 and in Maple notation (3*t-1)/(9*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 t - ------- 9 t - 1 and in Maple notation -6*t/(9*t-1) This theorem took, 0.036, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 29, :consider the sequence, (x y + x y + x + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 32, 7, 42, 86, 6, 36, 42, 36, 216, 178, 32, 192, 351, 7, 42, 149, 42, 252, 560, 86, 516, 824, 6, 36, 42, 36, 216, 192, 42, 252, 516, 36, 216 , 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 44, 10, 60, 86, 0, 0, 60, 0, 0, 235, 44, 264, 330, 10, 60, 140 , 60, 360, 545, 86, 516, 836, 0, 0, 60, 0, 0, 264, 60, 360, 516, 0, 0, 360, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [161882451752427645777103597992252045002160385605011820035879989175970403922774\ 2029729454523064232894701383681310720000, 1618824517524276457771035979922520450\ 0216038560501182003587998917597040392277420297294545230642328947013836813107200\ 00] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 86, 824, 7510, 67807, 603353, 5358482, 47524774, 421270888, 3733269764, 33080131676, 293101909486, 2596916891080, 23008674385319, 203855416752713, 1806140335046164, 16002212793116926, 141777804813824537, 1256134903974001094, 11129207356568800075, 98603457171311361544, 873614891958444024932, 7740123745471728248909, 68576572330468198436350, 607580239852574707970251, 5383088335752972843706541, 47693519439367862246442743, 422558883043642601574711097, 3743821207900250925650001394, 33169808506693792419079754936, 293880539463710306860247206286, 2603746459749225516426139296676, 23068882474484522326144943834620, 204387541889692203035453489024048, 1810849195903926431344483323451967, 16043907470934275583948774031707445, 142147102871673593492123144612000533, 1259406344208142235719905770541008768, 11158189704805234281446781881565969392, 98860227329296622258406325288539781603] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 86, 836, 7576, 67729, 603368, 5358596, 47523922, 421270345, 3733279121, 33080109917, 293101886872, 2596917269227, 23008674822740, 203855413532165, 1806140347207120, 16002212818030630, 141777804726360233, 1256134904102715470, 11129207357856841360, 98603457169267953877, 873614891958428544338, 7740123745512976547030, 68576572330459892468779, 607580239852427950932973, 5383088335754231466726302, 47693519439369052448926529, 422558883043636545290840443, 3743821207900279395738687547, 33169808506693880663418542069, 293880539463710119793893536695, 2603746459749226018713436042060, 23068882474484526025524120678775, 204387541889692200031184912229410, 1810849195903926433387493152938536, 16043907470934275712580587509441668, 142147102871673593522200278929860159, 1259406344208142235493464681737275074, 11158189704805234285128163254169418825, 98860227329296622263960413976059331324] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.951, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 30, :consider the sequence, (x y + x y + x + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 28, 7, 42, 77, 6, 36, 42, 36, 216, 147, 28, 168, 311, 7, 42, 149, 42, 252, 439, 77, 462, 724, 6, 36, 42, 36, 216, 168, 42, 252, 462, 36, 216 , 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 37, 10, 60, 92, 0, 0, 60, 0, 0, 180, 37, 222, 308, 10, 60, 140 , 60, 360, 445, 92, 552, 751, 0, 0, 60, 0, 0, 222, 60, 360, 552, 0, 0, 360, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [145275112114807399460676455560526626512925588704855610996528224210641548197272\ 35118751502088923778088374894592000000, 145275112114807399460676455560526626512\ 92558917415095024960381065523698591607418681608967497077190793197780992000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 77, 724, 6383, 56017, 488591, 4263307, 37187624, 324452392, 2830920443, 24702480682, 215561562704, 1881117258016, 16416048165728, 143260649829919, 1250226524661245, 10910704162226932, 95217836659887839, 830968923324633301, 7251900887307475499, 63287702159447822299, 552315264876608748212, 4820087551317588156352, 42065195448707741450579, 367105560238444852175530, 3203753228593799831582816, 27959356972481461239259084, 244003085966488991263843124, 2129430501397075687469084899, 18583676184069047636574636725, 162180932448388453071873640540, 1415363396075082143143608656387, 12351967162885212594737960059477, 107796410209679843216874794192387, 940746192305092199976547383607663, 8209952425012250828806577176767776, 71648782025216053898957064561260176, 625283522062025643801573608697844919, 5456889454882760219635030520485819678, 47622624742778064322271945899776851276] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 92, 751, 6500, 56188, 489392, 4264180, 37193015, 324456019, 2830958450, 24702490195, 215561848571, 1881117219685, 16416050426645, 143260648757866, 1250226543063896, 10910704147718491, 95217836811965276, 830968923158770636, 7251900888577170632, 63287702157694022188, 552315264887312522471, 4820087551299901060135, 42065195448798885123122, 367105560238272092127655, 3203753228594583566355839, 27959356972479811108306801, 244003085966495790304252589, 2129430501397060173005514850, 18583676184069107062716227768, 162180932448388308833858811991, 1415363396075082665732269787852, 12351967162885211264538880794820, 107796410209679847835000730550956, 940746192305092187781220744314436, 8209952425012250869775944110856259, 71648782025216053787637701934381403, 625283522062025644166152400044690226, 5456889454882760218622240981124395275, 47622624742778064325524196135169135471] Using the found enumerative automaton with, 36, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 13 12 11 10 9 8 - (1539 t - 31617 t - 37170 t + 124542 t - 56910 t - 31980 t 7 6 5 4 3 2 / + 29114 t - 2521 t - 3355 t + 1081 t - 31 t - 47 t + 12 t - 1) / ( / 5 4 3 2 (3 t + 1) (9 t + 21 t - 24 t + 2 t + 4 t - 1) 7 6 5 4 3 2 (927 t + 219 t - 1199 t + 445 t + 122 t - 99 t + 18 t - 1)) and in Maple notation -(1539*t^13-31617*t^12-37170*t^11+124542*t^10-56910*t^9-31980*t^8+29114*t^7-\ 2521*t^6-3355*t^5+1081*t^4-31*t^3-47*t^2+12*t-1)/(3*t+1)/(9*t^5+21*t^4-24*t^3+2 *t^2+4*t-1)/(927*t^7+219*t^6-1199*t^5+445*t^4+122*t^3-99*t^2+18*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 12 11 10 9 8 7 - t (51597 t + 138726 t - 172926 t - 126576 t + 179646 t - 1575 t 6 5 4 3 2 / - 53702 t + 12785 t + 5271 t - 2197 t - 33 t + 98 t - 10) / ( / 5 4 3 2 (3 t + 1) (9 t + 21 t - 24 t + 2 t + 4 t - 1) 7 6 5 4 3 2 (927 t + 219 t - 1199 t + 445 t + 122 t - 99 t + 18 t - 1)) and in Maple notation -t*(51597*t^12+138726*t^11-172926*t^10-126576*t^9+179646*t^8-1575*t^7-53702*t^6 +12785*t^5+5271*t^4-2197*t^3-33*t^2+98*t-10)/(3*t+1)/(9*t^5+21*t^4-24*t^3+2*t^2 +4*t-1)/(927*t^7+219*t^6-1199*t^5+445*t^4+122*t^3-99*t^2+18*t-1) This theorem took, 0.429, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 31, :consider the sequence, (x y + x y + x + x y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 5, 6, 36, 25, 5, 30, 82, 6, 36, 30, 36, 216, 143, 25, 150, 323, 5, 30, 146, 30, 180, 507, 82, 492, 724, 6, 36, 30, 36, 216, 150, 30, 180, 492, 36, 216 , 180, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 11, 0, 0, 43, 11, 66, 76, 0, 0, 66, 0, 0, 227, 43, 258, 290, 11, 66, 110 , 66, 396, 441, 76, 456, 736, 0, 0, 66, 0, 0, 258, 66, 396, 456, 0, 0, 396, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [503086733270672346209323293985685434235625055532136716009910498657345821060646\ 15508035631146245067660855591441530880, 503086733270672346209323293985690139937\ 76069663093003014872006434176589255583108619524517775106090115687989410529280] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 82, 724, 6534, 57098, 501416, 4375167, 38212114, 333385033, 2909086452, 25379304377, 221418974480, 1931675361528, 16852151839507, 147019096024795, 1282603597149657, 11189499073077515, 97617771493054865, 851622330339807585, 7429596034714861882, 64816166917535579962, 565459479576591520188, 4933096726210082546729, 43036582109025219820817, 375453290318868494817561, 3275473244237423718808546, 28575392060617970595614698, 249293146528610898110494401, 2174845852396421058987505658, 18973463761663239335977735829, 165525444810669031600099342923, 1444052241801668575917296042881, 12597983829222938082712601602945, 109905439683642301901625681181947, 958820541111698714038582565042585, 8364798254790607904545502454123080, 72974917456627524779443798832385399, 636636822023975655528879162700754482, 5554051409481583763661702287021414365, 48453821695539060226632889120210778448] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 11, 76, 736, 6468, 57338, 500429, 4378467, 38200405, 333432838, 2908916019, 25379969978, 221416687664, 1931684236644, 16852120470121, 147019217593036, 1282603164458514, 11189500729727546, 97617765557495858, 851622352934064306, 7429595953253171707, 64816167225871255798, 565459478459741760339, 4933096730419338240263, 43036582093715821023701, 375453290376365309852613, 3275473244027642030717713, 28575392061403538416713424, 249293146525737261870116952, 2174845852407156935726328554, 18973463761623884248838947628, 165525444810815783464137451845, 1444052241801129702922515738964, 12597983829224944414948944700054, 109905439683634924460044834178262, 958820541111726147589574946406427, 8364798254790506916301452041092184, 72974917456627899933088587787488234, 636636822023974273261917108673164388, 5554051409481588894349204600255753504, 48453821695539041308528422349131698946] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 7.360, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 32, :consider the sequence, (x y + x y + x + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 8, 6, 36, 37, 8, 48, 78, 6, 36, 48, 36, 216, 204, 37, 222, 306, 8, 48, 128, 48, 288, 522, 78, 468, 666, 6, 36, 48, 36, 216, 222, 48, 288, 468, 36, 216 , 288, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 37, 8, 48, 78, 0, 0, 48, 0, 0, 204, 37, 222, 306, 8, 48, 128, 48, 288, 534, 78, 468, 666, 0, 0, 48, 0, 0, 222, 48, 288, 468, 0, 0, 288, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [225543730303375062291505574400322835060982140646906869238277291440957520920349\ 742515280223237650424187197126082560000, 22554373030337506229150557440032283506\ 0982140646906869238277291440957520920349742515280223237650424187197126082560000 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 78, 666, 5778, 50706, 448578, 3990546, 35634978, 319035186, 2861238978, 25690684626, 230853364578, 2075503498866, 18666470795778, 167919872997906, 1510808671996578, 13594456938061746, 122333185783110978, 1100897112091330386, 9907464649081962978, 89163525683297603826, 802449794199038056578, 7221916526087500242066, 64996459004564448575778, 584963392659741715565106, 5264642103649645510384578, 47381608351118630015252946, 426433451669698592668026978, 3837894924085072869196745586, 34541017471112369033877726978, 310868936166091600571541643026, 2797819099051306080743727388578, 25180363932800644780292662105266, 226623227643239143344228652595778, 2039608762277352332027626035252306, 18356477141425371239826043288612578, 165208283958403542667898843425564146, 1486874493739083093067876313798382978, 13381870072332455091951607161995280786, 120436828423076339353608786484816530978] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 78, 666, 5778, 50706, 448578, 3990546, 35634978, 319035186, 2861238978, 25690684626, 230853364578, 2075503498866, 18666470795778, 167919872997906, 1510808671996578, 13594456938061746, 122333185783110978, 1100897112091330386, 9907464649081962978, 89163525683297603826, 802449794199038056578, 7221916526087500242066, 64996459004564448575778, 584963392659741715565106, 5264642103649645510384578, 47381608351118630015252946, 426433451669698592668026978, 3837894924085072869196745586, 34541017471112369033877726978, 310868936166091600571541643026, 2797819099051306080743727388578, 25180363932800644780292662105266, 226623227643239143344228652595778, 2039608762277352332027626035252306, 18356477141425371239826043288612578, 165208283958403542667898843425564146, 1486874493739083093067876313798382978, 13381870072332455091951607161995280786, 120436828423076339353608786484816530978] Using the found enumerative automaton with, 28, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 2 72 t - 12 t + 7 t - 1 - ----------------------- (6 t - 1) (9 t - 1) and in Maple notation -(72*t^3-12*t^2+7*t-1)/(6*t-1)/(9*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 2 (36 t + 21 t - 4) t - ---------------------- (6 t - 1) (9 t - 1) and in Maple notation -2*(36*t^2+21*t-4)*t/(6*t-1)/(9*t-1) This theorem took, 0.310, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 33, :consider the sequence, (x y + x y + x + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 4, 6, 36, 19, 4, 24, 93, 6, 36, 24, 36, 216, 108, 19, 114, 371, 4, 24, 185, 24, 144, 590, 93, 558, 818, 6, 36, 24, 36, 216, 114, 24, 144, 558, 36, 216 , 144, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 13, 0, 0, 52, 13, 78, 87, 0, 0, 78, 0, 0, 270, 52, 312, 323, 13, 78, 104 , 78, 468, 425, 87, 522, 878, 0, 0, 78, 0, 0, 312, 78, 468, 522, 0, 0, 468, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [444424515964846311530976905161491524088067166121010065389622414375724781253322\ 786587299751749537847434871098572800000, 44442451596484631153097695848625278223\ 1642616245215352420979552837094747664239281592981411720070808923733701427200000 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 4, 93, 818, 7709, 68049, 605008, 5332498, 47050902, 414392915, 3650435417, 32144960127, 283067993860, 2492489757103, 21947035078269, 193246047674354, 1701550799306063, 14982266539253220, 131919783409561771, 1161560844597982699, 10227604802565935526, 90054584978018082404, 792935209636557719126, 6981834622745269705209, 61475406301841307954691, 541294050462587302997893, 4766121378922965548342724, 41965938712356211000159703, 369512203055274680931925907, 3253573548561529614990953490, 28647878869670571201408206905, 252246015460080530812026259594, 2221038863019818577546956430453, 19556358985118539813482958073687, 172194725233471792767691051975466, 1516183223093526119326034188227414, 13350069596307880739747672741083666, 117548034770060671029659657764447279, 1035016362919598976768518059621378077, 9113371172954389280156693756686314731, 80243691898496211808647667262174553716] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 13, 87, 878, 7622, 68583, 604117, 5340031, 47032089, 414496385, 3650133719, 32146452990, 283063356532, 2492511273652, 21946963254828, 193246362293705, 1701549701418377, 14982271183129578, 131919766723846276, 1161560913341843221, 10227604550013470667, 90054585999417483857, 792935205826515114893, 6981834637958563591539, 61475406244476183900097, 541294050689550064777576, 4766121378060555291505320, 41965938715746349562546240, 369512203042323167516494250, 3253573548612210951113319465, 28647878869476208831473600631, 252246015460838655654977124640, 2221038863016903308298604253322, 19556358985129885218813426014381, 172194725233428082161841587993683, 1516183223093695955655814677016755, 13350069596307225525295681901307961, 117548034770063213959122741547838389, 1035016362919589156999703480521323995, 9113371172954427360768867118922406098, 80243691898496064657566415308317540028] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.062, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 34, :consider the sequence, (x y + x y + x + x y + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 29, 6, 36, 76, 6, 36, 36, 36, 216, 158, 29, 174, 285, 6, 36, 117, 36, 216, 412, 76, 456, 637, 6, 36, 36, 36, 216, 174, 36, 216, 456, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 32, 9, 54, 70, 0, 0, 54, 0, 0, 170, 32, 192, 252, 9, 54, 108, 54, 324, 400, 70, 420, 682, 0, 0, 54, 0, 0, 192, 54, 324, 420, 0, 0, 324, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [812700186500290082064287764591979046704967466797565197329519805077453539737539\ 471257710117776052308603711992627200, 81270018650029008206428776459197904670496\ 7468883797798707420054392833583935463611049961814844475344751808007372800] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 76, 637, 5818, 49828, 432421, 3720358, 32076625, 276218320, 2379381748, 20492498077, 176501517919, 1520161153012, 13092851232571, 112765673914825, 971225572762222, 8364943217572018, 72045350412047827, 620510122975955122, 5344311875503772908, 46029335963126912752, 396440145223143329365, 3414448307985299847079, 29407862431014726238672, 253283193755816887525030, 2181470223923728190494315, 18788504151964148248479706, 161821089464810376032845504, 1393728036221068268462646841, 12003860840226898522560583972, 103386508218576032813383835371, 890444351521448550628600794949, 7669193561309457420376930456258, 66053010253064298514255150860745, 568899471451885197053549659165165, 4899801044316797202490777787237692, 42200865844745664089299914011951497, 363466406480294133323095469217355426, 3130453036810063657817307637696704568, 26961876093505367179344835176052082245] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 70, 682, 5725, 50203, 431128, 3723817, 32060392, 276253753, 2379189907, 20492881981, 176499320434, 1520165469847, 13092826532797, 112765723861030, 971225299033783, 8364943806756745, 72045347409829444, 620510129994039925, 5344311842783450986, 46029336046863561232, 396440144867371548076, 3414448308980631752065, 29407862427141868402432, 253283193767566451304412, 2181470223881399892403969, 18788504152101701968261564, 161821089464344953789633022, 1393728036222664924884819325, 12003860840221744212089379277, 103386508218594416976111484804, 890444351521391044905411256822, 7669193561309667586851060599065, 66053010253063652484402366033673, 568899471451887585249169550331748, 4899801044316789902148509982786481, 42200865844745691098407632430954258, 363466406480294050452647967439779421, 3130453036810063962211184464352661655, 26961876093505366235709745899695033539] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 33 32 31 30 29 - (946161 t - 8152704 t + 39790476 t - 12187101 t - 85244409 t 28 27 26 25 24 + 91806720 t + 33954279 t - 86786249 t + 63910540 t - 2599941 t 23 22 21 20 19 - 40634567 t + 43512646 t - 35716423 t - 26368695 t + 49861891 t 18 17 16 15 14 + 3187633 t - 27939949 t + 7981717 t + 7152140 t - 5952856 t 13 12 11 10 9 8 + 452136 t + 1513311 t - 609999 t - 77139 t + 60603 t + 113 t 7 6 5 4 3 2 + 12019 t - 12116 t + 2952 t + 571 t - 511 t + 136 t - 18 t + 1) / 17 16 15 14 13 12 / ((459 t - 1689 t + 4936 t - 3369 t - 5367 t + 8678 t / 11 10 9 8 7 6 5 - 2730 t - 3996 t + 3494 t + 1144 t - 1742 t - 248 t + 598 t 4 3 2 17 16 15 - 50 t - 87 t + 17 t + 4 t - 1) (9585 t - 2655 t - 11562 t 14 13 12 11 10 9 + 7065 t + 8961 t + 11766 t + 8958 t - 7466 t - 3968 t 8 7 6 5 4 3 2 + 2522 t - 54 t - 1454 t + 922 t + 268 t - 411 t + 141 t - 20 t + 1 )) and in Maple notation -(946161*t^33-8152704*t^32+39790476*t^31-12187101*t^30-85244409*t^29+91806720*t ^28+33954279*t^27-86786249*t^26+63910540*t^25-2599941*t^24-40634567*t^23+ 43512646*t^22-35716423*t^21-26368695*t^20+49861891*t^19+3187633*t^18-27939949*t ^17+7981717*t^16+7152140*t^15-5952856*t^14+452136*t^13+1513311*t^12-609999*t^11 -77139*t^10+60603*t^9+113*t^8+12019*t^7-12116*t^6+2952*t^5+571*t^4-511*t^3+136* t^2-18*t+1)/(459*t^17-1689*t^16+4936*t^15-3369*t^14-5367*t^13+8678*t^12-2730*t^ 11-3996*t^10+3494*t^9+1144*t^8-1742*t^7-248*t^6+598*t^5-50*t^4-87*t^3+17*t^2+4* t-1)/(9585*t^17-2655*t^16-11562*t^15+7065*t^14+8961*t^13+11766*t^12+8958*t^11-\ 7466*t^10-3968*t^9+2522*t^8-54*t^7-1454*t^6+922*t^5+268*t^4-411*t^3+141*t^2-20* t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 32 31 30 29 28 - t (1003671 t + 2509056 t - 25031061 t + 17310141 t + 41754051 t 27 26 25 24 23 - 59920872 t - 13952226 t + 29784109 t - 32141563 t + 23652700 t 22 21 20 19 18 + 8119383 t - 30056505 t + 42076643 t + 14079002 t - 45055040 t 17 16 15 14 13 - 956559 t + 23329735 t - 5317102 t - 6484386 t + 4641306 t 12 11 10 9 8 7 - 56193 t - 1353762 t + 343053 t + 159216 t + 33588 t - 71059 t 6 5 4 3 2 / - 12846 t + 25724 t - 6004 t - 1295 t + 838 t - 146 t + 9) / (( / 17 16 15 14 13 12 11 459 t - 1689 t + 4936 t - 3369 t - 5367 t + 8678 t - 2730 t 10 9 8 7 6 5 4 3 - 3996 t + 3494 t + 1144 t - 1742 t - 248 t + 598 t - 50 t - 87 t 2 17 16 15 14 13 + 17 t + 4 t - 1) (9585 t - 2655 t - 11562 t + 7065 t + 8961 t 12 11 10 9 8 7 6 + 11766 t + 8958 t - 7466 t - 3968 t + 2522 t - 54 t - 1454 t 5 4 3 2 + 922 t + 268 t - 411 t + 141 t - 20 t + 1)) and in Maple notation -t*(1003671*t^32+2509056*t^31-25031061*t^30+17310141*t^29+41754051*t^28-\ 59920872*t^27-13952226*t^26+29784109*t^25-32141563*t^24+23652700*t^23+8119383*t ^22-30056505*t^21+42076643*t^20+14079002*t^19-45055040*t^18-956559*t^17+ 23329735*t^16-5317102*t^15-6484386*t^14+4641306*t^13-56193*t^12-1353762*t^11+ 343053*t^10+159216*t^9+33588*t^8-71059*t^7-12846*t^6+25724*t^5-6004*t^4-1295*t^ 3+838*t^2-146*t+9)/(459*t^17-1689*t^16+4936*t^15-3369*t^14-5367*t^13+8678*t^12-\ 2730*t^11-3996*t^10+3494*t^9+1144*t^8-1742*t^7-248*t^6+598*t^5-50*t^4-87*t^3+17 *t^2+4*t-1)/(9585*t^17-2655*t^16-11562*t^15+7065*t^14+8961*t^13+11766*t^12+8958 *t^11-7466*t^10-3968*t^9+2522*t^8-54*t^7-1454*t^6+922*t^5+268*t^4-411*t^3+141*t ^2-20*t+1) This theorem took, 0.609, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 35, :consider the sequence, (x y + x y + x + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 5, 6, 36, 25, 5, 30, 82, 6, 36, 30, 36, 216, 143, 25, 150, 323, 5, 30, 146, 30, 180, 507, 82, 492, 724, 6, 36, 30, 36, 216, 150, 30, 180, 492, 36, 216 , 180, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 11, 0, 0, 43, 11, 66, 76, 0, 0, 66, 0, 0, 227, 43, 258, 290, 11, 66, 110 , 66, 396, 441, 76, 456, 736, 0, 0, 66, 0, 0, 258, 66, 396, 456, 0, 0, 396, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [503086733270672346209323293985685434235625055532136716009910498657345821060646\ 15508035631146245067660855591441530880, 503086733270672346209323293985690139937\ 76069663093003014872006434176589255583108619524517775106090115687989410529280] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 82, 724, 6534, 57098, 501416, 4375167, 38212114, 333385033, 2909086452, 25379304377, 221418974480, 1931675361528, 16852151839507, 147019096024795, 1282603597149657, 11189499073077515, 97617771493054865, 851622330339807585, 7429596034714861882, 64816166917535579962, 565459479576591520188, 4933096726210082546729, 43036582109025219820817, 375453290318868494817561, 3275473244237423718808546, 28575392060617970595614698, 249293146528610898110494401, 2174845852396421058987505658, 18973463761663239335977735829, 165525444810669031600099342923, 1444052241801668575917296042881, 12597983829222938082712601602945, 109905439683642301901625681181947, 958820541111698714038582565042585, 8364798254790607904545502454123080, 72974917456627524779443798832385399, 636636822023975655528879162700754482, 5554051409481583763661702287021414365, 48453821695539060226632889120210778448] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 11, 76, 736, 6468, 57338, 500429, 4378467, 38200405, 333432838, 2908916019, 25379969978, 221416687664, 1931684236644, 16852120470121, 147019217593036, 1282603164458514, 11189500729727546, 97617765557495858, 851622352934064306, 7429595953253171707, 64816167225871255798, 565459478459741760339, 4933096730419338240263, 43036582093715821023701, 375453290376365309852613, 3275473244027642030717713, 28575392061403538416713424, 249293146525737261870116952, 2174845852407156935726328554, 18973463761623884248838947628, 165525444810815783464137451845, 1444052241801129702922515738964, 12597983829224944414948944700054, 109905439683634924460044834178262, 958820541111726147589574946406427, 8364798254790506916301452041092184, 72974917456627899933088587787488234, 636636822023974273261917108673164388, 5554051409481588894349204600255753504, 48453821695539041308528422349131698946] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 7.420, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 36, :consider the sequence, (x y + x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 36, 6, 36, 72, 6, 36, 36, 36, 216, 216, 36, 216, 288, 6, 36, 180, 36, 216, 486, 72, 432, 594, 6, 36, 36, 36, 216, 216, 36, 216, 432, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 12, 0, 0, 36, 12, 72, 72, 0, 0, 72, 0, 0, 180, 36, 216, 288, 12, 72, 144 , 72, 432, 594, 72, 432, 594, 0, 0, 72, 0, 0, 216, 72, 432, 432, 0, 0, 432, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [240547113170926548891412063901604051286575230271157810934411248487020110838051\ 861868500776290154664395770796441600000, 24054711317092654889141206390160405128\ 6575230271157810934411248487020110838051861868500776290154664395770796441600000 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 72, 594, 5022, 43254, 377622, 3328614, 29537622, 263319174, 2354756022, 21102104934, 189374748822, 1701107565894, 15290377052022, 137495847222054, 1236757347521622, 11126584462833414, 100113870176333622, 900872491651999974, 8106938385257984022, 72956961229661761734, 656579745640995288822, 5909020278213194198694, 53179997908584167384022, 478612873605250024030854, 4307473217015205321725622, 38767003080544578528218214, 348901492489347590550089622, 3140104220990806617727560774, 28260882720437329376208572022, 254347612873056383285840298534, 2289126526192229962972341588822, 20602126797738404747149747711494, 185419069551695653206739769876022, 1668771196197560941755010171719654, 15018938187171848853161205002489622, 135170428212909441942647525764483014, 1216533761086361791069007816332805622, 10948803292798317001132150853710001574, 98539226293311218299255840723678520022] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 12, 72, 594, 5022, 43254, 377622, 3328614, 29537622, 263319174, 2354756022, 21102104934, 189374748822, 1701107565894, 15290377052022, 137495847222054, 1236757347521622, 11126584462833414, 100113870176333622, 900872491651999974, 8106938385257984022, 72956961229661761734, 656579745640995288822, 5909020278213194198694, 53179997908584167384022, 478612873605250024030854, 4307473217015205321725622, 38767003080544578528218214, 348901492489347590550089622, 3140104220990806617727560774, 28260882720437329376208572022, 254347612873056383285840298534, 2289126526192229962972341588822, 20602126797738404747149747711494, 185419069551695653206739769876022, 1668771196197560941755010171719654, 15018938187171848853161205002489622, 135170428212909441942647525764483014, 1216533761086361791069007816332805622, 10948803292798317001132150853710001574, 98539226293311218299255840723678520022] Using the found enumerative automaton with, 18, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 3 2 162 t - 36 t + 9 t - 1 - ------------------------ (9 t - 1) (6 t - 1) and in Maple notation -(162*t^3-36*t^2+9*t-1)/(9*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 6 t (27 t - 18 t + 2) ---------------------- (9 t - 1) (6 t - 1) and in Maple notation 6*t*(27*t^2-18*t+2)/(9*t-1)/(6*t-1) This theorem took, 0.175, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 37, :consider the sequence, (x y + x y + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 28, 6, 36, 77, 6, 36, 36, 36, 216, 158, 28, 168, 278, 6, 36, 117, 36, 216, 392, 77, 462, 675, 6, 36, 36, 36, 216, 168, 36, 216, 462, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 31, 9, 54, 74, 0, 0, 54, 0, 0, 155, 31, 186, 239, 9, 54, 108, 54, 324, 389, 74, 444, 693, 0, 0, 54, 0, 0, 186, 54, 324, 444, 0, 0, 324, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [299613328556701823530503346293847151895738346180461719251586341044783223611642\ 420477973383027920171690927063040000, 29961332855670182353050334629384715189573\ 8346246482098268362802025778019661713375629549029608511802709072936960000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 77, 675, 6008, 51123, 437885, 3728367, 31781705, 270657096, 2305411847, 19634024127, 167218901039, 1424131240374, 12128787405905, 103295842120815, 879728753140760, 7492287071972223, 63808732900927568, 543432711515831550, 4628192874635482940, 39416414154644556363, 335693381308138687670, 2858962395611823954759, 24348606332265308720441, 207367061161355482791072, 1766059932873189011879561, 15040805751380445405161403, 128096353611566235670259540, 1090943934753636280122575421, 9291120591881330820609418535, 79128650980679560753914685515, 673906160630705513821943193431, 5739381471910396393893973079913, 48879950361766715601051600129659, 416290424161906335326880215484042, 3545374247852355649686925912433057, 30194493622187648132933992767879603, 257154077782512045316397790049647290, 2190075466991156096083962694928877438, 18651971582473286269213914227306180150] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 74, 693, 5921, 51279, 436925, 3730485, 31770341, 270683154, 2305273112, 19634338788, 167217144833, 1424134982826, 12128764736933, 103295885970105, 879728454664877, 7492287573941589, 63808728905198318, 543432717060143445, 4628192820335702552, 39416414212277856309, 335693380560116346638, 2858962396144448887455, 24348606321829202115566, 207367061164981153316613, 1766059932725842337550884, 15040805751376776885944916, 128096353609462281691896881, 1090943934752799827143364103, 9291120591850967197707422582, 79128650980656373193044368723, 673906160630262911633673251414, 5739381471909902023915030051311, 48879950361760203482766704832593, 416290424161896878540062986260340, 3545374247852259006814684014888206, 30194493622187477538101216138157414, 257154077782510599763184489971512887, 2190075466991153128092643602187551090, 18651971582473264492742934266050318820] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 33 32 31 30 29 (122688 t + 2102544 t + 617992 t - 12420845 t + 9805007 t 28 27 26 25 + 17375873 t - 18181150 t - 17054084 t - 33480173 t 24 23 22 21 + 155723414 t - 116864259 t - 92129126 t + 164836827 t 20 19 18 17 16 - 18331059 t - 83845136 t + 32768709 t + 24262132 t - 16929582 t 15 14 13 12 11 - 375168 t + 1902016 t - 775410 t + 838228 t - 36292 t 10 9 8 7 6 5 - 385443 t + 83768 t + 91321 t - 38580 t - 5409 t + 5963 t 4 3 2 / 17 16 15 - 1051 t - 111 t + 77 t - 14 t + 1) / ((1008 t - 932 t + 4237 t / 14 13 12 11 10 9 + 1830 t - 18173 t + 20725 t + 487 t - 14044 t + 9005 t 8 7 6 5 4 3 2 - 1360 t - 1135 t + 1079 t - 530 t + 7 t + 95 t - 19 t - 4 t + 1) ( 17 16 15 14 13 12 11 1080 t + 126 t - 1219 t + 1728 t + 183 t + 2221 t - 9585 t 10 9 8 7 6 5 4 + 3806 t + 4425 t - 1606 t - 2715 t + 1531 t + 382 t - 325 t 3 2 - 65 t + 75 t - 16 t + 1)) and in Maple notation (122688*t^33+2102544*t^32+617992*t^31-12420845*t^30+9805007*t^29+17375873*t^28-\ 18181150*t^27-17054084*t^26-33480173*t^25+155723414*t^24-116864259*t^23-\ 92129126*t^22+164836827*t^21-18331059*t^20-83845136*t^19+32768709*t^18+24262132 *t^17-16929582*t^16-375168*t^15+1902016*t^14-775410*t^13+838228*t^12-36292*t^11 -385443*t^10+83768*t^9+91321*t^8-38580*t^7-5409*t^6+5963*t^5-1051*t^4-111*t^3+ 77*t^2-14*t+1)/(1008*t^17-932*t^16+4237*t^15+1830*t^14-18173*t^13+20725*t^12+ 487*t^11-14044*t^10+9005*t^9-1360*t^8-1135*t^7+1079*t^6-530*t^5+7*t^4+95*t^3-19 *t^2-4*t+1)/(1080*t^17+126*t^16-1219*t^15+1728*t^14+183*t^13+2221*t^12-9585*t^ 11+3806*t^10+4425*t^9-1606*t^8-2715*t^7+1531*t^6+382*t^5-325*t^4-65*t^3+75*t^2-\ 16*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 32 31 30 29 28 - t (412992 t + 2033712 t + 2700016 t - 12129143 t + 6729889 t 27 26 25 24 23 + 24953665 t - 45709397 t + 3245682 t + 15033849 t + 64438613 t 22 21 20 19 - 76383606 t - 63073078 t + 116848389 t + 10207855 t 18 17 16 15 14 - 81835973 t + 14465720 t + 35670075 t - 15204248 t - 6610764 t 13 12 11 10 9 + 5594997 t - 1289457 t + 81262 t + 560897 t - 540268 t 8 7 6 5 4 3 2 + 48591 t + 135182 t - 52083 t - 7805 t + 8492 t - 1247 t - 293 t / 17 16 15 14 13 + 106 t - 9) / ((1008 t - 932 t + 4237 t + 1830 t - 18173 t / 12 11 10 9 8 7 6 + 20725 t + 487 t - 14044 t + 9005 t - 1360 t - 1135 t + 1079 t 5 4 3 2 17 16 15 - 530 t + 7 t + 95 t - 19 t - 4 t + 1) (1080 t + 126 t - 1219 t 14 13 12 11 10 9 8 + 1728 t + 183 t + 2221 t - 9585 t + 3806 t + 4425 t - 1606 t 7 6 5 4 3 2 - 2715 t + 1531 t + 382 t - 325 t - 65 t + 75 t - 16 t + 1)) and in Maple notation -t*(412992*t^32+2033712*t^31+2700016*t^30-12129143*t^29+6729889*t^28+24953665*t ^27-45709397*t^26+3245682*t^25+15033849*t^24+64438613*t^23-76383606*t^22-\ 63073078*t^21+116848389*t^20+10207855*t^19-81835973*t^18+14465720*t^17+35670075 *t^16-15204248*t^15-6610764*t^14+5594997*t^13-1289457*t^12+81262*t^11+560897*t^ 10-540268*t^9+48591*t^8+135182*t^7-52083*t^6-7805*t^5+8492*t^4-1247*t^3-293*t^2 +106*t-9)/(1008*t^17-932*t^16+4237*t^15+1830*t^14-18173*t^13+20725*t^12+487*t^ 11-14044*t^10+9005*t^9-1360*t^8-1135*t^7+1079*t^6-530*t^5+7*t^4+95*t^3-19*t^2-4 *t+1)/(1080*t^17+126*t^16-1219*t^15+1728*t^14+183*t^13+2221*t^12-9585*t^11+3806 *t^10+4425*t^9-1606*t^8-2715*t^7+1531*t^6+382*t^5-325*t^4-65*t^3+75*t^2-16*t+1) This theorem took, 0.593, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 38, :consider the sequence, (x y + x y + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 32, 7, 42, 86, 6, 36, 42, 36, 216, 178, 32, 192, 351, 7, 42, 149, 42, 252, 560, 86, 516, 824, 6, 36, 42, 36, 216, 192, 42, 252, 516, 36, 216 , 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 44, 10, 60, 86, 0, 0, 60, 0, 0, 235, 44, 264, 330, 10, 60, 140 , 60, 360, 545, 86, 516, 836, 0, 0, 60, 0, 0, 264, 60, 360, 516, 0, 0, 360, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [161882451752427645777103597992252045002160385605011820035879989175970403922774\ 2029729454523064232894701383681310720000, 1618824517524276457771035979922520450\ 0216038560501182003587998917597040392277420297294545230642328947013836813107200\ 00] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 86, 824, 7510, 67807, 603353, 5358482, 47524774, 421270888, 3733269764, 33080131676, 293101909486, 2596916891080, 23008674385319, 203855416752713, 1806140335046164, 16002212793116926, 141777804813824537, 1256134903974001094, 11129207356568800075, 98603457171311361544, 873614891958444024932, 7740123745471728248909, 68576572330468198436350, 607580239852574707970251, 5383088335752972843706541, 47693519439367862246442743, 422558883043642601574711097, 3743821207900250925650001394, 33169808506693792419079754936, 293880539463710306860247206286, 2603746459749225516426139296676, 23068882474484522326144943834620, 204387541889692203035453489024048, 1810849195903926431344483323451967, 16043907470934275583948774031707445, 142147102871673593492123144612000533, 1259406344208142235719905770541008768, 11158189704805234281446781881565969392, 98860227329296622258406325288539781603] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 86, 836, 7576, 67729, 603368, 5358596, 47523922, 421270345, 3733279121, 33080109917, 293101886872, 2596917269227, 23008674822740, 203855413532165, 1806140347207120, 16002212818030630, 141777804726360233, 1256134904102715470, 11129207357856841360, 98603457169267953877, 873614891958428544338, 7740123745512976547030, 68576572330459892468779, 607580239852427950932973, 5383088335754231466726302, 47693519439369052448926529, 422558883043636545290840443, 3743821207900279395738687547, 33169808506693880663418542069, 293880539463710119793893536695, 2603746459749226018713436042060, 23068882474484526025524120678775, 204387541889692200031184912229410, 1810849195903926433387493152938536, 16043907470934275712580587509441668, 142147102871673593522200278929860159, 1259406344208142235493464681737275074, 11158189704805234285128163254169418825, 98860227329296622263960413976059331324] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.123, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 2 n Theorem number, 39, :consider the sequence, (x y + x y + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 34, 7, 42, 77, 6, 36, 42, 36, 216, 188, 34, 204, 254, 7, 42, 149, 42, 252, 482, 77, 462, 658, 6, 36, 42, 36, 216, 204, 42, 252, 462, 36, 216 , 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 10, 0, 0, 34, 10, 60, 77, 0, 0, 60, 0, 0, 182, 34, 204, 278, 10, 60, 140 , 60, 360, 479, 77, 462, 616, 0, 0, 60, 0, 0, 204, 60, 360, 462, 0, 0, 360, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [229781516295022559458128548631867534118914955817804486710471405971837227876448\ 60087653467753218993940040553332736000, 229781516295022559458128548631867534118\ 91495581780448671047140597183722787644860087653467753218993940040553332736000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 77, 658, 5453, 45577, 383042, 3221155, 27055826, 227216059, 1908180023, 16024507120, 134567698754, 1130038007608, 9489505568165, 79688082972061, 669179749589021, 5619426749810446, 47189041619211893, 396269149112027083, 3327663091978623812, 27943990770473022151, 234659156243507250128, 1970546005494631142419, 16547624285577279039755, 138958374228701036732140, 1166900421932692988161220, 9799025082320178502342846, 82287135004282145960537189, 691004720375385955250467555, 5802699578105778252044016869, 48728064223920716891427911914, 409193033525454114007153634849, 3436191060577574803939338323809, 28855351966909247449613030297858, 242312293599288378604299065189575, 2034813080660802541733194209593114, 17087305855290546797668698292345039, 143490340300652094053767061372543627, 1204957524256039523416056980422676812, 10118608905791517807017831176908672890] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 10, 77, 616, 5354, 45529, 383015, 3219214, 27048365, 227203681, 1908160439, 16024394875, 134567198066, 1130036740981, 9489502927574, 79688074280584, 669179714989853, 5619426643986976, 47189041353138638, 396269148353501695, 3327663089370753905, 27943990762067497834, 234659156219802665027, 1970546005427624341369, 16547624285367017909411, 138958374228030030246139, 1166900421930690703058972, 9799025082314378840824303, 82287135004264632564771782, 691004720375331406017812674, 5802699578105612181936634217, 48728064223920224802143886916, 409193033525452641977255098346, 3436191060577570302807817348849, 28855351966909233712933569859043, 242312293599288337344369848717110, 2034813080660802418113077976626309, 17087305855290546423237020752907281, 143490340300652092914778423888316519, 1204957524256039519974641037087677719, 10118608905791517796667976326828715542] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.436, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 40, :consider the sequence, 2 2 2 2 n (x y + x y + x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 8, 6, 36, 33, 8, 48, 73, 6, 36, 48, 36, 216, 186, 33, 198, 300, 8, 48, 128, 48, 288, 478, 73, 438, 659, 6, 36, 48, 36, 216, 198, 48, 288, 438, 36, 216 , 288, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 33, 8, 48, 70, 0, 0, 48, 0, 0, 186, 33, 198, 276, 8, 48, 128, 48, 288, 478, 70, 420, 671, 0, 0, 48, 0, 0, 198, 48, 288, 420, 0, 0, 288, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [282546496429954267733407208961192302243875569498352220724859563129772361065402\ 73485577767126453613384842541531136000, 282546496429954267733407208961192302243\ 87556949835222072485956312977236106540273485577767126453613384842541531136000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 73, 659, 5977, 53882, 485275, 4368551, 39319189, 353883314, 3184963543, 28664770679, 257983013137, 2321848020722, 20896632549835, 188069701137311, 1692627311185669, 15233645874760874, 137102812865138263, 1233925316455595999, 11105327847917178097, 99947950637296115162, 899531555733220041595, 8095784001653474550071, 72862056014853693550549, 655758504134174532853634, 5901826537207283601308983, 53116438834869979939038119, 478047949513826961761351857, 4302431545624482545010967202, 38721883910620315275498652555, 348496955195583196772085206831, 3136472596760248508910469917829, 28228253370842239815860106689594, 254054280337580155890796522371703, 2286488523038221432152395095001039, 20578396707343992866648872113550417, 185205570366095936062156547055936842, 1666850133294863424350318542999134715, 15001651199653770821514414902988467591, 135014860796883937391715817358290271509] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 70, 671, 5974, 53984, 485308, 4369319, 39319582, 353888852, 3184966780, 28664809931, 257983037254, 2321848296944, 20896632723028, 188069703075239, 1692627312411142, 15233645888339492, 137102812873755940, 1233925316550685691, 11105327847977619934, 99947950637961861104, 899531555733643488748, 8095784001658135125959, 72862056014856658743502, 655758504134207157947732, 5901826537207304360848300, 53116438834870208317885451, 478047949513827107087693014, 4302431545624484143672464464, 38721883910620316292811738468, 348496955195583207962744385479, 3136472596760248516031747612662, 28228253370842239894194807033572, 254054280337580155940645724515860, 2286488523038221432700738255689211, 20578396707343992866997817303400494, 185205570366095936065994949955595024, 1666850133294863424352761161652608188, 15001651199653770821541283725610597799, 135014860796883937391732915695838154622] Using the found enumerative automaton with, 28, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (1458 t - 432 t - 2844 t + 1647 t + 1362 t - 591 t - 393 t + 112 t 3 2 / + 60 t - 17 t - 3 t + 1) / ((3 t - 1) (9 t - 1) (t + 1) (3 t + 1) / 2 2 (t - 1) (3 t - 1) (2 t - 1) (7 t - 1)) and in Maple notation (1458*t^11-432*t^10-2844*t^9+1647*t^8+1362*t^7-591*t^6-393*t^5+112*t^4+60*t^3-\ 17*t^2-3*t+1)/(3*t-1)/(9*t-1)/(t+1)/(3*t+1)/(t-1)/(3*t^2-1)/(2*t-1)/(7*t^2-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 10 9 8 7 6 5 4 3 - t (1458 t - 2322 t - 1584 t + 1413 t + 867 t + 384 t - 488 t - 213 t 2 / + 115 t + 18 t - 8) / ((3 t - 1) (9 t - 1) (t + 1) (3 t + 1) (t - 1) / 2 2 (3 t - 1) (2 t - 1) (7 t - 1)) and in Maple notation -t*(1458*t^10-2322*t^9-1584*t^8+1413*t^7+867*t^6+384*t^5-488*t^4-213*t^3+115*t^ 2+18*t-8)/(3*t-1)/(9*t-1)/(t+1)/(3*t+1)/(t-1)/(3*t^2-1)/(2*t-1)/(7*t^2-1) This theorem took, 0.973, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 41, :consider the sequence, 2 2 2 2 n (x y + x y + x y + x y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 8, 6, 36, 33, 8, 48, 73, 6, 36, 48, 36, 216, 186, 33, 198, 300, 8, 48, 128, 48, 288, 478, 73, 438, 659, 6, 36, 48, 36, 216, 198, 48, 288, 438, 36, 216 , 288, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 33, 8, 48, 70, 0, 0, 48, 0, 0, 186, 33, 198, 276, 8, 48, 128, 48, 288, 478, 70, 420, 671, 0, 0, 48, 0, 0, 198, 48, 288, 420, 0, 0, 288, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [282546496429954267733407208961192302243875569498352220724859563129772361065402\ 73485577767126453613384842541531136000, 282546496429954267733407208961192302243\ 87556949835222072485956312977236106540273485577767126453613384842541531136000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 73, 659, 5977, 53882, 485275, 4368551, 39319189, 353883314, 3184963543, 28664770679, 257983013137, 2321848020722, 20896632549835, 188069701137311, 1692627311185669, 15233645874760874, 137102812865138263, 1233925316455595999, 11105327847917178097, 99947950637296115162, 899531555733220041595, 8095784001653474550071, 72862056014853693550549, 655758504134174532853634, 5901826537207283601308983, 53116438834869979939038119, 478047949513826961761351857, 4302431545624482545010967202, 38721883910620315275498652555, 348496955195583196772085206831, 3136472596760248508910469917829, 28228253370842239815860106689594, 254054280337580155890796522371703, 2286488523038221432152395095001039, 20578396707343992866648872113550417, 185205570366095936062156547055936842, 1666850133294863424350318542999134715, 15001651199653770821514414902988467591, 135014860796883937391715817358290271509] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 70, 671, 5974, 53984, 485308, 4369319, 39319582, 353888852, 3184966780, 28664809931, 257983037254, 2321848296944, 20896632723028, 188069703075239, 1692627312411142, 15233645888339492, 137102812873755940, 1233925316550685691, 11105327847977619934, 99947950637961861104, 899531555733643488748, 8095784001658135125959, 72862056014856658743502, 655758504134207157947732, 5901826537207304360848300, 53116438834870208317885451, 478047949513827107087693014, 4302431545624484143672464464, 38721883910620316292811738468, 348496955195583207962744385479, 3136472596760248516031747612662, 28228253370842239894194807033572, 254054280337580155940645724515860, 2286488523038221432700738255689211, 20578396707343992866997817303400494, 185205570366095936065994949955595024, 1666850133294863424352761161652608188, 15001651199653770821541283725610597799, 135014860796883937391732915695838154622] Using the found enumerative automaton with, 28, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 11 10 9 8 7 6 5 4 (1458 t - 432 t - 2844 t + 1647 t + 1362 t - 591 t - 393 t + 112 t 3 2 / + 60 t - 17 t - 3 t + 1) / ((3 t - 1) (9 t - 1) (t + 1) (3 t + 1) / 2 2 (t - 1) (3 t - 1) (2 t - 1) (7 t - 1)) and in Maple notation (1458*t^11-432*t^10-2844*t^9+1647*t^8+1362*t^7-591*t^6-393*t^5+112*t^4+60*t^3-\ 17*t^2-3*t+1)/(3*t-1)/(9*t-1)/(t+1)/(3*t+1)/(t-1)/(3*t^2-1)/(2*t-1)/(7*t^2-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 10 9 8 7 6 5 4 3 - t (1458 t - 2322 t - 1584 t + 1413 t + 867 t + 384 t - 488 t - 213 t 2 / + 115 t + 18 t - 8) / ((3 t - 1) (9 t - 1) (t + 1) (3 t + 1) (t - 1) / 2 2 (3 t - 1) (2 t - 1) (7 t - 1)) and in Maple notation -t*(1458*t^10-2322*t^9-1584*t^8+1413*t^7+867*t^6+384*t^5-488*t^4-213*t^3+115*t^ 2+18*t-8)/(3*t-1)/(9*t-1)/(t+1)/(3*t+1)/(t-1)/(3*t^2-1)/(2*t-1)/(7*t^2-1) This theorem took, 0.468, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 42, :consider the sequence, 2 2 2 2 2 n (x y + x y + x y + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 29, 6, 36, 76, 6, 36, 36, 36, 216, 158, 29, 174, 285, 6, 36, 117, 36, 216, 412, 76, 456, 637, 6, 36, 36, 36, 216, 174, 36, 216, 456, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 32, 9, 54, 70, 0, 0, 54, 0, 0, 170, 32, 192, 252, 9, 54, 108, 54, 324, 400, 70, 420, 682, 0, 0, 54, 0, 0, 192, 54, 324, 420, 0, 0, 324, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [812700186500290082064287764591979046704967466797565197329519805077453539737539\ 471257710117776052308603711992627200, 81270018650029008206428776459197904670496\ 7468883797798707420054392833583935463611049961814844475344751808007372800] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 76, 637, 5818, 49828, 432421, 3720358, 32076625, 276218320, 2379381748, 20492498077, 176501517919, 1520161153012, 13092851232571, 112765673914825, 971225572762222, 8364943217572018, 72045350412047827, 620510122975955122, 5344311875503772908, 46029335963126912752, 396440145223143329365, 3414448307985299847079, 29407862431014726238672, 253283193755816887525030, 2181470223923728190494315, 18788504151964148248479706, 161821089464810376032845504, 1393728036221068268462646841, 12003860840226898522560583972, 103386508218576032813383835371, 890444351521448550628600794949, 7669193561309457420376930456258, 66053010253064298514255150860745, 568899471451885197053549659165165, 4899801044316797202490777787237692, 42200865844745664089299914011951497, 363466406480294133323095469217355426, 3130453036810063657817307637696704568, 26961876093505367179344835176052082245] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 70, 682, 5725, 50203, 431128, 3723817, 32060392, 276253753, 2379189907, 20492881981, 176499320434, 1520165469847, 13092826532797, 112765723861030, 971225299033783, 8364943806756745, 72045347409829444, 620510129994039925, 5344311842783450986, 46029336046863561232, 396440144867371548076, 3414448308980631752065, 29407862427141868402432, 253283193767566451304412, 2181470223881399892403969, 18788504152101701968261564, 161821089464344953789633022, 1393728036222664924884819325, 12003860840221744212089379277, 103386508218594416976111484804, 890444351521391044905411256822, 7669193561309667586851060599065, 66053010253063652484402366033673, 568899471451887585249169550331748, 4899801044316789902148509982786481, 42200865844745691098407632430954258, 363466406480294050452647967439779421, 3130453036810063962211184464352661655, 26961876093505366235709745899695033539] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 33 32 31 30 29 - (946161 t - 8152704 t + 39790476 t - 12187101 t - 85244409 t 28 27 26 25 24 + 91806720 t + 33954279 t - 86786249 t + 63910540 t - 2599941 t 23 22 21 20 19 - 40634567 t + 43512646 t - 35716423 t - 26368695 t + 49861891 t 18 17 16 15 14 + 3187633 t - 27939949 t + 7981717 t + 7152140 t - 5952856 t 13 12 11 10 9 8 + 452136 t + 1513311 t - 609999 t - 77139 t + 60603 t + 113 t 7 6 5 4 3 2 + 12019 t - 12116 t + 2952 t + 571 t - 511 t + 136 t - 18 t + 1) / 17 16 15 14 13 12 / ((459 t - 1689 t + 4936 t - 3369 t - 5367 t + 8678 t / 11 10 9 8 7 6 5 - 2730 t - 3996 t + 3494 t + 1144 t - 1742 t - 248 t + 598 t 4 3 2 17 16 15 - 50 t - 87 t + 17 t + 4 t - 1) (9585 t - 2655 t - 11562 t 14 13 12 11 10 9 + 7065 t + 8961 t + 11766 t + 8958 t - 7466 t - 3968 t 8 7 6 5 4 3 2 + 2522 t - 54 t - 1454 t + 922 t + 268 t - 411 t + 141 t - 20 t + 1 )) and in Maple notation -(946161*t^33-8152704*t^32+39790476*t^31-12187101*t^30-85244409*t^29+91806720*t ^28+33954279*t^27-86786249*t^26+63910540*t^25-2599941*t^24-40634567*t^23+ 43512646*t^22-35716423*t^21-26368695*t^20+49861891*t^19+3187633*t^18-27939949*t ^17+7981717*t^16+7152140*t^15-5952856*t^14+452136*t^13+1513311*t^12-609999*t^11 -77139*t^10+60603*t^9+113*t^8+12019*t^7-12116*t^6+2952*t^5+571*t^4-511*t^3+136* t^2-18*t+1)/(459*t^17-1689*t^16+4936*t^15-3369*t^14-5367*t^13+8678*t^12-2730*t^ 11-3996*t^10+3494*t^9+1144*t^8-1742*t^7-248*t^6+598*t^5-50*t^4-87*t^3+17*t^2+4* t-1)/(9585*t^17-2655*t^16-11562*t^15+7065*t^14+8961*t^13+11766*t^12+8958*t^11-\ 7466*t^10-3968*t^9+2522*t^8-54*t^7-1454*t^6+922*t^5+268*t^4-411*t^3+141*t^2-20* t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 32 31 30 29 28 - t (1003671 t + 2509056 t - 25031061 t + 17310141 t + 41754051 t 27 26 25 24 23 - 59920872 t - 13952226 t + 29784109 t - 32141563 t + 23652700 t 22 21 20 19 18 + 8119383 t - 30056505 t + 42076643 t + 14079002 t - 45055040 t 17 16 15 14 13 - 956559 t + 23329735 t - 5317102 t - 6484386 t + 4641306 t 12 11 10 9 8 7 - 56193 t - 1353762 t + 343053 t + 159216 t + 33588 t - 71059 t 6 5 4 3 2 / - 12846 t + 25724 t - 6004 t - 1295 t + 838 t - 146 t + 9) / (( / 17 16 15 14 13 12 11 459 t - 1689 t + 4936 t - 3369 t - 5367 t + 8678 t - 2730 t 10 9 8 7 6 5 4 3 - 3996 t + 3494 t + 1144 t - 1742 t - 248 t + 598 t - 50 t - 87 t 2 17 16 15 14 13 + 17 t + 4 t - 1) (9585 t - 2655 t - 11562 t + 7065 t + 8961 t 12 11 10 9 8 7 6 + 11766 t + 8958 t - 7466 t - 3968 t + 2522 t - 54 t - 1454 t 5 4 3 2 + 922 t + 268 t - 411 t + 141 t - 20 t + 1)) and in Maple notation -t*(1003671*t^32+2509056*t^31-25031061*t^30+17310141*t^29+41754051*t^28-\ 59920872*t^27-13952226*t^26+29784109*t^25-32141563*t^24+23652700*t^23+8119383*t ^22-30056505*t^21+42076643*t^20+14079002*t^19-45055040*t^18-956559*t^17+ 23329735*t^16-5317102*t^15-6484386*t^14+4641306*t^13-56193*t^12-1353762*t^11+ 343053*t^10+159216*t^9+33588*t^8-71059*t^7-12846*t^6+25724*t^5-6004*t^4-1295*t^ 3+838*t^2-146*t+9)/(459*t^17-1689*t^16+4936*t^15-3369*t^14-5367*t^13+8678*t^12-\ 2730*t^11-3996*t^10+3494*t^9+1144*t^8-1742*t^7-248*t^6+598*t^5-50*t^4-87*t^3+17 *t^2+4*t-1)/(9585*t^17-2655*t^16-11562*t^15+7065*t^14+8961*t^13+11766*t^12+8958 *t^11-7466*t^10-3968*t^9+2522*t^8-54*t^7-1454*t^6+922*t^5+268*t^4-411*t^3+141*t ^2-20*t+1) This theorem took, 0.578, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 43, :consider the sequence, 2 2 2 2 2 n (x y + x y + x y + x y + y + x) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 7, 6, 36, 31, 7, 42, 64, 6, 36, 42, 36, 216, 174, 31, 186, 241, 7, 42, 98, 42, 252, 368, 64, 384, 543, 6, 36, 42, 36, 216, 186, 42, 252, 384, 36, 216, 252, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 25, 7, 42, 61, 0, 0, 42, 0, 0, 138, 25, 150, 220, 7, 42, 98, 42 , 252, 362, 61, 366, 555, 0, 0, 42, 0, 0, 150, 42, 252, 366, 0, 0, 252, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [279735559493400782277913806361690352230628998324154502650854005715091248998327\ 02793355970939788984320000000000000, 279735559493400782277913806361690352230628\ 99832415450265085400571509124899832702793355970939788984320000000000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 64, 543, 4671, 39270, 330420, 2773815, 23285985, 195439821, 1640315118, 13766732514, 115540222512, 969692984508, 8138325714381, 68302362288702, 573239809645785, 4811017556242698, 40377324152220621, 338873902716074970, 2844059731845134271, 23869279062671259360, 200327185962789626034, 1681281672780125288100, 14110456598822827605621, 118424526149899166888862, 993898978074120766320783, 8341474614503945976901599, 70007314907563792947609963 , 587548888783168414005023559, 4931108944345248862814044041, 41385212167380798288338171235, 347332781625756745824371541078, 2915052379191889965478137403620, 24465097517308823923113930478494, 205327698673242011113872448832031, 1723249368314278247570095208031210, 14462677975665397192876765417003314, 121380606935844035801115112614282369, 1018708413815458897284388233510859368, 8549692233182867385837819107982774828] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 61, 555, 4662, 39363, 330372, 2774358, 23285604, 195443238, 1640313600, 13766754654, 115540209798, 969693084597, 8138325497826, 68302362467127, 573239807229756, 4811017554550527, 40377324130160721, 338873902682564370, 2844059731633707138, 23869279062206859654, 200327185960680725988, 1681281672774814106538, 14110456598802858240909, 118424526149846525841027, 993898978073941483900452, 8341474614503457444379107, 70007314907562207359916969 , 587548888783163974030741407, 4931108944345234909949085924, 41385212167380758767709518539, 347332781625756625162392978927, 2915052379191889623344458211697, 24465097517308822901285880728247, 205327698673242008215909708556073, 1723249368314278239035366253197142, 14462677975665397168655304106789866, 121380606935844035730524771496389115, 1018708413815458897084379239507175121, 8549692233182867385260851027182627231] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.162, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 44, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 5, 6, 36, 27, 5, 30, 97, 6, 36, 30, 36, 216, 155, 27, 162, 387, 5, 30, 146, 30, 180, 574, 97, 582, 887, 6, 36, 30, 36, 216, 162, 30, 180, 582, 36, 216 , 180, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 11, 0, 0, 48, 11, 66, 85, 0, 0, 66, 0, 0, 254, 48, 288, 330, 11, 66, 110 , 66, 396, 490, 85, 510, 893, 0, 0, 66, 0, 0, 288, 66, 396, 510, 0, 0, 396, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [130379139191452975111121482686061764938153519792692781573006129910641636654328\ 5259529012474710578340369955417384550400, 1303791391914529751111214826861262156\ 1669447133892196186354012978397682147731155713603986051188993246438045826154496\ 00] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 5, 97, 887, 8263, 74891, 670936, 5991884, 53381032, 474926597, 4224204247, 37558277354, 333908004007, 2968379264507, 26387540188447, 234569819804780, 2085174528625132, 18535786251383531, 164770310647709080, 1464692659709630006, 13020086945990863795, 115739385486233062511, 1028841368945122419928, 9145672452475779710879, 81298560925011837580108, 722686709218150461044660, 6424173698961090170125399, 57106360292307572004036920, 507635150350645691966428618, 4512517421455396761370122506, 40113088028679707782968516730, 356576979245023064845681413773, 3169717126412637485410150812244, 28176543203523716056087758516386, 250469538833579668231382637776389, 2226497034422318469407376956815328, 19791983757207527934072179402346133, 175936736042554151394250166003458976, 1563953137239324315611506454073775468, 13902437151544103152070720801869287800, 123582832599291603126143616160899272506] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 11, 85, 893, 8254, 74693, 671470, 5990759, 53372281, 474963335, 4224054931, 37558375406, 333908699305, 2968373782892, 26387554078978, 234569814284912, 2085174378845353, 18535787038264727, 164770308818456026, 1464692659791222362, 13020086969832260581, 115739385384351774872, 1028841369179068520656, 9145672452699694851836, 81298560921583341914806, 722686709232767803484972, 6424173698936035948578910, 57106360292262176781162176, 507635150351176321257028747, 4512517421453492551785488183, 40113088028682512325935696704, 356576979245032444073227187882, 3169717126412560315120082777197, 28176543203523965981594936620556, 250469538833579391536287153605907, 2226497034422316780900984117037253, 19791983757207538942952842885128091, 175936736042554118953100563190786675, 1563953137239324336401582742829595494, 13902437151544103429837672170142060645, 123582832599291601558243741556643874894] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.725, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 45, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 28, 6, 36, 77, 6, 36, 36, 36, 216, 158, 28, 168, 278, 6, 36, 117, 36, 216, 392, 77, 462, 675, 6, 36, 36, 36, 216, 168, 36, 216, 462, 36, 216 , 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 31, 9, 54, 74, 0, 0, 54, 0, 0, 155, 31, 186, 239, 9, 54, 108, 54, 324, 389, 74, 444, 693, 0, 0, 54, 0, 0, 186, 54, 324, 444, 0, 0, 324, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [299613328556701823530503346293847151895738346180461719251586341044783223611642\ 420477973383027920171690927063040000, 29961332855670182353050334629384715189573\ 8346246482098268362802025778019661713375629549029608511802709072936960000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 77, 675, 6008, 51123, 437885, 3728367, 31781705, 270657096, 2305411847, 19634024127, 167218901039, 1424131240374, 12128787405905, 103295842120815, 879728753140760, 7492287071972223, 63808732900927568, 543432711515831550, 4628192874635482940, 39416414154644556363, 335693381308138687670, 2858962395611823954759, 24348606332265308720441, 207367061161355482791072, 1766059932873189011879561, 15040805751380445405161403, 128096353611566235670259540, 1090943934753636280122575421, 9291120591881330820609418535, 79128650980679560753914685515, 673906160630705513821943193431, 5739381471910396393893973079913, 48879950361766715601051600129659, 416290424161906335326880215484042, 3545374247852355649686925912433057, 30194493622187648132933992767879603, 257154077782512045316397790049647290, 2190075466991156096083962694928877438, 18651971582473286269213914227306180150] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 74, 693, 5921, 51279, 436925, 3730485, 31770341, 270683154, 2305273112, 19634338788, 167217144833, 1424134982826, 12128764736933, 103295885970105, 879728454664877, 7492287573941589, 63808728905198318, 543432717060143445, 4628192820335702552, 39416414212277856309, 335693380560116346638, 2858962396144448887455, 24348606321829202115566, 207367061164981153316613, 1766059932725842337550884, 15040805751376776885944916, 128096353609462281691896881, 1090943934752799827143364103, 9291120591850967197707422582, 79128650980656373193044368723, 673906160630262911633673251414, 5739381471909902023915030051311, 48879950361760203482766704832593, 416290424161896878540062986260340, 3545374247852259006814684014888206, 30194493622187477538101216138157414, 257154077782510599763184489971512887, 2190075466991153128092643602187551090, 18651971582473264492742934266050318820] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 33 32 31 30 29 (122688 t + 2102544 t + 617992 t - 12420845 t + 9805007 t 28 27 26 25 + 17375873 t - 18181150 t - 17054084 t - 33480173 t 24 23 22 21 + 155723414 t - 116864259 t - 92129126 t + 164836827 t 20 19 18 17 16 - 18331059 t - 83845136 t + 32768709 t + 24262132 t - 16929582 t 15 14 13 12 11 - 375168 t + 1902016 t - 775410 t + 838228 t - 36292 t 10 9 8 7 6 5 - 385443 t + 83768 t + 91321 t - 38580 t - 5409 t + 5963 t 4 3 2 / 17 16 15 - 1051 t - 111 t + 77 t - 14 t + 1) / ((1008 t - 932 t + 4237 t / 14 13 12 11 10 9 + 1830 t - 18173 t + 20725 t + 487 t - 14044 t + 9005 t 8 7 6 5 4 3 2 - 1360 t - 1135 t + 1079 t - 530 t + 7 t + 95 t - 19 t - 4 t + 1) ( 17 16 15 14 13 12 11 1080 t + 126 t - 1219 t + 1728 t + 183 t + 2221 t - 9585 t 10 9 8 7 6 5 4 + 3806 t + 4425 t - 1606 t - 2715 t + 1531 t + 382 t - 325 t 3 2 - 65 t + 75 t - 16 t + 1)) and in Maple notation (122688*t^33+2102544*t^32+617992*t^31-12420845*t^30+9805007*t^29+17375873*t^28-\ 18181150*t^27-17054084*t^26-33480173*t^25+155723414*t^24-116864259*t^23-\ 92129126*t^22+164836827*t^21-18331059*t^20-83845136*t^19+32768709*t^18+24262132 *t^17-16929582*t^16-375168*t^15+1902016*t^14-775410*t^13+838228*t^12-36292*t^11 -385443*t^10+83768*t^9+91321*t^8-38580*t^7-5409*t^6+5963*t^5-1051*t^4-111*t^3+ 77*t^2-14*t+1)/(1008*t^17-932*t^16+4237*t^15+1830*t^14-18173*t^13+20725*t^12+ 487*t^11-14044*t^10+9005*t^9-1360*t^8-1135*t^7+1079*t^6-530*t^5+7*t^4+95*t^3-19 *t^2-4*t+1)/(1080*t^17+126*t^16-1219*t^15+1728*t^14+183*t^13+2221*t^12-9585*t^ 11+3806*t^10+4425*t^9-1606*t^8-2715*t^7+1531*t^6+382*t^5-325*t^4-65*t^3+75*t^2-\ 16*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 32 31 30 29 28 - t (412992 t + 2033712 t + 2700016 t - 12129143 t + 6729889 t 27 26 25 24 23 + 24953665 t - 45709397 t + 3245682 t + 15033849 t + 64438613 t 22 21 20 19 - 76383606 t - 63073078 t + 116848389 t + 10207855 t 18 17 16 15 14 - 81835973 t + 14465720 t + 35670075 t - 15204248 t - 6610764 t 13 12 11 10 9 + 5594997 t - 1289457 t + 81262 t + 560897 t - 540268 t 8 7 6 5 4 3 2 + 48591 t + 135182 t - 52083 t - 7805 t + 8492 t - 1247 t - 293 t / 17 16 15 14 13 + 106 t - 9) / ((1008 t - 932 t + 4237 t + 1830 t - 18173 t / 12 11 10 9 8 7 6 + 20725 t + 487 t - 14044 t + 9005 t - 1360 t - 1135 t + 1079 t 5 4 3 2 17 16 15 - 530 t + 7 t + 95 t - 19 t - 4 t + 1) (1080 t + 126 t - 1219 t 14 13 12 11 10 9 8 + 1728 t + 183 t + 2221 t - 9585 t + 3806 t + 4425 t - 1606 t 7 6 5 4 3 2 - 2715 t + 1531 t + 382 t - 325 t - 65 t + 75 t - 16 t + 1)) and in Maple notation -t*(412992*t^32+2033712*t^31+2700016*t^30-12129143*t^29+6729889*t^28+24953665*t ^27-45709397*t^26+3245682*t^25+15033849*t^24+64438613*t^23-76383606*t^22-\ 63073078*t^21+116848389*t^20+10207855*t^19-81835973*t^18+14465720*t^17+35670075 *t^16-15204248*t^15-6610764*t^14+5594997*t^13-1289457*t^12+81262*t^11+560897*t^ 10-540268*t^9+48591*t^8+135182*t^7-52083*t^6-7805*t^5+8492*t^4-1247*t^3-293*t^2 +106*t-9)/(1008*t^17-932*t^16+4237*t^15+1830*t^14-18173*t^13+20725*t^12+487*t^ 11-14044*t^10+9005*t^9-1360*t^8-1135*t^7+1079*t^6-530*t^5+7*t^4+95*t^3-19*t^2-4 *t+1)/(1080*t^17+126*t^16-1219*t^15+1728*t^14+183*t^13+2221*t^12-9585*t^11+3806 *t^10+4425*t^9-1606*t^8-2715*t^7+1531*t^6+382*t^5-325*t^4-65*t^3+75*t^2-16*t+1) This theorem took, 0.603, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 46, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + x y + y ) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 6, 6, 6, 36, 27, 6, 36, 42, 6, 36, 36, 36, 216, 153, 27, 162, 144, 6, 36, 72, 36, 216, 186, 42, 252, 264, 6, 36, 36, 36, 216, 162, 36, 216, 252, 36, 216, 216, 216, 1296] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 9, 6, 36, 42, 0, 0, 36, 0, 0, 27, 9, 54, 72, 6, 36, 72, 36, 216 , 186, 42, 252, 264, 0, 0, 36, 0, 0, 54, 36, 216, 252, 0, 0, 216, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [881081690996846456728566157801394786708421517753374882404256386202655813702315\ 18305558709567695618048000, 881081690996846456728566157801394786708421517753374\ 88240425638620265581370231518305558709567695618048000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 42, 264, 1608, 9696, 58272, 349824, 2099328, 12596736, 75581952, 453494784, 2720974848, 16325861376, 97955192832, 587731206144, 3526387335168, 21158324207616, 126949945638912, 761699674619904, 4570198049292288, 27421188298899456, 164527129799688192, 987162778810712064, 5922976672889438208, 35537860037386960896, 213227160224422428672, 1279362961346735898624, 7676177768080818044928, 46057066608485713575936, 276342399650915892068352, 1658054397905498573635584, 9948326387432997884264448, 59689958324598000190488576, 358139749947588026912735232, 2148838499685528213016018944, 12893030998113169381175328768, 77358185988679016493210402816, 464149115932074099371579277312, 2784894695592444597054109384704, 16709368173554667583973923749888] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 42, 264, 1608, 9696, 58272, 349824, 2099328, 12596736, 75581952, 453494784, 2720974848, 16325861376, 97955192832, 587731206144, 3526387335168, 21158324207616, 126949945638912, 761699674619904, 4570198049292288, 27421188298899456, 164527129799688192, 987162778810712064, 5922976672889438208, 35537860037386960896, 213227160224422428672, 1279362961346735898624, 7676177768080818044928, 46057066608485713575936, 276342399650915892068352, 1658054397905498573635584, 9948326387432997884264448, 59689958324598000190488576, 358139749947588026912735232, 2148838499685528213016018944, 12893030998113169381175328768, 77358185988679016493210402816, 464149115932074099371579277312, 2784894695592444597054109384704, 16709368173554667583973923749888] Using the found enumerative automaton with, 8, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 6 t - 2 t + 1 ------------------- (2 t - 1) (6 t - 1) and in Maple notation (6*t^2-2*t+1)/(2*t-1)/(6*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 (t - 1) t - ------------------- (2 t - 1) (6 t - 1) and in Maple notation -6*(t-1)*t/(2*t-1)/(6*t-1) This theorem took, 0.090, seconds. to state and prove ------------------------------------------------------------------ This concludes this webbook, that took, 125.419, seconds. to generate. k is , 7 Counting the Occurrences of Coefficients that Appear in the Expansion of, n P(x, y) , modolu , 3 For all Polynomials that are Sums of, 7, Monomials taken from, 2 2 2 2 2 2 {1, x, y, x , y , x y, x y , x y, x y } By Shalosh B. Ekhad In this webbook, we will consider the sequences described in the title, that\ after normalization and weeding out obvious symmetry, concerns the following set of, 21, polynomials 2 2 2 2 2 2 {x y + x + x y + y + x + y + 1, x y + x y + x y + y + x + y + 1, 2 2 2 2 2 2 2 2 x y + x y + x + y + x + y + 1, x y + x y + x + x y + y + y + 1, 2 2 2 2 2 2 2 2 x y + x y + x + x y + y + x + y, x y + x + x y + y + x + y + 1, 2 2 2 2 2 2 2 2 2 x y + x y + x y + y + x + y + 1, x y + x y + x + y + x + y + 1, 2 2 2 2 2 2 2 2 2 x y + x y + x + x y + x + y + 1, x y + x y + x + x y + y + y + 1, 2 2 2 2 2 2 2 2 2 2 x y + x y + x + x y + y + x + 1, x y + x y + x + x y + y + x + y, 2 2 2 2 2 2 2 2 2 x y + x y + x y + y + x + y + 1, x y + x y + x y + x y + x + y + 1, 2 2 2 2 2 x y + x y + x y + x y + y + y + 1, 2 2 2 2 2 x y + x y + x y + x y + y + x + 1, 2 2 2 2 2 x y + x y + x y + x y + y + x + y, 2 2 2 2 2 2 x y + x y + x y + x + y + y + 1, 2 2 2 2 2 2 x y + x y + x y + x + y + x + y, 2 2 2 2 2 2 x y + x y + x y + x + x y + y + 1, 2 2 2 2 2 2 x y + x y + x y + x + x y + y + y} by finding enumerative automata with at most, 500, states . ----------------------------------------------------------------------------\ ---- 2 2 2 n Theorem number, 1, :consider the sequence, (x y + x + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 21, 7, 49, 81, 7, 49, 49, 49, 343, 99, 21, 147, 298, 7, 49, 130, 49, 343, 455, 81, 567, 624, 7, 49, 49, 49, 343, 147, 49, 343, 567, 49, 343 , 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 41, 9, 63, 71, 0, 0, 63, 0, 0, 269, 41, 287, 186, 9, 63, 126, 63, 441, 391, 71, 497, 659, 0, 0, 63, 0, 0, 287, 63, 441, 497, 0, 0, 441, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [701829909586202358220405128773680326529304566036156000331514484497476575819536\ 921209737694374779656400837015083417600, 70182990958620235822040513180305155679\ 5925835788071317484923688152887416825990775485167714082906331274582423083417600 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 81, 624, 5274, 40735, 323587, 2508220, 19613499, 152276865, 1185521616, 9209868160, 71609782399, 556428616111, 4324778184390, 33607215723393, 261178742366592, 2029630556491372, 15772754928467869, 122571679267852618, 952524728856309120, 7402185089524606677, 57523422930203255871, 447021851686862509534, 3473866514809219528960, 26995866405501753190258, 209788423657881871595928, 1630293131055607684320975, 12669220954867389245322996 , 98454166651748220612289048, 765100173964643964845608774, 5945693180497749631190383795, 46204757041916789209028369289, 359063190525343146702768271473, 2790326869582125236655981590166, 21683993885526074281451247313828, 168509143597369407187510954650241, 1309506524176998378956018374120831, 10176345927687512365348864696280811, 79081710945200360983898146272429144, 614554286073690117860627198859533418] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 71, 659, 5102, 41388, 320589, 2520696, 19559993, 152513105, 1184546177, 9214298064, 71591778582, 556511227752, 4324443932660, 33608752490897, 261172521752504, 2029659116944650, 15772639058824632, 122572209909189561, 952522570058465606, 7402194948333769655, 57523382708302084574, 447022034863195652622, 3473865765448185751341, 26995869809139410409771, 209788409697720057966245, 1630293194303480960132126, 12669220694815135907477357 , 98454167827119668113633437, 765100169120654031077430573, 5945693202341590251235821753, 46204756951693219173142317938, 359063190931324356575915828567, 2790326867901726005209142914079, 21683993893071863098735757682501, 168509143566073901665098991084941, 1309506524317255210723436403976524, 10176345927104698727504411988820130, 79081710947807493232276547463853739, 614554286062836943182540234070881104] Using the found enumerative automaton with, 20, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 19 18 17 16 15 14 - (30312 t + 283584 t + 302412 t - 1260696 t - 140827 t + 1187455 t 13 12 11 10 9 8 + 245487 t - 801748 t - 122131 t + 392697 t - 50439 t - 81405 t 7 6 5 4 3 2 / + 32648 t + 323 t - 3365 t + 1056 t - 62 t - 49 t + 13 t - 1) / (( / 10 9 8 7 6 5 4 1179 t + 3590 t - 2772 t - 4149 t + 2684 t + 1241 t - 1030 t 3 2 10 9 8 7 6 + 76 t + 68 t - 16 t + 1) (357 t - 192 t - 1470 t + 1315 t + 518 t 5 4 3 2 - 801 t + 78 t + 114 t - 24 t - 4 t + 1)) and in Maple notation -(30312*t^19+283584*t^18+302412*t^17-1260696*t^16-140827*t^15+1187455*t^14+ 245487*t^13-801748*t^12-122131*t^11+392697*t^10-50439*t^9-81405*t^8+32648*t^7+ 323*t^6-3365*t^5+1056*t^4-62*t^3-49*t^2+13*t-1)/(1179*t^10+3590*t^9-2772*t^8-\ 4149*t^7+2684*t^6+1241*t^5-1030*t^4+76*t^3+68*t^2-16*t+1)/(357*t^10-192*t^9-\ 1470*t^8+1315*t^7+518*t^6-801*t^5+78*t^4+114*t^3-24*t^2-4*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 18 17 16 15 14 - (23238 t - 272043 t - 893397 t + 1742172 t + 1781952 t 13 12 11 10 9 - 3812348 t - 290781 t + 2992959 t - 831693 t - 983777 t 8 7 6 5 4 3 + 519829 t + 109339 t - 118134 t + 8529 t + 10446 t - 2308 t 2 / 10 9 8 7 - 211 t + 109 t - 9) t / ((1179 t + 3590 t - 2772 t - 4149 t / 6 5 4 3 2 10 + 2684 t + 1241 t - 1030 t + 76 t + 68 t - 16 t + 1) (357 t 9 8 7 6 5 4 3 2 - 192 t - 1470 t + 1315 t + 518 t - 801 t + 78 t + 114 t - 24 t - 4 t + 1)) and in Maple notation -(23238*t^18-272043*t^17-893397*t^16+1742172*t^15+1781952*t^14-3812348*t^13-\ 290781*t^12+2992959*t^11-831693*t^10-983777*t^9+519829*t^8+109339*t^7-118134*t^ 6+8529*t^5+10446*t^4-2308*t^3-211*t^2+109*t-9)*t/(1179*t^10+3590*t^9-2772*t^8-\ 4149*t^7+2684*t^6+1241*t^5-1030*t^4+76*t^3+68*t^2-16*t+1)/(357*t^10-192*t^9-\ 1470*t^8+1315*t^7+518*t^6-801*t^5+78*t^4+114*t^3-24*t^2-4*t+1) This theorem took, 0.803, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 2, :consider the sequence, 2 2 2 n (x y + x y + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 9, 7, 49, 33, 9, 63, 85, 7, 49, 63, 49, 343, 159, 33, 231, 313, 9, 63, 145, 63, 441, 511, 85, 595, 753, 7, 49, 63, 49, 343, 231, 63, 441, 595, 49, 343 , 441, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 38, 8, 56, 84, 0, 0, 56, 0, 0, 266, 38, 266, 270, 8, 56, 144, 56, 392, 516, 84, 588, 752, 0, 0, 56, 0, 0, 266, 56, 392, 588, 0, 0, 392, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [120490515368853116055012272117064194840063460904515273772464835041785853822731\ 913375827838054587096052116773295891405625, 12049051536885311605501227211706419\ 4840063461050565743921121482467485634024081434077826112074506564050793533176679\ 296250] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 9, 85, 753, 6433, 54183, 453397, 3781737, 31490149, 261986907, 2178668347, 18113652603, 150582215095, 1251753123489, 10405262795257, 86493312850875, 718968411139237, 5976351346186185, 49677761950737979, 412940761118069799, 3432522716520092821, 28532449853842264107, 237172700135679437053, 1971470721262991275779, 16387623047252463847765, 136220227162413422364171, 1132314934001417937398623, 9412237356075981864624555, 78238138020374050756971505, 650345503516580517091146699, 5405922031236453915541270111, 44936103735784783980336784665, 373526182465878899992902349507, 3104893335007019597393162539161, 25809067943317346641156507077205, 214534902244742905969596449711019, 1783296645285773229895390370992687, 14823447801808384098344104340113971, 123218201141087531659071478357220317, 1024237093518602896621286249867274045, 8513852775194635155075915528035860471] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 84, 752, 6432, 54182, 453396, 3781736, 31490148, 261986906, 2178668346, 18113652602, 150582215094, 1251753123488, 10405262795256, 86493312850874, 718968411139236, 5976351346186184, 49677761950737978, 412940761118069798, 3432522716520092820, 28532449853842264106, 237172700135679437052, 1971470721262991275778, 16387623047252463847764, 136220227162413422364170, 1132314934001417937398622, 9412237356075981864624554, 78238138020374050756971504, 650345503516580517091146698, 5405922031236453915541270110, 44936103735784783980336784664, 373526182465878899992902349506, 3104893335007019597393162539160, 25809067943317346641156507077204, 214534902244742905969596449711018, 1783296645285773229895390370992686, 14823447801808384098344104340113970, 123218201141087531659071478357220316, 1024237093518602896621286249867274044, 8513852775194635155075915528035860470] Using the found enumerative automaton with, 22, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 8 7 6 5 4 3 2 / - (90 t + 81 t - 540 t + 313 t + 66 t - 117 t + 51 t - 11 t + 1) / ( / 8 7 6 5 4 3 2 (90 t + 225 t - 1004 t + 523 t + 282 t - 357 t + 127 t - 19 t + 1) (t - 1)) and in Maple notation -(90*t^8+81*t^7-540*t^6+313*t^5+66*t^4-117*t^3+51*t^2-11*t+1)/(90*t^8+225*t^7-\ 1004*t^6+523*t^5+282*t^4-357*t^3+127*t^2-19*t+1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 / 2 t (72 t - 232 t + 105 t + 108 t - 120 t + 38 t - 4) / ( / 8 7 6 5 4 3 2 (90 t + 225 t - 1004 t + 523 t + 282 t - 357 t + 127 t - 19 t + 1) (t - 1)) and in Maple notation 2*t*(72*t^6-232*t^5+105*t^4+108*t^3-120*t^2+38*t-4)/(90*t^8+225*t^7-1004*t^6+ 523*t^5+282*t^4-357*t^3+127*t^2-19*t+1)/(t-1) This theorem took, 0.294, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 3, :consider the sequence, (x y + x y + x + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 38, 7, 49, 94, 7, 49, 49, 49, 343, 224, 38, 266, 338, 7, 49, 193, 49, 343, 594, 94, 658, 753, 7, 49, 49, 49, 343, 266, 49, 343, 658, 49, 343 , 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 12, 0, 0, 37, 12, 84, 72, 0, 0, 84, 0, 0, 223, 37, 259, 277, 12, 84, 168 , 84, 588, 545, 72, 504, 788, 0, 0, 84, 0, 0, 259, 84, 588, 504, 0, 0, 588, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [166878279185797473775922545788465213220012767928933764865219620818446560520976\ 8324213831048613416398815212770500000000000, 1668782791857974737759225457884652\ 1322001276775585442399229840474477844574421859563304563106315046854121523545000\ 00000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 94, 753, 6780, 59563, 521542, 4572468, 40003056, 349974001, 3060886612, 26768207391, 234084121248, 2046974621152, 17899785746074, 156523679981607, 1368707493311598, 11968516886090461, 104657299285252390, 915162952932467544, 8002527604882250130, 69977084250369348031, 611905638025516602850, 5350730024195478915375, 46788767699161492544466, 409138328417426786317492, 3577657161648034264136830, 31284359820851117459453169, 273562032797571953546106060, 2392127761641476875691433283, 20917651337709620097806079562, 182911692348572759729461782090, 1599447597875345796137018559306, 13986162311227743370194693718975, 122300184408985618530959522011216, 1069438118379102454086069282620997, 9351563078384847133160217602200968, 81773531824056280193602559041126744, 715058055091875060724863594423014668, 6252732525340880683836986604177813297, 54676209511801435973278597618156464324] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 12, 72, 788, 6797, 59871, 524511, 4581167, 40060610, 350208552, 3062039427, 26773881533, 234111132305, 2047109862345, 17900446249218, 156526948294754, 1368723634139387, 11968596517760337, 104657692915663875, 915164896062219971, 8002537202726522216, 69977131644720702990, 611905872040398033021, 5350731179688690913169, 46788773404050767413295, 409138356583781600911161, 3577657300707884615527674, 31284360507397929211935284, 273562036187091835282221017, 2392127778375669060879431295, 20917651420327166054961445935, 182911692756459801865720249193, 1599447599889107259104742010472, 13986162321169806491441443691502, 122300184458070213965290195203699, 1069438118621436327178674600893819, 9351563079581265631466823387142337, 81773531829963079021177056347996877, 715058055121037326876017007771206360, 6252732525484856771148675436543671224, 54676209512512255692034692301175057945] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.183, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 4, :consider the sequence, 2 2 2 2 n (x y + x y + x + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 10, 7, 49, 43, 10, 70, 90, 7, 49, 70, 49, 343, 246, 43, 301, 357, 10, 70 , 181, 70, 490, 631, 90, 630, 838, 7, 49, 70, 49, 343, 301, 70, 490, 630, 49, 343, 490, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 39, 9, 63, 95, 0, 0, 63, 0, 0, 245, 39, 273, 347, 9, 63, 180, 63, 441, 636, 95, 665, 846, 0, 0, 63, 0, 0, 273, 63, 441, 665, 0, 0, 441, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [365381315957325060464479714217866779830677024911792414041376933136459215107774\ 43492474371379285269628413324297609440000000, 365381315957325060464479714217866\ 7798306770249117924140414536133979085922590650245531772030383642995536651993060\ 9440000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 10, 90, 838, 7551, 66715, 593103, 5255629, 46561707, 412426051, 3652790190, 32351573941, 286525074663, 2537629123027, 22474661908662, 199048063275106, 1762879668965754, 15613035826427320, 138277664414291667, 1224663326354086516, 10846294437744449496, 96060770785013111503, 850767212775119130141, 7534864072083330932386, 66732915585196771078971, 591023538076476943395097, 5234430707971688801545017, 46359007842772664690028499, 410580964403021912870362695, 3636331668320131484265284965, 32205360570647217567328696959, 285228451112400274103864862235, 2526140613936980552907754794996, 22372895748986084342268610738291, 198146714966480173976765410474308, 1754896688051101358994746891207425, 15542333801767671694111700285527581, 137651487777231819418757552860934332, 1219117561683717748493559926042762583, 10797178099599781468427496615892426333, 95625769473347902948968645670016336286] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 95, 846, 7529, 67164, 594038, 5260713, 46579844, 412485192, 3653007152, 32352337707, 286527422885, 2537636651025, 22474683708026, 199048123550529, 1762879823627300, 15613036160813487, 138277664850564386, 1224663325582492476, 10846294427505680246, 96060770727070358292, 850767212509432475312, 7534864070987950400163, 66732915580985319771650, 591023538061132793988687, 5234430707918166584928515, 46359007842593626340972418, 410580964402447051024602824, 3636331668318367219760302485, 32205360570642089198688771728, 285228451112386419366226978128, 2526140613936947195060013525743, 22372895748986021060857178092920, 198146714966480133611283094763114, 1754896688051101807032828836502501, 15542333801767675098329411761837985, 137651487777231836772865613003262102, 1219117561683717824327510414966728193, 10797178099599781771864999983396554625, 95625769473347904090997685265625772159] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.302, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 5, :consider the sequence, 2 2 2 2 n (x y + x y + x + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 13, 7, 49, 49, 13, 91, 55, 7, 49, 91, 49, 343, 319, 49, 343, 235, 13, 91 , 205, 91, 637, 721, 55, 385, 655, 7, 49, 91, 49, 343, 343, 91, 637, 385, 49, 343, 637, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 36, 6, 42, 96, 0, 0, 42, 0, 0, 228, 36, 252, 432, 6, 42, 156, 42, 294, 630, 96, 672, 870, 0, 0, 42, 0, 0, 252, 42, 294, 672, 0, 0, 294, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [618869548076643352989836641621052118974015163355333768515913304439539891999626\ 13592381230512100677962156190340105751894169, 618869548076643352989836637788924\ 1116069842386726482113690025162978595737525321067171308799888115962480082935415\ 1231121456] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 13, 55, 655, 7213, 62065, 500011, 4577251, 41728753, 365250373, 3200827711, 28532169271, 253627443061, 2241364306009, 19835129242387, 175916646389707, 1559006691715129, 13805908941462445, 122301753196289191, 1083707134247078815, 9601151037002326333, 85054756877761727041, 753533764885069263355, 6676026918540481658035, 59145466792867130320513, 523988365119940359343381, 4642229962406886822895375, 41127513517153403860234759, 364364746311640786641290245, 3228048962524087973082053929, 28598589384536668017118500835, 253366444750613695407404397211, 2244674143535372457461102632393, 19886462711820809583797390619517, 176182139568174044870360683882615, 1560868078685852898119191919084911, 13828353741462361805281442599260109, 122510912150645029892384956107606289, 1085373164086547444025217159312053451, 9615754793375804292148436733286789507, 85189815201621151995327245821632728593] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 96, 870, 5892, 56634, 543084, 4704378, 40344576, 362545854, 3244255800, 28578986286, 252295179036, 2241006544482, 19875154514532, 175896095816130, 1557828623816664, 13807441113323046, 122335704127383792, 1083634937028333366, 9600194475145555764, 85057657051629256458, 753560040226895496924, 6675919896762152910474, 59144766391440977816112, 523992104401352714910222, 4642247940879885918724392, 41127387934336246200565950, 364364307917744208621311436, 3228053051148785171743366194, 28598599273787410940554553556, 253366315056303986211910943442, 2244673949681400614414648283912, 19886466732330886187832283444854, 176182142215004693012232014023392, 1560867956673786321146073046885830, 13828353757684173267962081142314724, 122510915778075302511735643855476186, 1085373160772674214030059870915801164, 9615754687741010440237208046270594714, 85189815382896893009202849212769446496] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 8 7 6 5 4 3 2 - (15066 t - 1323 t + 1611 t + 183 t + 265 t + 215 t - 39 t + 5 t + 1) / 2 3 2 / ((9 t - 2 t - 1) (30 t - 11 t - 8 t + 1) / 4 3 2 (189 t - 78 t + 34 t - 2 t + 1)) and in Maple notation -(15066*t^8-1323*t^7+1611*t^6+183*t^5+265*t^4+215*t^3-39*t^2+5*t+1)/(9*t^2-2*t-\ 1)/(30*t^3-11*t^2-8*t+1)/(189*t^4-78*t^3+34*t^2-2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 / - 6 t (3726 t - 1881 t + 690 t + 385 t - 148 t + 27 t + 8 t + 1) / ( / 2 3 2 (9 t - 2 t - 1) (30 t - 11 t - 8 t + 1) 4 3 2 (189 t - 78 t + 34 t - 2 t + 1)) and in Maple notation -6*t*(3726*t^7-1881*t^6+690*t^5+385*t^4-148*t^3+27*t^2+8*t+1)/(9*t^2-2*t-1)/(30 *t^3-11*t^2-8*t+1)/(189*t^4-78*t^3+34*t^2-2*t+1) This theorem took, 0.471, seconds. to state and prove ----------------------------------------------------------------------------\ ---- 2 2 2 2 n Theorem number, 6, :consider the sequence, (x y + x + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 28, 7, 49, 110, 7, 49, 49, 49, 343, 145, 28, 196, 413, 7, 49, 193, 49, 343, 667, 110, 770, 908, 7, 49, 49, 49, 343, 196, 49, 343, 770, 49, 343, 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 12, 0, 0, 48, 12, 84, 91, 0, 0, 84, 0, 0, 306, 48, 336, 310, 12, 84, 168 , 84, 588, 519, 91, 637, 946, 0, 0, 84, 0, 0, 336, 84, 588, 637, 0, 0, 588, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [144895898103912891577807319050136579652271255465112090939717087186435479335511\ 07988064342929281001993146771387915121459200, 144895898103912891577807321301208\ 9290986288104860323289374842339564811866574289751613782952939798711186677138791\ 5121459200] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 110, 908, 8411, 74264, 657677, 5797025, 51149930, 450744785, 3972691757, 35003485757, 308422145885, 2717415644432, 23942460175712, 210948940181366, 1858601875036523, 16375499067410987, 144278876370215354, 1271190891736003721, 11200019858523465170, 98679462909643088519, 869430280658173694714, 7660246410568406200715, 67491754529359296497057, 594646264819440822373709, 5239220445406574750915168, 46160940514986042686184770, 406707916060462992010208783, 3583361324955874685020634255, 31571744433829072240854156524, 278167607499943911792856421276, 2450837584355667814871535077159, 21593473513490457759441809860916, 190252549313726917688355387109493, 1676248728474448048536840416044427, 14768841783446618411538705239731652, 130123103999573681281381205349567998, 1146469197974705642639498088115000832, 10101139471042886131462194259342664990, 88997610048055613553952340817014750138] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 12, 91, 946, 8266, 74194, 656095, 5801266, 51133453, 450827080, 3972332623, 35003793529, 308413645543, 2717414109400, 23942348253571, 210949067433676, 1858600820550781, 16375502045038447, 144278858177210470, 1271190905436397789, 11200019461622337910, 98679462827835927994, 869430275017637628403, 7660246414243971952099, 67491754468634850177034, 594646264923449302185298, 5239220444455071322318156, 46160940515485121636224957, 406707916041485216078152645, 3583361324950397403559419091, 31571744433546275284888437430, 278167607500015034718668205397, 2450837584352310443731722583081, 21593473513493597492519389424041, 190252549313676694512383632913941, 1676248728474459059435729110236790, 14768841783445688550899659027150687, 130123103999573280603591574486084678, 1146469197974691382983707273109933286, 10101139471042884509535671178340874254, 88997610048055431472036997816783418601] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.329, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 7, :consider the sequence, 2 2 2 2 n (x y + x y + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 21, 7, 49, 81, 7, 49, 49, 49, 343, 99, 21, 147, 298, 7, 49, 130, 49, 343, 455, 81, 567, 624, 7, 49, 49, 49, 343, 147, 49, 343, 567, 49, 343 , 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 41, 9, 63, 71, 0, 0, 63, 0, 0, 269, 41, 287, 186, 9, 63, 126, 63, 441, 391, 71, 497, 659, 0, 0, 63, 0, 0, 287, 63, 441, 497, 0, 0, 441, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [701829909586202358220405128773680326529304566036156000331514484497476575819536\ 921209737694374779656400837015083417600, 70182990958620235822040513180305155679\ 5925835788071317484923688152887416825990775485167714082906331274582423083417600 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 81, 624, 5274, 40735, 323587, 2508220, 19613499, 152276865, 1185521616, 9209868160, 71609782399, 556428616111, 4324778184390, 33607215723393, 261178742366592, 2029630556491372, 15772754928467869, 122571679267852618, 952524728856309120, 7402185089524606677, 57523422930203255871, 447021851686862509534, 3473866514809219528960, 26995866405501753190258, 209788423657881871595928, 1630293131055607684320975, 12669220954867389245322996 , 98454166651748220612289048, 765100173964643964845608774, 5945693180497749631190383795, 46204757041916789209028369289, 359063190525343146702768271473, 2790326869582125236655981590166, 21683993885526074281451247313828, 168509143597369407187510954650241, 1309506524176998378956018374120831, 10176345927687512365348864696280811, 79081710945200360983898146272429144, 614554286073690117860627198859533418] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 71, 659, 5102, 41388, 320589, 2520696, 19559993, 152513105, 1184546177, 9214298064, 71591778582, 556511227752, 4324443932660, 33608752490897, 261172521752504, 2029659116944650, 15772639058824632, 122572209909189561, 952522570058465606, 7402194948333769655, 57523382708302084574, 447022034863195652622, 3473865765448185751341, 26995869809139410409771, 209788409697720057966245, 1630293194303480960132126, 12669220694815135907477357 , 98454167827119668113633437, 765100169120654031077430573, 5945693202341590251235821753, 46204756951693219173142317938, 359063190931324356575915828567, 2790326867901726005209142914079, 21683993893071863098735757682501, 168509143566073901665098991084941, 1309506524317255210723436403976524, 10176345927104698727504411988820130, 79081710947807493232276547463853739, 614554286062836943182540234070881104] Using the found enumerative automaton with, 20, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 19 18 17 16 15 14 - (30312 t + 283584 t + 302412 t - 1260696 t - 140827 t + 1187455 t 13 12 11 10 9 8 + 245487 t - 801748 t - 122131 t + 392697 t - 50439 t - 81405 t 7 6 5 4 3 2 / + 32648 t + 323 t - 3365 t + 1056 t - 62 t - 49 t + 13 t - 1) / (( / 10 9 8 7 6 5 4 1179 t + 3590 t - 2772 t - 4149 t + 2684 t + 1241 t - 1030 t 3 2 10 9 8 7 6 + 76 t + 68 t - 16 t + 1) (357 t - 192 t - 1470 t + 1315 t + 518 t 5 4 3 2 - 801 t + 78 t + 114 t - 24 t - 4 t + 1)) and in Maple notation -(30312*t^19+283584*t^18+302412*t^17-1260696*t^16-140827*t^15+1187455*t^14+ 245487*t^13-801748*t^12-122131*t^11+392697*t^10-50439*t^9-81405*t^8+32648*t^7+ 323*t^6-3365*t^5+1056*t^4-62*t^3-49*t^2+13*t-1)/(1179*t^10+3590*t^9-2772*t^8-\ 4149*t^7+2684*t^6+1241*t^5-1030*t^4+76*t^3+68*t^2-16*t+1)/(357*t^10-192*t^9-\ 1470*t^8+1315*t^7+518*t^6-801*t^5+78*t^4+114*t^3-24*t^2-4*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 18 17 16 15 14 - (23238 t - 272043 t - 893397 t + 1742172 t + 1781952 t 13 12 11 10 9 - 3812348 t - 290781 t + 2992959 t - 831693 t - 983777 t 8 7 6 5 4 3 + 519829 t + 109339 t - 118134 t + 8529 t + 10446 t - 2308 t 2 / 10 9 8 7 - 211 t + 109 t - 9) t / ((1179 t + 3590 t - 2772 t - 4149 t / 6 5 4 3 2 10 + 2684 t + 1241 t - 1030 t + 76 t + 68 t - 16 t + 1) (357 t 9 8 7 6 5 4 3 2 - 192 t - 1470 t + 1315 t + 518 t - 801 t + 78 t + 114 t - 24 t - 4 t + 1)) and in Maple notation -(23238*t^18-272043*t^17-893397*t^16+1742172*t^15+1781952*t^14-3812348*t^13-\ 290781*t^12+2992959*t^11-831693*t^10-983777*t^9+519829*t^8+109339*t^7-118134*t^ 6+8529*t^5+10446*t^4-2308*t^3-211*t^2+109*t-9)*t/(1179*t^10+3590*t^9-2772*t^8-\ 4149*t^7+2684*t^6+1241*t^5-1030*t^4+76*t^3+68*t^2-16*t+1)/(357*t^10-192*t^9-\ 1470*t^8+1315*t^7+518*t^6-801*t^5+78*t^4+114*t^3-24*t^2-4*t+1) This theorem took, 0.260, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 8, :consider the sequence, 2 2 2 2 2 n (x y + x y + x + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 6, 7, 49, 22, 6, 42, 90, 7, 49, 42, 49, 343, 114, 22, 154, 316, 6, 42, 157, 42, 294, 487, 90, 630, 804, 7, 49, 42, 49, 343, 154, 42, 294, 630, 49, 343 , 294, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 11, 0, 0, 39, 11, 77, 86, 0, 0, 77, 0, 0, 239, 39, 273, 246, 11, 77, 132 , 77, 539, 396, 86, 602, 848, 0, 0, 77, 0, 0, 273, 77, 539, 602, 0, 0, 539, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [582782783056213983684007504344730512587373900790558172971843362727271506851161\ 7427488590584213688692762180776687042560, 5827827830562139836840075043784000036\ 5385665986214277910663934454671088740475012882050777969794518345221807766870425\ 60] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 90, 804, 7632, 67275, 596997, 5240430, 46041330, 403645065, 3538952607, 31012832571, 271769305281, 2381283217365, 20864922999027, 182814567963534, 1601779612673376, 14034331684209936, 122964561836502180, 1077376273579609011, 9439621889969844075, 82706868230640400770, 724650297006304507356, 6349145999556338246169, 55629112180634901019389, 487403818539215388430851, 4270470424648611321867177, 37416443531234242389611847, 327830449646080279948983051, 2872341496087922543259923670, 25166501973523337172914295402, 220500529586272116054730559730, 1931952385677763569675763661802, 16927124967073097723065354006623, 148309845387075686361371495794869, 1299441593299827468497502860388162, 11385275535345336860569639646191458, 99754001781612836499612529313500437, 874011422951448636615785941526602167, 7657797720412083535430484954175772337, 67095079522579643508581373520551072453] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 11, 86, 848, 7469, 67439, 594260, 5242955, 46007195, 403703819, 3538486217, 31013846192, 271762282067, 2381299901522, 20864816975966, 182814849426167, 1601778023596640, 14034336411598790, 122964537835594739, 1077376351618301771, 9439621523730434234, 82706869502415835391, 724650291376340312075, 6349146020104742342093, 55629112093575545010215, 487403818868962871487632, 4270470423295189750049951, 37416443536496588811675038, 327830449624945937637410048, 2872341496171549657868553179, 25166501973192169129403849942, 220500529587596998459199238068, 1931952385672560878782094017355, 16927124967094041246913698042125, 148309845386993799602857635533168, 1299441593300158033383309946149239, 11385275535344046378468837896896409, 99754001781618048733990871049280211, 874011422951428282853959190937825071, 7657797720412165669076122295136236162, 67095079522579322332043498750670324767] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 20 19 18 17 16 - (2763396 t - 9858804 t - 6326567 t + 19223858 t + 5691465 t 15 14 13 12 11 - 10576320 t - 6803829 t + 4893649 t + 2831047 t - 1778960 t 10 9 8 7 6 5 - 228344 t + 179233 t - 9152 t + 43515 t - 33810 t + 7491 t 4 3 2 / 11 10 + 644 t - 670 t + 158 t - 19 t + 1) / ((4290 t - 13239 t / 9 8 7 6 5 4 3 - 1666 t + 9764 t + 2370 t - 3093 t - 782 t + 500 t + 170 t 2 10 9 8 7 6 - 114 t + 19 t - 1) (2622 t + 529 t - 3257 t - 49 t + 1753 t 5 4 3 2 - 582 t - 188 t + 132 t - 10 t - 6 t + 1)) and in Maple notation -(2763396*t^20-9858804*t^19-6326567*t^18+19223858*t^17+5691465*t^16-10576320*t^ 15-6803829*t^14+4893649*t^13+2831047*t^12-1778960*t^11-228344*t^10+179233*t^9-\ 9152*t^8+43515*t^7-33810*t^6+7491*t^5+644*t^4-670*t^3+158*t^2-19*t+1)/(4290*t^ 11-13239*t^10-1666*t^9+9764*t^8+2370*t^7-3093*t^6-782*t^5+500*t^4+170*t^3-114*t ^2+19*t-1)/(2622*t^10+529*t^9-3257*t^8-49*t^7+1753*t^6-582*t^5-188*t^4+132*t^3-\ 10*t^2-6*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 19 18 17 16 15 t (4006224 t - 7733328 t - 12227482 t + 11797341 t + 9566691 t 14 13 12 11 10 - 819030 t - 8162025 t - 2182576 t + 4088827 t + 1650657 t 9 8 7 6 5 4 - 1945558 t - 94244 t + 383964 t - 21141 t - 58935 t + 16650 t 3 2 / 11 10 9 + 835 t - 1096 t + 189 t - 11) / ((4290 t - 13239 t - 1666 t / 8 7 6 5 4 3 2 + 9764 t + 2370 t - 3093 t - 782 t + 500 t + 170 t - 114 t + 19 t 10 9 8 7 6 5 4 - 1) (2622 t + 529 t - 3257 t - 49 t + 1753 t - 582 t - 188 t 3 2 + 132 t - 10 t - 6 t + 1)) and in Maple notation t*(4006224*t^19-7733328*t^18-12227482*t^17+11797341*t^16+9566691*t^15-819030*t^ 14-8162025*t^13-2182576*t^12+4088827*t^11+1650657*t^10-1945558*t^9-94244*t^8+ 383964*t^7-21141*t^6-58935*t^5+16650*t^4+835*t^3-1096*t^2+189*t-11)/(4290*t^11-\ 13239*t^10-1666*t^9+9764*t^8+2370*t^7-3093*t^6-782*t^5+500*t^4+170*t^3-114*t^2+ 19*t-1)/(2622*t^10+529*t^9-3257*t^8-49*t^7+1753*t^6-582*t^5-188*t^4+132*t^3-10* t^2-6*t+1) This theorem took, 1.173, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 9, :consider the sequence, 2 2 2 2 n (x y + x y + x + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 10, 7, 49, 43, 10, 70, 90, 7, 49, 70, 49, 343, 246, 43, 301, 357, 10, 70 , 181, 70, 490, 631, 90, 630, 838, 7, 49, 70, 49, 343, 301, 70, 490, 630, 49, 343, 490, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 39, 9, 63, 95, 0, 0, 63, 0, 0, 245, 39, 273, 347, 9, 63, 180, 63, 441, 636, 95, 665, 846, 0, 0, 63, 0, 0, 273, 63, 441, 665, 0, 0, 441, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [365381315957325060464479714217866779830677024911792414041376933136459215107774\ 43492474371379285269628413324297609440000000, 365381315957325060464479714217866\ 7798306770249117924140414536133979085922590650245531772030383642995536651993060\ 9440000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 10, 90, 838, 7551, 66715, 593103, 5255629, 46561707, 412426051, 3652790190, 32351573941, 286525074663, 2537629123027, 22474661908662, 199048063275106, 1762879668965754, 15613035826427320, 138277664414291667, 1224663326354086516, 10846294437744449496, 96060770785013111503, 850767212775119130141, 7534864072083330932386, 66732915585196771078971, 591023538076476943395097, 5234430707971688801545017, 46359007842772664690028499, 410580964403021912870362695, 3636331668320131484265284965, 32205360570647217567328696959, 285228451112400274103864862235, 2526140613936980552907754794996, 22372895748986084342268610738291, 198146714966480173976765410474308, 1754896688051101358994746891207425, 15542333801767671694111700285527581, 137651487777231819418757552860934332, 1219117561683717748493559926042762583, 10797178099599781468427496615892426333, 95625769473347902948968645670016336286] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 95, 846, 7529, 67164, 594038, 5260713, 46579844, 412485192, 3653007152, 32352337707, 286527422885, 2537636651025, 22474683708026, 199048123550529, 1762879823627300, 15613036160813487, 138277664850564386, 1224663325582492476, 10846294427505680246, 96060770727070358292, 850767212509432475312, 7534864070987950400163, 66732915580985319771650, 591023538061132793988687, 5234430707918166584928515, 46359007842593626340972418, 410580964402447051024602824, 3636331668318367219760302485, 32205360570642089198688771728, 285228451112386419366226978128, 2526140613936947195060013525743, 22372895748986021060857178092920, 198146714966480133611283094763114, 1754896688051101807032828836502501, 15542333801767675098329411761837985, 137651487777231836772865613003262102, 1219117561683717824327510414966728193, 10797178099599781771864999983396554625, 95625769473347904090997685265625772159] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.320, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 10, :consider the sequence, 2 2 2 2 2 n (x y + x y + x + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 28, 7, 49, 110, 7, 49, 49, 49, 343, 145, 28, 196, 413, 7, 49, 193, 49, 343, 667, 110, 770, 908, 7, 49, 49, 49, 343, 196, 49, 343, 770, 49, 343, 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 12, 0, 0, 48, 12, 84, 91, 0, 0, 84, 0, 0, 306, 48, 336, 310, 12, 84, 168 , 84, 588, 519, 91, 637, 946, 0, 0, 84, 0, 0, 336, 84, 588, 637, 0, 0, 588, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [144895898103912891577807319050136579652271255465112090939717087186435479335511\ 07988064342929281001993146771387915121459200, 144895898103912891577807321301208\ 9290986288104860323289374842339564811866574289751613782952939798711186677138791\ 5121459200] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 110, 908, 8411, 74264, 657677, 5797025, 51149930, 450744785, 3972691757, 35003485757, 308422145885, 2717415644432, 23942460175712, 210948940181366, 1858601875036523, 16375499067410987, 144278876370215354, 1271190891736003721, 11200019858523465170, 98679462909643088519, 869430280658173694714, 7660246410568406200715, 67491754529359296497057, 594646264819440822373709, 5239220445406574750915168, 46160940514986042686184770, 406707916060462992010208783, 3583361324955874685020634255, 31571744433829072240854156524, 278167607499943911792856421276, 2450837584355667814871535077159, 21593473513490457759441809860916, 190252549313726917688355387109493, 1676248728474448048536840416044427, 14768841783446618411538705239731652, 130123103999573681281381205349567998, 1146469197974705642639498088115000832, 10101139471042886131462194259342664990, 88997610048055613553952340817014750138] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 12, 91, 946, 8266, 74194, 656095, 5801266, 51133453, 450827080, 3972332623, 35003793529, 308413645543, 2717414109400, 23942348253571, 210949067433676, 1858600820550781, 16375502045038447, 144278858177210470, 1271190905436397789, 11200019461622337910, 98679462827835927994, 869430275017637628403, 7660246414243971952099, 67491754468634850177034, 594646264923449302185298, 5239220444455071322318156, 46160940515485121636224957, 406707916041485216078152645, 3583361324950397403559419091, 31571744433546275284888437430, 278167607500015034718668205397, 2450837584352310443731722583081, 21593473513493597492519389424041, 190252549313676694512383632913941, 1676248728474459059435729110236790, 14768841783445688550899659027150687, 130123103999573280603591574486084678, 1146469197974691382983707273109933286, 10101139471042884509535671178340874254, 88997610048055431472036997816783418601] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.312, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 11, :consider the sequence, 2 2 2 2 2 n (x y + x y + x + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 25, 7, 49, 103, 7, 49, 49, 49, 343, 127, 25, 175, 373, 7, 49, 193, 49, 343, 607, 103, 721, 961, 7, 49, 49, 49, 343, 175, 49, 343, 721, 49, 343, 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 12, 0, 0, 48, 12, 84, 102, 0, 0, 84, 0, 0, 276, 48, 336, 342, 12, 84, 168, 84, 588, 516, 102, 714, 978, 0, 0, 84, 0, 0, 336, 84, 588, 714, 0, 0, 588, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [210232289663574469691826877418253727220399378137801784162744127395612946978128\ 8086435383090989183557839858637357068359375, 2102322896635744696918268774182537\ 3047475123333092922808133779871896753640372676831982749335913304297652278851754\ 76562500] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 103, 961, 8869, 80035, 722299, 6501229, 58529425, 526747255, 4740884167, 42667400545, 384007406845, 3456053691019, 31104469057459, 279939934452085, 2519458679649001, 22675121698308127, 204076072953902623, 1836684509676114697, 16530159988512167125, 148771436462101915315, 1338942912915234604171, 12050486134714178312605, 108454374833137231827649, 976089371544219832271431, 8784804334567258762638583, 79063238963993873786330929, 711569150447733635471372461, 6404122352889861653719658203, 57637101170444096233878428515, 518733910506370291542624939973, 4668605194421864298668228926105, 42017446749126401691942968467567, 378157020738842582401752209834959, 3403413186633306296523593028378457, 30630718679619648316507673247513733, 275676468116181494366281000829373187, 2481088213043686363169759114758920763, 22329793917383573331939832966698983917, 200968145256404841343844773170395972209] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 12, 102, 978, 8844, 80172, 721890, 6502854, 58523736, 526768368, 4740807822 , 42667679898, 384006390084, 3456057400644, 31104455537658, 279939983749422, 2519458499935536, 22675122353519064, 204076070565199974, 1836684518384785602, 16530159956762609436, 148771436577853056108, 1338942912493234976466, 12050486136252684412758, 108454374827528221861128, 976089371564668888805376, 8784804334492706589771582, 79063238964265672471268586, 711569150446742725064295156, 6404122352893474266643438740, 57637101170430925545588403626, 518733910506418308596747672958, 4668605194421689240430243310816, 42017446749127039910743676096040, 378157020738840255614722880886294, 3403413186633314779410455932046610, 30630718679619617390011771466485452, 275676468116181607116593981813438652, 2481088213043685952110157742104620162, 22329793917383574830560913597911977894, 200968145256404835880245003672044180664] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 4644 t - 2196 t + 1224 t - 480 t - 95 t + 41 t - 7 t + 1 --------------------------------------------------------------------- 3 2 2 (3 t - 1) (9 t - 1) (18 t - 4 t + 5 t - 1) (t + 1) (6 t - 2 t - 1) and in Maple notation (4644*t^7-2196*t^6+1224*t^5-480*t^4-95*t^3+41*t^2-7*t+1)/(3*t-1)/(9*t-1)/(18*t^ 3-4*t^2+5*t-1)/(t+1)/(6*t^2-2*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 6 t (684 t - 444 t + 210 t - 72 t + 3 t + 11 t - 2) - --------------------------------------------------------------------- 3 2 2 (3 t - 1) (9 t - 1) (18 t - 4 t + 5 t - 1) (t + 1) (6 t - 2 t - 1) and in Maple notation -6*t*(684*t^6-444*t^5+210*t^4-72*t^3+3*t^2+11*t-2)/(3*t-1)/(9*t-1)/(18*t^3-4*t^ 2+5*t-1)/(t+1)/(6*t^2-2*t-1) This theorem took, 1.020, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 12, :consider the sequence, 2 2 2 2 2 n (x y + x y + x + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 10, 7, 49, 43, 10, 70, 90, 7, 49, 70, 49, 343, 246, 43, 301, 357, 10, 70 , 181, 70, 490, 631, 90, 630, 838, 7, 49, 70, 49, 343, 301, 70, 490, 630, 49, 343, 490, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 39, 9, 63, 95, 0, 0, 63, 0, 0, 245, 39, 273, 347, 9, 63, 180, 63, 441, 636, 95, 665, 846, 0, 0, 63, 0, 0, 273, 63, 441, 665, 0, 0, 441, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [365381315957325060464479714217866779830677024911792414041376933136459215107774\ 43492474371379285269628413324297609440000000, 365381315957325060464479714217866\ 7798306770249117924140414536133979085922590650245531772030383642995536651993060\ 9440000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 10, 90, 838, 7551, 66715, 593103, 5255629, 46561707, 412426051, 3652790190, 32351573941, 286525074663, 2537629123027, 22474661908662, 199048063275106, 1762879668965754, 15613035826427320, 138277664414291667, 1224663326354086516, 10846294437744449496, 96060770785013111503, 850767212775119130141, 7534864072083330932386, 66732915585196771078971, 591023538076476943395097, 5234430707971688801545017, 46359007842772664690028499, 410580964403021912870362695, 3636331668320131484265284965, 32205360570647217567328696959, 285228451112400274103864862235, 2526140613936980552907754794996, 22372895748986084342268610738291, 198146714966480173976765410474308, 1754896688051101358994746891207425, 15542333801767671694111700285527581, 137651487777231819418757552860934332, 1219117561683717748493559926042762583, 10797178099599781468427496615892426333, 95625769473347902948968645670016336286] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 95, 846, 7529, 67164, 594038, 5260713, 46579844, 412485192, 3653007152, 32352337707, 286527422885, 2537636651025, 22474683708026, 199048123550529, 1762879823627300, 15613036160813487, 138277664850564386, 1224663325582492476, 10846294427505680246, 96060770727070358292, 850767212509432475312, 7534864070987950400163, 66732915580985319771650, 591023538061132793988687, 5234430707918166584928515, 46359007842593626340972418, 410580964402447051024602824, 3636331668318367219760302485, 32205360570642089198688771728, 285228451112386419366226978128, 2526140613936947195060013525743, 22372895748986021060857178092920, 198146714966480133611283094763114, 1754896688051101807032828836502501, 15542333801767675098329411761837985, 137651487777231836772865613003262102, 1219117561683717824327510414966728193, 10797178099599781771864999983396554625, 95625769473347904090997685265625772159] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.347, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 13, :consider the sequence, 2 2 2 2 2 n (x y + x y + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 38, 7, 49, 94, 7, 49, 49, 49, 343, 224, 38, 266, 338, 7, 49, 193, 49, 343, 594, 94, 658, 753, 7, 49, 49, 49, 343, 266, 49, 343, 658, 49, 343 , 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 12, 0, 0, 37, 12, 84, 72, 0, 0, 84, 0, 0, 223, 37, 259, 277, 12, 84, 168 , 84, 588, 545, 72, 504, 788, 0, 0, 84, 0, 0, 259, 84, 588, 504, 0, 0, 588, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [166878279185797473775922545788465213220012767928933764865219620818446560520976\ 8324213831048613416398815212770500000000000, 1668782791857974737759225457884652\ 1322001276775585442399229840474477844574421859563304563106315046854121523545000\ 00000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 94, 753, 6780, 59563, 521542, 4572468, 40003056, 349974001, 3060886612, 26768207391, 234084121248, 2046974621152, 17899785746074, 156523679981607, 1368707493311598, 11968516886090461, 104657299285252390, 915162952932467544, 8002527604882250130, 69977084250369348031, 611905638025516602850, 5350730024195478915375, 46788767699161492544466, 409138328417426786317492, 3577657161648034264136830, 31284359820851117459453169, 273562032797571953546106060, 2392127761641476875691433283, 20917651337709620097806079562, 182911692348572759729461782090, 1599447597875345796137018559306, 13986162311227743370194693718975, 122300184408985618530959522011216, 1069438118379102454086069282620997, 9351563078384847133160217602200968, 81773531824056280193602559041126744, 715058055091875060724863594423014668, 6252732525340880683836986604177813297, 54676209511801435973278597618156464324] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 12, 72, 788, 6797, 59871, 524511, 4581167, 40060610, 350208552, 3062039427, 26773881533, 234111132305, 2047109862345, 17900446249218, 156526948294754, 1368723634139387, 11968596517760337, 104657692915663875, 915164896062219971, 8002537202726522216, 69977131644720702990, 611905872040398033021, 5350731179688690913169, 46788773404050767413295, 409138356583781600911161, 3577657300707884615527674, 31284360507397929211935284, 273562036187091835282221017, 2392127778375669060879431295, 20917651420327166054961445935, 182911692756459801865720249193, 1599447599889107259104742010472, 13986162321169806491441443691502, 122300184458070213965290195203699, 1069438118621436327178674600893819, 9351563079581265631466823387142337, 81773531829963079021177056347996877, 715058055121037326876017007771206360, 6252732525484856771148675436543671224, 54676209512512255692034692301175057945] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.360, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 14, :consider the sequence, 2 2 2 2 n (x y + x y + x y + x y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 13, 7, 49, 49, 13, 91, 55, 7, 49, 91, 49, 343, 319, 49, 343, 235, 13, 91 , 205, 91, 637, 721, 55, 385, 655, 7, 49, 91, 49, 343, 343, 91, 637, 385, 49, 343, 637, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 6, 0, 0, 36, 6, 42, 96, 0, 0, 42, 0, 0, 228, 36, 252, 432, 6, 42, 156, 42, 294, 630, 96, 672, 870, 0, 0, 42, 0, 0, 252, 42, 294, 672, 0, 0, 294, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [618869548076643352989836641621052118974015163355333768515913304439539891999626\ 13592381230512100677962156190340105751894169, 618869548076643352989836637788924\ 1116069842386726482113690025162978595737525321067171308799888115962480082935415\ 1231121456] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 13, 55, 655, 7213, 62065, 500011, 4577251, 41728753, 365250373, 3200827711, 28532169271, 253627443061, 2241364306009, 19835129242387, 175916646389707, 1559006691715129, 13805908941462445, 122301753196289191, 1083707134247078815, 9601151037002326333, 85054756877761727041, 753533764885069263355, 6676026918540481658035, 59145466792867130320513, 523988365119940359343381, 4642229962406886822895375, 41127513517153403860234759, 364364746311640786641290245, 3228048962524087973082053929, 28598589384536668017118500835, 253366444750613695407404397211, 2244674143535372457461102632393, 19886462711820809583797390619517, 176182139568174044870360683882615, 1560868078685852898119191919084911, 13828353741462361805281442599260109, 122510912150645029892384956107606289, 1085373164086547444025217159312053451, 9615754793375804292148436733286789507, 85189815201621151995327245821632728593] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 6, 96, 870, 5892, 56634, 543084, 4704378, 40344576, 362545854, 3244255800, 28578986286, 252295179036, 2241006544482, 19875154514532, 175896095816130, 1557828623816664, 13807441113323046, 122335704127383792, 1083634937028333366, 9600194475145555764, 85057657051629256458, 753560040226895496924, 6675919896762152910474, 59144766391440977816112, 523992104401352714910222, 4642247940879885918724392, 41127387934336246200565950, 364364307917744208621311436, 3228053051148785171743366194, 28598599273787410940554553556, 253366315056303986211910943442, 2244673949681400614414648283912, 19886466732330886187832283444854, 176182142215004693012232014023392, 1560867956673786321146073046885830, 13828353757684173267962081142314724, 122510915778075302511735643855476186, 1085373160772674214030059870915801164, 9615754687741010440237208046270594714, 85189815382896893009202849212769446496] Using the found enumerative automaton with, 30, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 8 7 6 5 4 3 2 - (15066 t - 1323 t + 1611 t + 183 t + 265 t + 215 t - 39 t + 5 t + 1) / 2 3 2 / ((9 t - 2 t - 1) (30 t - 11 t - 8 t + 1) / 4 3 2 (189 t - 78 t + 34 t - 2 t + 1)) and in Maple notation -(15066*t^8-1323*t^7+1611*t^6+183*t^5+265*t^4+215*t^3-39*t^2+5*t+1)/(9*t^2-2*t-\ 1)/(30*t^3-11*t^2-8*t+1)/(189*t^4-78*t^3+34*t^2-2*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 7 6 5 4 3 2 / - 6 t (3726 t - 1881 t + 690 t + 385 t - 148 t + 27 t + 8 t + 1) / ( / 2 3 2 (9 t - 2 t - 1) (30 t - 11 t - 8 t + 1) 4 3 2 (189 t - 78 t + 34 t - 2 t + 1)) and in Maple notation -6*t*(3726*t^7-1881*t^6+690*t^5+385*t^4-148*t^3+27*t^2+8*t+1)/(9*t^2-2*t-1)/(30 *t^3-11*t^2-8*t+1)/(189*t^4-78*t^3+34*t^2-2*t+1) This theorem took, 1.031, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 15, :consider the sequence, 2 2 2 2 2 n (x y + x y + x y + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 21, 7, 49, 81, 7, 49, 49, 49, 343, 99, 21, 147, 298, 7, 49, 130, 49, 343, 455, 81, 567, 624, 7, 49, 49, 49, 343, 147, 49, 343, 567, 49, 343 , 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 41, 9, 63, 71, 0, 0, 63, 0, 0, 269, 41, 287, 186, 9, 63, 126, 63, 441, 391, 71, 497, 659, 0, 0, 63, 0, 0, 287, 63, 441, 497, 0, 0, 441, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [701829909586202358220405128773680326529304566036156000331514484497476575819536\ 921209737694374779656400837015083417600, 70182990958620235822040513180305155679\ 5925835788071317484923688152887416825990775485167714082906331274582423083417600 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 81, 624, 5274, 40735, 323587, 2508220, 19613499, 152276865, 1185521616, 9209868160, 71609782399, 556428616111, 4324778184390, 33607215723393, 261178742366592, 2029630556491372, 15772754928467869, 122571679267852618, 952524728856309120, 7402185089524606677, 57523422930203255871, 447021851686862509534, 3473866514809219528960, 26995866405501753190258, 209788423657881871595928, 1630293131055607684320975, 12669220954867389245322996 , 98454166651748220612289048, 765100173964643964845608774, 5945693180497749631190383795, 46204757041916789209028369289, 359063190525343146702768271473, 2790326869582125236655981590166, 21683993885526074281451247313828, 168509143597369407187510954650241, 1309506524176998378956018374120831, 10176345927687512365348864696280811, 79081710945200360983898146272429144, 614554286073690117860627198859533418] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 71, 659, 5102, 41388, 320589, 2520696, 19559993, 152513105, 1184546177, 9214298064, 71591778582, 556511227752, 4324443932660, 33608752490897, 261172521752504, 2029659116944650, 15772639058824632, 122572209909189561, 952522570058465606, 7402194948333769655, 57523382708302084574, 447022034863195652622, 3473865765448185751341, 26995869809139410409771, 209788409697720057966245, 1630293194303480960132126, 12669220694815135907477357 , 98454167827119668113633437, 765100169120654031077430573, 5945693202341590251235821753, 46204756951693219173142317938, 359063190931324356575915828567, 2790326867901726005209142914079, 21683993893071863098735757682501, 168509143566073901665098991084941, 1309506524317255210723436403976524, 10176345927104698727504411988820130, 79081710947807493232276547463853739, 614554286062836943182540234070881104] Using the found enumerative automaton with, 20, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 19 18 17 16 15 14 - (30312 t + 283584 t + 302412 t - 1260696 t - 140827 t + 1187455 t 13 12 11 10 9 8 + 245487 t - 801748 t - 122131 t + 392697 t - 50439 t - 81405 t 7 6 5 4 3 2 / + 32648 t + 323 t - 3365 t + 1056 t - 62 t - 49 t + 13 t - 1) / (( / 10 9 8 7 6 5 4 1179 t + 3590 t - 2772 t - 4149 t + 2684 t + 1241 t - 1030 t 3 2 10 9 8 7 6 + 76 t + 68 t - 16 t + 1) (357 t - 192 t - 1470 t + 1315 t + 518 t 5 4 3 2 - 801 t + 78 t + 114 t - 24 t - 4 t + 1)) and in Maple notation -(30312*t^19+283584*t^18+302412*t^17-1260696*t^16-140827*t^15+1187455*t^14+ 245487*t^13-801748*t^12-122131*t^11+392697*t^10-50439*t^9-81405*t^8+32648*t^7+ 323*t^6-3365*t^5+1056*t^4-62*t^3-49*t^2+13*t-1)/(1179*t^10+3590*t^9-2772*t^8-\ 4149*t^7+2684*t^6+1241*t^5-1030*t^4+76*t^3+68*t^2-16*t+1)/(357*t^10-192*t^9-\ 1470*t^8+1315*t^7+518*t^6-801*t^5+78*t^4+114*t^3-24*t^2-4*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 18 17 16 15 14 - (23238 t - 272043 t - 893397 t + 1742172 t + 1781952 t 13 12 11 10 9 - 3812348 t - 290781 t + 2992959 t - 831693 t - 983777 t 8 7 6 5 4 3 + 519829 t + 109339 t - 118134 t + 8529 t + 10446 t - 2308 t 2 / 10 9 8 7 - 211 t + 109 t - 9) t / ((1179 t + 3590 t - 2772 t - 4149 t / 6 5 4 3 2 10 + 2684 t + 1241 t - 1030 t + 76 t + 68 t - 16 t + 1) (357 t 9 8 7 6 5 4 3 2 - 192 t - 1470 t + 1315 t + 518 t - 801 t + 78 t + 114 t - 24 t - 4 t + 1)) and in Maple notation -(23238*t^18-272043*t^17-893397*t^16+1742172*t^15+1781952*t^14-3812348*t^13-\ 290781*t^12+2992959*t^11-831693*t^10-983777*t^9+519829*t^8+109339*t^7-118134*t^ 6+8529*t^5+10446*t^4-2308*t^3-211*t^2+109*t-9)*t/(1179*t^10+3590*t^9-2772*t^8-\ 4149*t^7+2684*t^6+1241*t^5-1030*t^4+76*t^3+68*t^2-16*t+1)/(357*t^10-192*t^9-\ 1470*t^8+1315*t^7+518*t^6-801*t^5+78*t^4+114*t^3-24*t^2-4*t+1) This theorem took, 0.266, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 16, :consider the sequence, 2 2 2 2 2 n (x y + x y + x y + x y + y + x + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 10, 7, 49, 43, 10, 70, 90, 7, 49, 70, 49, 343, 246, 43, 301, 357, 10, 70 , 181, 70, 490, 631, 90, 630, 838, 7, 49, 70, 49, 343, 301, 70, 490, 630, 49, 343, 490, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 39, 9, 63, 95, 0, 0, 63, 0, 0, 245, 39, 273, 347, 9, 63, 180, 63, 441, 636, 95, 665, 846, 0, 0, 63, 0, 0, 273, 63, 441, 665, 0, 0, 441, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [365381315957325060464479714217866779830677024911792414041376933136459215107774\ 43492474371379285269628413324297609440000000, 365381315957325060464479714217866\ 7798306770249117924140414536133979085922590650245531772030383642995536651993060\ 9440000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 10, 90, 838, 7551, 66715, 593103, 5255629, 46561707, 412426051, 3652790190, 32351573941, 286525074663, 2537629123027, 22474661908662, 199048063275106, 1762879668965754, 15613035826427320, 138277664414291667, 1224663326354086516, 10846294437744449496, 96060770785013111503, 850767212775119130141, 7534864072083330932386, 66732915585196771078971, 591023538076476943395097, 5234430707971688801545017, 46359007842772664690028499, 410580964403021912870362695, 3636331668320131484265284965, 32205360570647217567328696959, 285228451112400274103864862235, 2526140613936980552907754794996, 22372895748986084342268610738291, 198146714966480173976765410474308, 1754896688051101358994746891207425, 15542333801767671694111700285527581, 137651487777231819418757552860934332, 1219117561683717748493559926042762583, 10797178099599781468427496615892426333, 95625769473347902948968645670016336286] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 95, 846, 7529, 67164, 594038, 5260713, 46579844, 412485192, 3653007152, 32352337707, 286527422885, 2537636651025, 22474683708026, 199048123550529, 1762879823627300, 15613036160813487, 138277664850564386, 1224663325582492476, 10846294427505680246, 96060770727070358292, 850767212509432475312, 7534864070987950400163, 66732915580985319771650, 591023538061132793988687, 5234430707918166584928515, 46359007842593626340972418, 410580964402447051024602824, 3636331668318367219760302485, 32205360570642089198688771728, 285228451112386419366226978128, 2526140613936947195060013525743, 22372895748986021060857178092920, 198146714966480133611283094763114, 1754896688051101807032828836502501, 15542333801767675098329411761837985, 137651487777231836772865613003262102, 1219117561683717824327510414966728193, 10797178099599781771864999983396554625, 95625769473347904090997685265625772159] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 5.367, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 17, :consider the sequence, 2 2 2 2 2 n (x y + x y + x y + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 9, 7, 49, 33, 9, 63, 85, 7, 49, 63, 49, 343, 159, 33, 231, 313, 9, 63, 145, 63, 441, 511, 85, 595, 753, 7, 49, 63, 49, 343, 231, 63, 441, 595, 49, 343 , 441, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 38, 8, 56, 84, 0, 0, 56, 0, 0, 266, 38, 266, 270, 8, 56, 144, 56, 392, 516, 84, 588, 752, 0, 0, 56, 0, 0, 266, 56, 392, 588, 0, 0, 392, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [120490515368853116055012272117064194840063460904515273772464835041785853822731\ 913375827838054587096052116773295891405625, 12049051536885311605501227211706419\ 4840063461050565743921121482467485634024081434077826112074506564050793533176679\ 296250] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 9, 85, 753, 6433, 54183, 453397, 3781737, 31490149, 261986907, 2178668347, 18113652603, 150582215095, 1251753123489, 10405262795257, 86493312850875, 718968411139237, 5976351346186185, 49677761950737979, 412940761118069799, 3432522716520092821, 28532449853842264107, 237172700135679437053, 1971470721262991275779, 16387623047252463847765, 136220227162413422364171, 1132314934001417937398623, 9412237356075981864624555, 78238138020374050756971505, 650345503516580517091146699, 5405922031236453915541270111, 44936103735784783980336784665, 373526182465878899992902349507, 3104893335007019597393162539161, 25809067943317346641156507077205, 214534902244742905969596449711019, 1783296645285773229895390370992687, 14823447801808384098344104340113971, 123218201141087531659071478357220317, 1024237093518602896621286249867274045, 8513852775194635155075915528035860471] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 84, 752, 6432, 54182, 453396, 3781736, 31490148, 261986906, 2178668346, 18113652602, 150582215094, 1251753123488, 10405262795256, 86493312850874, 718968411139236, 5976351346186184, 49677761950737978, 412940761118069798, 3432522716520092820, 28532449853842264106, 237172700135679437052, 1971470721262991275778, 16387623047252463847764, 136220227162413422364170, 1132314934001417937398622, 9412237356075981864624554, 78238138020374050756971504, 650345503516580517091146698, 5405922031236453915541270110, 44936103735784783980336784664, 373526182465878899992902349506, 3104893335007019597393162539160, 25809067943317346641156507077204, 214534902244742905969596449711018, 1783296645285773229895390370992686, 14823447801808384098344104340113970, 123218201141087531659071478357220316, 1024237093518602896621286249867274044, 8513852775194635155075915528035860470] Using the found enumerative automaton with, 22, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 8 7 6 5 4 3 2 / - (90 t + 81 t - 540 t + 313 t + 66 t - 117 t + 51 t - 11 t + 1) / ( / 8 7 6 5 4 3 2 (90 t + 225 t - 1004 t + 523 t + 282 t - 357 t + 127 t - 19 t + 1) (t - 1)) and in Maple notation -(90*t^8+81*t^7-540*t^6+313*t^5+66*t^4-117*t^3+51*t^2-11*t+1)/(90*t^8+225*t^7-\ 1004*t^6+523*t^5+282*t^4-357*t^3+127*t^2-19*t+1)/(t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 6 5 4 3 2 / 2 t (72 t - 232 t + 105 t + 108 t - 120 t + 38 t - 4) / ( / 8 7 6 5 4 3 2 (90 t + 225 t - 1004 t + 523 t + 282 t - 357 t + 127 t - 19 t + 1) (t - 1)) and in Maple notation 2*t*(72*t^6-232*t^5+105*t^4+108*t^3-120*t^2+38*t-4)/(90*t^8+225*t^7-1004*t^6+ 523*t^5+282*t^4-357*t^3+127*t^2-19*t+1)/(t-1) This theorem took, 0.887, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 18, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 6, 7, 49, 22, 6, 42, 90, 7, 49, 42, 49, 343, 114, 22, 154, 316, 6, 42, 157, 42, 294, 487, 90, 630, 804, 7, 49, 42, 49, 343, 154, 42, 294, 630, 49, 343 , 294, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 11, 0, 0, 39, 11, 77, 86, 0, 0, 77, 0, 0, 239, 39, 273, 246, 11, 77, 132 , 77, 539, 396, 86, 602, 848, 0, 0, 77, 0, 0, 273, 77, 539, 602, 0, 0, 539, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [582782783056213983684007504344730512587373900790558172971843362727271506851161\ 7427488590584213688692762180776687042560, 5827827830562139836840075043784000036\ 5385665986214277910663934454671088740475012882050777969794518345221807766870425\ 60] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 6, 90, 804, 7632, 67275, 596997, 5240430, 46041330, 403645065, 3538952607, 31012832571, 271769305281, 2381283217365, 20864922999027, 182814567963534, 1601779612673376, 14034331684209936, 122964561836502180, 1077376273579609011, 9439621889969844075, 82706868230640400770, 724650297006304507356, 6349145999556338246169, 55629112180634901019389, 487403818539215388430851, 4270470424648611321867177, 37416443531234242389611847, 327830449646080279948983051, 2872341496087922543259923670, 25166501973523337172914295402, 220500529586272116054730559730, 1931952385677763569675763661802, 16927124967073097723065354006623, 148309845387075686361371495794869, 1299441593299827468497502860388162, 11385275535345336860569639646191458, 99754001781612836499612529313500437, 874011422951448636615785941526602167, 7657797720412083535430484954175772337, 67095079522579643508581373520551072453] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 11, 86, 848, 7469, 67439, 594260, 5242955, 46007195, 403703819, 3538486217, 31013846192, 271762282067, 2381299901522, 20864816975966, 182814849426167, 1601778023596640, 14034336411598790, 122964537835594739, 1077376351618301771, 9439621523730434234, 82706869502415835391, 724650291376340312075, 6349146020104742342093, 55629112093575545010215, 487403818868962871487632, 4270470423295189750049951, 37416443536496588811675038, 327830449624945937637410048, 2872341496171549657868553179, 25166501973192169129403849942, 220500529587596998459199238068, 1931952385672560878782094017355, 16927124967094041246913698042125, 148309845386993799602857635533168, 1299441593300158033383309946149239, 11385275535344046378468837896896409, 99754001781618048733990871049280211, 874011422951428282853959190937825071, 7657797720412165669076122295136236162, 67095079522579322332043498750670324767] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 20 19 18 17 16 - (2763396 t - 9858804 t - 6326567 t + 19223858 t + 5691465 t 15 14 13 12 11 - 10576320 t - 6803829 t + 4893649 t + 2831047 t - 1778960 t 10 9 8 7 6 5 - 228344 t + 179233 t - 9152 t + 43515 t - 33810 t + 7491 t 4 3 2 / 11 10 + 644 t - 670 t + 158 t - 19 t + 1) / ((4290 t - 13239 t / 9 8 7 6 5 4 3 - 1666 t + 9764 t + 2370 t - 3093 t - 782 t + 500 t + 170 t 2 10 9 8 7 6 - 114 t + 19 t - 1) (2622 t + 529 t - 3257 t - 49 t + 1753 t 5 4 3 2 - 582 t - 188 t + 132 t - 10 t - 6 t + 1)) and in Maple notation -(2763396*t^20-9858804*t^19-6326567*t^18+19223858*t^17+5691465*t^16-10576320*t^ 15-6803829*t^14+4893649*t^13+2831047*t^12-1778960*t^11-228344*t^10+179233*t^9-\ 9152*t^8+43515*t^7-33810*t^6+7491*t^5+644*t^4-670*t^3+158*t^2-19*t+1)/(4290*t^ 11-13239*t^10-1666*t^9+9764*t^8+2370*t^7-3093*t^6-782*t^5+500*t^4+170*t^3-114*t ^2+19*t-1)/(2622*t^10+529*t^9-3257*t^8-49*t^7+1753*t^6-582*t^5-188*t^4+132*t^3-\ 10*t^2-6*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 19 18 17 16 15 t (4006224 t - 7733328 t - 12227482 t + 11797341 t + 9566691 t 14 13 12 11 10 - 819030 t - 8162025 t - 2182576 t + 4088827 t + 1650657 t 9 8 7 6 5 4 - 1945558 t - 94244 t + 383964 t - 21141 t - 58935 t + 16650 t 3 2 / 11 10 9 + 835 t - 1096 t + 189 t - 11) / ((4290 t - 13239 t - 1666 t / 8 7 6 5 4 3 2 + 9764 t + 2370 t - 3093 t - 782 t + 500 t + 170 t - 114 t + 19 t 10 9 8 7 6 5 4 - 1) (2622 t + 529 t - 3257 t - 49 t + 1753 t - 582 t - 188 t 3 2 + 132 t - 10 t - 6 t + 1)) and in Maple notation t*(4006224*t^19-7733328*t^18-12227482*t^17+11797341*t^16+9566691*t^15-819030*t^ 14-8162025*t^13-2182576*t^12+4088827*t^11+1650657*t^10-1945558*t^9-94244*t^8+ 383964*t^7-21141*t^6-58935*t^5+16650*t^4+835*t^3-1096*t^2+189*t-11)/(4290*t^11-\ 13239*t^10-1666*t^9+9764*t^8+2370*t^7-3093*t^6-782*t^5+500*t^4+170*t^3-114*t^2+ 19*t-1)/(2622*t^10+529*t^9-3257*t^8-49*t^7+1753*t^6-582*t^5-188*t^4+132*t^3-10* t^2-6*t+1) This theorem took, 0.587, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 19, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 38, 7, 49, 94, 7, 49, 49, 49, 343, 224, 38, 266, 338, 7, 49, 193, 49, 343, 594, 94, 658, 753, 7, 49, 49, 49, 343, 266, 49, 343, 658, 49, 343 , 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 12, 0, 0, 37, 12, 84, 72, 0, 0, 84, 0, 0, 223, 37, 259, 277, 12, 84, 168 , 84, 588, 545, 72, 504, 788, 0, 0, 84, 0, 0, 259, 84, 588, 504, 0, 0, 588, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [166878279185797473775922545788465213220012767928933764865219620818446560520976\ 8324213831048613416398815212770500000000000, 1668782791857974737759225457884652\ 1322001276775585442399229840474477844574421859563304563106315046854121523545000\ 00000000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 94, 753, 6780, 59563, 521542, 4572468, 40003056, 349974001, 3060886612, 26768207391, 234084121248, 2046974621152, 17899785746074, 156523679981607, 1368707493311598, 11968516886090461, 104657299285252390, 915162952932467544, 8002527604882250130, 69977084250369348031, 611905638025516602850, 5350730024195478915375, 46788767699161492544466, 409138328417426786317492, 3577657161648034264136830, 31284359820851117459453169, 273562032797571953546106060, 2392127761641476875691433283, 20917651337709620097806079562, 182911692348572759729461782090, 1599447597875345796137018559306, 13986162311227743370194693718975, 122300184408985618530959522011216, 1069438118379102454086069282620997, 9351563078384847133160217602200968, 81773531824056280193602559041126744, 715058055091875060724863594423014668, 6252732525340880683836986604177813297, 54676209511801435973278597618156464324] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 12, 72, 788, 6797, 59871, 524511, 4581167, 40060610, 350208552, 3062039427, 26773881533, 234111132305, 2047109862345, 17900446249218, 156526948294754, 1368723634139387, 11968596517760337, 104657692915663875, 915164896062219971, 8002537202726522216, 69977131644720702990, 611905872040398033021, 5350731179688690913169, 46788773404050767413295, 409138356583781600911161, 3577657300707884615527674, 31284360507397929211935284, 273562036187091835282221017, 2392127778375669060879431295, 20917651420327166054961445935, 182911692756459801865720249193, 1599447599889107259104742010472, 13986162321169806491441443691502, 122300184458070213965290195203699, 1069438118621436327178674600893819, 9351563079581265631466823387142337, 81773531829963079021177056347996877, 715058055121037326876017007771206360, 6252732525484856771148675436543671224, 54676209512512255692034692301175057945] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.893, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 20, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + x y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 28, 7, 49, 110, 7, 49, 49, 49, 343, 145, 28, 196, 413, 7, 49, 193, 49, 343, 667, 110, 770, 908, 7, 49, 49, 49, 343, 196, 49, 343, 770, 49, 343, 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 12, 0, 0, 48, 12, 84, 91, 0, 0, 84, 0, 0, 306, 48, 336, 310, 12, 84, 168 , 84, 588, 519, 91, 637, 946, 0, 0, 84, 0, 0, 336, 84, 588, 637, 0, 0, 588, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [144895898103912891577807319050136579652271255465112090939717087186435479335511\ 07988064342929281001993146771387915121459200, 144895898103912891577807321301208\ 9290986288104860323289374842339564811866574289751613782952939798711186677138791\ 5121459200] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 110, 908, 8411, 74264, 657677, 5797025, 51149930, 450744785, 3972691757, 35003485757, 308422145885, 2717415644432, 23942460175712, 210948940181366, 1858601875036523, 16375499067410987, 144278876370215354, 1271190891736003721, 11200019858523465170, 98679462909643088519, 869430280658173694714, 7660246410568406200715, 67491754529359296497057, 594646264819440822373709, 5239220445406574750915168, 46160940514986042686184770, 406707916060462992010208783, 3583361324955874685020634255, 31571744433829072240854156524, 278167607499943911792856421276, 2450837584355667814871535077159, 21593473513490457759441809860916, 190252549313726917688355387109493, 1676248728474448048536840416044427, 14768841783446618411538705239731652, 130123103999573681281381205349567998, 1146469197974705642639498088115000832, 10101139471042886131462194259342664990, 88997610048055613553952340817014750138] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 12, 91, 946, 8266, 74194, 656095, 5801266, 51133453, 450827080, 3972332623, 35003793529, 308413645543, 2717414109400, 23942348253571, 210949067433676, 1858600820550781, 16375502045038447, 144278858177210470, 1271190905436397789, 11200019461622337910, 98679462827835927994, 869430275017637628403, 7660246414243971952099, 67491754468634850177034, 594646264923449302185298, 5239220444455071322318156, 46160940515485121636224957, 406707916041485216078152645, 3583361324950397403559419091, 31571744433546275284888437430, 278167607500015034718668205397, 2450837584352310443731722583081, 21593473513493597492519389424041, 190252549313676694512383632913941, 1676248728474459059435729110236790, 14768841783445688550899659027150687, 130123103999573280603591574486084678, 1146469197974691382983707273109933286, 10101139471042884509535671178340874254, 88997610048055431472036997816783418601] No generating functions were found with only , 40, terms to guess, but they definitely exist. This theorem took, 4.925, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 21, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + x y + y + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 7, 7, 7, 49, 21, 7, 49, 81, 7, 49, 49, 49, 343, 99, 21, 147, 298, 7, 49, 130, 49, 343, 455, 81, 567, 624, 7, 49, 49, 49, 343, 147, 49, 343, 567, 49, 343 , 343, 343, 2401] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 41, 9, 63, 71, 0, 0, 63, 0, 0, 269, 41, 287, 186, 9, 63, 126, 63, 441, 391, 71, 497, 659, 0, 0, 63, 0, 0, 287, 63, 441, 497, 0, 0, 441, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [701829909586202358220405128773680326529304566036156000331514484497476575819536\ 921209737694374779656400837015083417600, 70182990958620235822040513180305155679\ 5925835788071317484923688152887416825990775485167714082906331274582423083417600 ] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 7, 81, 624, 5274, 40735, 323587, 2508220, 19613499, 152276865, 1185521616, 9209868160, 71609782399, 556428616111, 4324778184390, 33607215723393, 261178742366592, 2029630556491372, 15772754928467869, 122571679267852618, 952524728856309120, 7402185089524606677, 57523422930203255871, 447021851686862509534, 3473866514809219528960, 26995866405501753190258, 209788423657881871595928, 1630293131055607684320975, 12669220954867389245322996 , 98454166651748220612289048, 765100173964643964845608774, 5945693180497749631190383795, 46204757041916789209028369289, 359063190525343146702768271473, 2790326869582125236655981590166, 21683993885526074281451247313828, 168509143597369407187510954650241, 1309506524176998378956018374120831, 10176345927687512365348864696280811, 79081710945200360983898146272429144, 614554286073690117860627198859533418] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 71, 659, 5102, 41388, 320589, 2520696, 19559993, 152513105, 1184546177, 9214298064, 71591778582, 556511227752, 4324443932660, 33608752490897, 261172521752504, 2029659116944650, 15772639058824632, 122572209909189561, 952522570058465606, 7402194948333769655, 57523382708302084574, 447022034863195652622, 3473865765448185751341, 26995869809139410409771, 209788409697720057966245, 1630293194303480960132126, 12669220694815135907477357 , 98454167827119668113633437, 765100169120654031077430573, 5945693202341590251235821753, 46204756951693219173142317938, 359063190931324356575915828567, 2790326867901726005209142914079, 21683993893071863098735757682501, 168509143566073901665098991084941, 1309506524317255210723436403976524, 10176345927104698727504411988820130, 79081710947807493232276547463853739, 614554286062836943182540234070881104] Using the found enumerative automaton with, 20, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 19 18 17 16 15 14 - (30312 t + 283584 t + 302412 t - 1260696 t - 140827 t + 1187455 t 13 12 11 10 9 8 + 245487 t - 801748 t - 122131 t + 392697 t - 50439 t - 81405 t 7 6 5 4 3 2 / + 32648 t + 323 t - 3365 t + 1056 t - 62 t - 49 t + 13 t - 1) / (( / 10 9 8 7 6 5 4 1179 t + 3590 t - 2772 t - 4149 t + 2684 t + 1241 t - 1030 t 3 2 10 9 8 7 6 + 76 t + 68 t - 16 t + 1) (357 t - 192 t - 1470 t + 1315 t + 518 t 5 4 3 2 - 801 t + 78 t + 114 t - 24 t - 4 t + 1)) and in Maple notation -(30312*t^19+283584*t^18+302412*t^17-1260696*t^16-140827*t^15+1187455*t^14+ 245487*t^13-801748*t^12-122131*t^11+392697*t^10-50439*t^9-81405*t^8+32648*t^7+ 323*t^6-3365*t^5+1056*t^4-62*t^3-49*t^2+13*t-1)/(1179*t^10+3590*t^9-2772*t^8-\ 4149*t^7+2684*t^6+1241*t^5-1030*t^4+76*t^3+68*t^2-16*t+1)/(357*t^10-192*t^9-\ 1470*t^8+1315*t^7+518*t^6-801*t^5+78*t^4+114*t^3-24*t^2-4*t+1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 18 17 16 15 14 - (23238 t - 272043 t - 893397 t + 1742172 t + 1781952 t 13 12 11 10 9 - 3812348 t - 290781 t + 2992959 t - 831693 t - 983777 t 8 7 6 5 4 3 + 519829 t + 109339 t - 118134 t + 8529 t + 10446 t - 2308 t 2 / 10 9 8 7 - 211 t + 109 t - 9) t / ((1179 t + 3590 t - 2772 t - 4149 t / 6 5 4 3 2 10 + 2684 t + 1241 t - 1030 t + 76 t + 68 t - 16 t + 1) (357 t 9 8 7 6 5 4 3 2 - 192 t - 1470 t + 1315 t + 518 t - 801 t + 78 t + 114 t - 24 t - 4 t + 1)) and in Maple notation -(23238*t^18-272043*t^17-893397*t^16+1742172*t^15+1781952*t^14-3812348*t^13-\ 290781*t^12+2992959*t^11-831693*t^10-983777*t^9+519829*t^8+109339*t^7-118134*t^ 6+8529*t^5+10446*t^4-2308*t^3-211*t^2+109*t-9)*t/(1179*t^10+3590*t^9-2772*t^8-\ 4149*t^7+2684*t^6+1241*t^5-1030*t^4+76*t^3+68*t^2-16*t+1)/(357*t^10-192*t^9-\ 1470*t^8+1315*t^7+518*t^6-801*t^5+78*t^4+114*t^3-24*t^2-4*t+1) This theorem took, 0.253, seconds. to state and prove ------------------------------------------------------------------ This concludes this webbook, that took, 66.354, seconds. to generate. k is , 8 Counting the Occurrences of Coefficients that Appear in the Expansion of, n P(x, y) , modolu , 3 For all Polynomials that are Sums of, 8, Monomials taken from, 2 2 2 2 2 2 {1, x, y, x , y , x y, x y , x y, x y } By Shalosh B. Ekhad In this webbook, we will consider the sequences described in the title, that\ after normalization and weeding out obvious symmetry, concerns the following set of, 6, polynomials 2 2 2 2 {x y + x y + x + x y + y + x + y + 1, 2 2 2 2 2 x y + x y + x + x y + y + x + y + 1, 2 2 2 2 2 x y + x y + x y + x y + y + x + y + 1, 2 2 2 2 2 2 x y + x y + x y + x + y + x + y + 1, 2 2 2 2 2 2 x y + x y + x y + x + x y + y + y + 1, 2 2 2 2 2 2 x y + x y + x y + x + x y + y + x + y} by finding enumerative automata with at most, 500, states . ----------------------------------------------------------------------------\ ---- Theorem number, 1, :consider the sequence, 2 2 2 2 n (x y + x y + x + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 8, 11, 8, 64, 62, 11, 88, 101, 8, 64, 88, 64, 512, 466, 62, 496, 462, 11, 88, 170, 88, 704, 727, 101, 808, 857, 8, 64, 88, 64, 512, 496, 88, 704, 808, 64 , 512, 704, 512, 4096] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 15, 7, 56, 85, 0, 0, 56, 0, 0, 90, 15, 120, 199, 7, 56, 154, 56 , 448, 539, 85, 680, 793, 0, 0, 56, 0, 0, 120, 56, 448, 680, 0, 0, 448, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [106150699881236298477783228703773002143655376392264918939127077116392487076271\ 76791625303103568762662844928936671113910419456, 106150699881236298459220189634\ 8133089814022479089977763434854516480771925108087045261959962397796792998645897\ 0411169596047360] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 11, 101, 857, 6989, 55841, 441701, 3479417, 27385949, 215753681, 1702772501 , 13466108777, 106708746509, 847126553921, 6735621765701, 53625912238937, 427399068692669, 3409272810992561, 27213307945501301, 217334418573773897, 1736409021251288429, 13877503644111565601, 110936564088661819301, 886987448381672537657, 7092847813784273237789, 56724364863773924529041, 453683877715286756319701, 3628802096847450945874217, 29026389832499872985484749 , 232186890023152798118298881, 1857349413376248784932362501, 14857919390928435876120247577, 118858091253646675068050813309, 950833104895419415276115851121, 7606474878887039098411355067701, 60850658221995666907271432841737, 486798415683852569386705777447469, 3894346198729418915287388501121761, 31154522698438593894091374019315301, 249234699584367579170702171241160697, 1993868701382214906545806445248651229] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 85, 793, 6733, 54817, 437605, 3463033, 27320413, 215491537, 1701723925, 13461914473, 106691969293, 847059445057, 6735353330245, 53624838497113, 427394773725373, 3409255631123377, 27213239226024565, 217334143695866953, 1736407921739660653, 13877499246065054497, 110936546496475774885, 886987378012928359993, 7092847532309296527133, 56724363737874017686417, 453683873211687128949205, 3628802078833052436392233, 29026389760442278947556813 , 232186889734922421966587137, 1857349412223327280325515525, 14857919386316749857692859673, 118858091235199930994341261693, 950833104821632438981277644657, 7606474878591891193232002241845, 60850658220815075286554021538313, 486798415679130202903836132233773, 3894346198710529449355909920266977, 31154522698363036030365459695896165, 249234699584065347715798513947484153, 1993868701381005980726191816073945053] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (2 t - 1) (22 t - 10 t + 1) - --------------------------------------- (6 t - 1) (4 t - 1) (8 t - 1) (5 t - 1) and in Maple notation -(2*t-1)*(22*t^2-10*t+1)/(6*t-1)/(4*t-1)/(8*t-1)/(5*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (196 t - 76 t + 7) --------------------------------------- (6 t - 1) (4 t - 1) (8 t - 1) (5 t - 1) and in Maple notation t*(196*t^2-76*t+7)/(6*t-1)/(4*t-1)/(8*t-1)/(5*t-1) This theorem took, 0.065, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 2, :consider the sequence, 2 2 2 2 2 n (x y + x y + x + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 8, 10, 8, 64, 56, 10, 80, 103, 8, 64, 80, 64, 512, 424, 56, 448, 434, 10, 80, 181, 80, 640, 698, 103, 824, 1006, 8, 64, 80, 64, 512, 448, 80, 640, 824, 64, 512, 640, 512, 4096] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 24, 9, 72, 102, 0, 0, 72, 0, 0, 144, 24, 192, 294, 9, 72, 180, 72, 576, 666, 102, 816, 993, 0, 0, 72, 0, 0, 192, 72, 576, 816, 0, 0, 576, 0, 0 ] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [105727155086779941914652362890948886050400778666105951998227382726668106467198\ 172869823854315008950859559059984406672310272000, 10572715508677994191465236289\ 0948886044769867766747060152597544320684994663919015187153426344974724799987123\ 171174255689728000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 10, 103, 1006, 9433, 86614, 787471, 7125610, 64322017, 579869650, 5223776071, 47039172022, 423480409561, 3811971242302, 34311014926543, 308815663672210, 2779424349843841, 25015239382298458, 225139271022104935, 2026264093013258686, 18236430432049356697, 164128143359125402342, 1477154644449759509455, 13294398602622788216314, 119649621580903930283233, 1076846765677093634511010, 9691621751377451510928583, 87224600077755081115717126, 785021422340533893514303513, 7065192909561852599249311054, 63586736729887662973717424911, 572280633294317439162708076450, 5150525713303819795486513721857, 46354731488139109847869939815850, 417192583735871630605416028309351, 3754733255338676900241372926853262, 33792599306639795246720051361781081, 304133393802774208215144144733484662, 2737200544440312066263670257057039503, 24634804901040740344492528466682261130, 221713244114761876204244128186837997281 ] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 102, 993, 9312, 85845, 783414, 7106109, 64232208, 579462129, 5221928694, 47030764617, 423441988944, 3811795187997, 34310207164902, 308811956244549, 2779407334875744, 25015161307521561, 225138912822993990, 2026262449776225393, 18236422893973968768, 164128108779674370597, 1477154485822387665750, 13294397874941282454285, 119649618242746436220912, 1076846750363628316370049, 9691621681128389308719702, 87224599755494202569711193, 785021420862193241197520112, 7065192902780109520871945325, 63586736698777085645948101254, 572280633151600746107152752789, 5150525712649121110406589210048, 46354731485135744497489763915049, 417192583722093989464430191987494, 3754733255275473335195287489412097, 33792599306349855146250114552575904, 304133393801444136691788459335571957, 2737200544434210494749728733763688822, 24634804901012749988883090221700387549, 221713244114633473217411578863586309584] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 4 3 2 102 t - 395 t + 268 t - 84 t + 14 t - 1 - ----------------------------------------------------- 3 2 (t - 1) (31 t - 27 t + 9 t - 1) (9 t - 1) (5 t - 1) and in Maple notation -(102*t^5-395*t^4+268*t^3-84*t^2+14*t-1)/(t-1)/(31*t^3-27*t^2+9*t-1)/(9*t-1)/(5 *t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 3 2 3 t (251 t - 362 t + 178 t - 38 t + 3) ----------------------------------------------------- 3 2 (t - 1) (31 t - 27 t + 9 t - 1) (9 t - 1) (5 t - 1) and in Maple notation 3*t*(251*t^4-362*t^3+178*t^2-38*t+3)/(t-1)/(31*t^3-27*t^2+9*t-1)/(9*t-1)/(5*t-1 ) This theorem took, 0.154, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 3, :consider the sequence, 2 2 2 2 2 n (x y + x y + x y + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 8, 11, 8, 64, 62, 11, 88, 101, 8, 64, 88, 64, 512, 466, 62, 496, 462, 11, 88, 170, 88, 704, 727, 101, 808, 857, 8, 64, 88, 64, 512, 496, 88, 704, 808, 64 , 512, 704, 512, 4096] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 15, 7, 56, 85, 0, 0, 56, 0, 0, 90, 15, 120, 199, 7, 56, 154, 56 , 448, 539, 85, 680, 793, 0, 0, 56, 0, 0, 120, 56, 448, 680, 0, 0, 448, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [106150699881236298477783228703773002143655376392264918939127077116392487076271\ 76791625303103568762662844928936671113910419456, 106150699881236298459220189634\ 8133089814022479089977763434854516480771925108087045261959962397796792998645897\ 0411169596047360] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 11, 101, 857, 6989, 55841, 441701, 3479417, 27385949, 215753681, 1702772501 , 13466108777, 106708746509, 847126553921, 6735621765701, 53625912238937, 427399068692669, 3409272810992561, 27213307945501301, 217334418573773897, 1736409021251288429, 13877503644111565601, 110936564088661819301, 886987448381672537657, 7092847813784273237789, 56724364863773924529041, 453683877715286756319701, 3628802096847450945874217, 29026389832499872985484749 , 232186890023152798118298881, 1857349413376248784932362501, 14857919390928435876120247577, 118858091253646675068050813309, 950833104895419415276115851121, 7606474878887039098411355067701, 60850658221995666907271432841737, 486798415683852569386705777447469, 3894346198729418915287388501121761, 31154522698438593894091374019315301, 249234699584367579170702171241160697, 1993868701382214906545806445248651229] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 85, 793, 6733, 54817, 437605, 3463033, 27320413, 215491537, 1701723925, 13461914473, 106691969293, 847059445057, 6735353330245, 53624838497113, 427394773725373, 3409255631123377, 27213239226024565, 217334143695866953, 1736407921739660653, 13877499246065054497, 110936546496475774885, 886987378012928359993, 7092847532309296527133, 56724363737874017686417, 453683873211687128949205, 3628802078833052436392233, 29026389760442278947556813 , 232186889734922421966587137, 1857349412223327280325515525, 14857919386316749857692859673, 118858091235199930994341261693, 950833104821632438981277644657, 7606474878591891193232002241845, 60850658220815075286554021538313, 486798415679130202903836132233773, 3894346198710529449355909920266977, 31154522698363036030365459695896165, 249234699584065347715798513947484153, 1993868701381005980726191816073945053] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (2 t - 1) (22 t - 10 t + 1) - --------------------------------------- (6 t - 1) (4 t - 1) (8 t - 1) (5 t - 1) and in Maple notation -(2*t-1)*(22*t^2-10*t+1)/(6*t-1)/(4*t-1)/(8*t-1)/(5*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (196 t - 76 t + 7) --------------------------------------- (6 t - 1) (4 t - 1) (8 t - 1) (5 t - 1) and in Maple notation t*(196*t^2-76*t+7)/(6*t-1)/(4*t-1)/(8*t-1)/(5*t-1) This theorem took, 0.063, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 4, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 8, 8, 8, 64, 52, 8, 64, 101, 8, 64, 64, 64, 512, 404, 52, 416, 448, 8, 64, 233, 64, 512, 700, 101, 808, 992, 8, 64, 64, 64, 512, 416, 64, 512, 808, 64, 512, 512, 512, 4096] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 13, 0, 0, 32, 13, 104, 112, 0, 0, 104, 0, 0, 184, 32, 256, 296, 13, 104, 208, 104, 832, 836, 112, 896, 1081, 0, 0, 104, 0, 0, 256, 104, 832, 896, 0, 0, 832, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [902314629962584286953312931444880639871786314037801714880910348223623666833062\ 990802753261456214553849366429566911453478256640, 90231462996258428695331293626\ 6081495296811905475899801305009859853885904524589590205542570195748031353555107\ 710911453478256640] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 101, 992, 9473, 87872, 804725, 7314056, 66200033, 597750680, 5389981925, 48563606480, 437356129697, 3937706587664, 35447332075445, 319068462920888, 2871843169896257, 25847805569488808, 232636795108626149, 2093766458219657792, 18844089082520614337, 169597837520671320416, 1526386170543042359669, 13737506243908075575656, 123637724004922563381281, 1112740435054259360093240, 10014668958764760887351909, 90132048357822423874486064, 811188587949369271718827361, 7300698134139256023240939824, 65706287862786774468804570293, 591356616523534408642299334808, 5322209691410250484732503391361, 47899888014136227640821549666632, 431098996521408554665947597042533, 3879890993113006002707160591166112, 34919019073849028758382739268576385, 314271172420767199548324430302375296, 2828440555998971734884523181458923893, 25455965027469523853524637638367502920, 229103685378176315176821873436495756385] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 13, 112, 1081, 9928, 90493, 819304, 7395961, 66658864, 600323149, 5404401280, 48644436121, 437809222456, 3940246425469, 35461569261496, 319148270205337, 2872290533708512, 25850313290297869, 232650852266172688, 2093845256337125497, 18844530789408404008, 169600313531279688445, 1526400049947465938824, 13737584045621245204537, 123638160126415054109200, 1112742879755647849664845, 10014682662663747262189600, 90132125175730865779981273, 811189018556093916714347416, 7300700547927330580372476925, 65706301393399594502290399192, 591356692370078359790426484313, 5322210116571941669626384449472, 47899890397401887940730218244621, 431099009880926416529627200877488, 3879891068000468392861057023707833, 34919019493634539075771441433257864, 314271174773896402329264306853976701, 2828440569189558138864781007147462824, 25455965101410032294474127830086071289, 229103685792653626576442154634566880240 ] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 4 3 2 72 t + 95 t - 39 t + 9 t - 1 -------------------------------------------- 2 (9 t - 1) (t + 1) (9 t + 4 t - 1) (5 t - 1) and in Maple notation (72*t^4+95*t^3-39*t^2+9*t-1)/(9*t-1)/(t+1)/(9*t^2+4*t-1)/(5*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 3 2 (333 t + 139 t - 109 t + 13) t - -------------------------------------------- 2 (9 t - 1) (t + 1) (9 t + 4 t - 1) (5 t - 1) and in Maple notation -(333*t^3+139*t^2-109*t+13)*t/(9*t-1)/(t+1)/(9*t^2+4*t-1)/(5*t-1) This theorem took, 0.141, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 5, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + x y + y + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 8, 10, 8, 64, 56, 10, 80, 103, 8, 64, 80, 64, 512, 424, 56, 448, 434, 10, 80, 181, 80, 640, 698, 103, 824, 1006, 8, 64, 80, 64, 512, 448, 80, 640, 824, 64, 512, 640, 512, 4096] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 9, 0, 0, 24, 9, 72, 102, 0, 0, 72, 0, 0, 144, 24, 192, 294, 9, 72, 180, 72, 576, 666, 102, 816, 993, 0, 0, 72, 0, 0, 192, 72, 576, 816, 0, 0, 576, 0, 0 ] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [105727155086779941914652362890948886050400778666105951998227382726668106467198\ 172869823854315008950859559059984406672310272000, 10572715508677994191465236289\ 0948886044769867766747060152597544320684994663919015187153426344974724799987123\ 171174255689728000] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 10, 103, 1006, 9433, 86614, 787471, 7125610, 64322017, 579869650, 5223776071, 47039172022, 423480409561, 3811971242302, 34311014926543, 308815663672210, 2779424349843841, 25015239382298458, 225139271022104935, 2026264093013258686, 18236430432049356697, 164128143359125402342, 1477154644449759509455, 13294398602622788216314, 119649621580903930283233, 1076846765677093634511010, 9691621751377451510928583, 87224600077755081115717126, 785021422340533893514303513, 7065192909561852599249311054, 63586736729887662973717424911, 572280633294317439162708076450, 5150525713303819795486513721857, 46354731488139109847869939815850, 417192583735871630605416028309351, 3754733255338676900241372926853262, 33792599306639795246720051361781081, 304133393802774208215144144733484662, 2737200544440312066263670257057039503, 24634804901040740344492528466682261130, 221713244114761876204244128186837997281 ] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 9, 102, 993, 9312, 85845, 783414, 7106109, 64232208, 579462129, 5221928694, 47030764617, 423441988944, 3811795187997, 34310207164902, 308811956244549, 2779407334875744, 25015161307521561, 225138912822993990, 2026262449776225393, 18236422893973968768, 164128108779674370597, 1477154485822387665750, 13294397874941282454285, 119649618242746436220912, 1076846750363628316370049, 9691621681128389308719702, 87224599755494202569711193, 785021420862193241197520112, 7065192902780109520871945325, 63586736698777085645948101254, 572280633151600746107152752789, 5150525712649121110406589210048, 46354731485135744497489763915049, 417192583722093989464430191987494, 3754733255275473335195287489412097, 33792599306349855146250114552575904, 304133393801444136691788459335571957, 2737200544434210494749728733763688822, 24634804901012749988883090221700387549, 221713244114633473217411578863586309584] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 5 4 3 2 102 t - 395 t + 268 t - 84 t + 14 t - 1 - ----------------------------------------------------- 3 2 (t - 1) (31 t - 27 t + 9 t - 1) (9 t - 1) (5 t - 1) and in Maple notation -(102*t^5-395*t^4+268*t^3-84*t^2+14*t-1)/(t-1)/(31*t^3-27*t^2+9*t-1)/(9*t-1)/(5 *t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 4 3 2 3 t (251 t - 362 t + 178 t - 38 t + 3) ----------------------------------------------------- 3 2 (t - 1) (31 t - 27 t + 9 t - 1) (9 t - 1) (5 t - 1) and in Maple notation 3*t*(251*t^4-362*t^3+178*t^2-38*t+3)/(t-1)/(31*t^3-27*t^2+9*t-1)/(9*t-1)/(5*t-1 ) This theorem took, 0.143, seconds. to state and prove ----------------------------------------------------------------------------\ ---- Theorem number, 6, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + x y + y + x + y) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 8, 11, 8, 64, 62, 11, 88, 101, 8, 64, 88, 64, 512, 466, 62, 496, 462, 11, 88, 170, 88, 704, 727, 101, 808, 857, 8, 64, 88, 64, 512, 496, 88, 704, 808, 64 , 512, 704, 512, 4096] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 7, 0, 0, 15, 7, 56, 85, 0, 0, 56, 0, 0, 90, 15, 120, 199, 7, 56, 154, 56 , 448, 539, 85, 680, 793, 0, 0, 56, 0, 0, 120, 56, 448, 680, 0, 0, 448, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [106150699881236298477783228703773002143655376392264918939127077116392487076271\ 76791625303103568762662844928936671113910419456, 106150699881236298459220189634\ 8133089814022479089977763434854516480771925108087045261959962397796792998645897\ 0411169596047360] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 11, 101, 857, 6989, 55841, 441701, 3479417, 27385949, 215753681, 1702772501 , 13466108777, 106708746509, 847126553921, 6735621765701, 53625912238937, 427399068692669, 3409272810992561, 27213307945501301, 217334418573773897, 1736409021251288429, 13877503644111565601, 110936564088661819301, 886987448381672537657, 7092847813784273237789, 56724364863773924529041, 453683877715286756319701, 3628802096847450945874217, 29026389832499872985484749 , 232186890023152798118298881, 1857349413376248784932362501, 14857919390928435876120247577, 118858091253646675068050813309, 950833104895419415276115851121, 7606474878887039098411355067701, 60850658221995666907271432841737, 486798415683852569386705777447469, 3894346198729418915287388501121761, 31154522698438593894091374019315301, 249234699584367579170702171241160697, 1993868701382214906545806445248651229] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 7, 85, 793, 6733, 54817, 437605, 3463033, 27320413, 215491537, 1701723925, 13461914473, 106691969293, 847059445057, 6735353330245, 53624838497113, 427394773725373, 3409255631123377, 27213239226024565, 217334143695866953, 1736407921739660653, 13877499246065054497, 110936546496475774885, 886987378012928359993, 7092847532309296527133, 56724363737874017686417, 453683873211687128949205, 3628802078833052436392233, 29026389760442278947556813 , 232186889734922421966587137, 1857349412223327280325515525, 14857919386316749857692859673, 118858091235199930994341261693, 950833104821632438981277644657, 7606474878591891193232002241845, 60850658220815075286554021538313, 486798415679130202903836132233773, 3894346198710529449355909920266977, 31154522698363036030365459695896165, 249234699584065347715798513947484153, 1993868701381005980726191816073945053] Using the found enumerative automaton with, 6, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is 2 (2 t - 1) (22 t - 10 t + 1) - --------------------------------------- (6 t - 1) (4 t - 1) (8 t - 1) (5 t - 1) and in Maple notation -(2*t-1)*(22*t^2-10*t+1)/(6*t-1)/(4*t-1)/(8*t-1)/(5*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 2 t (196 t - 76 t + 7) --------------------------------------- (6 t - 1) (4 t - 1) (8 t - 1) (5 t - 1) and in Maple notation t*(196*t^2-76*t+7)/(6*t-1)/(4*t-1)/(8*t-1)/(5*t-1) This theorem took, 0.062, seconds. to state and prove ------------------------------------------------------------------ This concludes this webbook, that took, 1.006, seconds. to generate. k is , 9 Counting the Occurrences of Coefficients that Appear in the Expansion of, n P(x, y) , modolu , 3 For all Polynomials that are Sums of, 9, Monomials taken from, 2 2 2 2 2 2 {1, x, y, x , y , x y, x y , x y, x y } By Shalosh B. Ekhad In this webbook, we will consider the sequences described in the title, that\ after normalization and weeding out obvious symmetry, concerns the following set of, 1, polynomials 2 2 2 2 2 2 {x y + x y + x y + x + x y + y + x + y + 1} by finding enumerative automata with at most, 500, states . ----------------------------------------------------------------------------\ ---- Theorem number, 1, :consider the sequence, 2 2 2 2 2 2 n (x y + x y + x y + x + x y + y + x + y + 1) , modulo , 3 We have the following facts. The first, 41, terms staring at n=0, for the number of occurrences of, 1, are: [1, 9, 8, 9, 81, 8, 8, 72, 72, 9, 81, 72, 81, 729, 8, 8, 72, 72, 8, 72, 128, 72 , 648, 72, 72, 648, 648, 9, 81, 72, 81, 729, 72, 72, 648, 648, 81, 729, 648, 729, 6561] The first, 41, terms staring at n=0, for the number of occurrences of, 2, are: [0, 0, 8, 0, 0, 8, 8, 72, 72, 0, 0, 72, 0, 0, 8, 8, 72, 72, 8, 72, 128, 72, 648 , 72, 72, 648, 648, 0, 0, 72, 0, 0, 72, 72, 648, 648, 0, 0, 648, 0, 0] Just for kicks, for the googol-th term, the number of occurrences of i from \ 1 to, 2, are: [171981820753927918384132947003705581819718899508822540550228550353886133541617\ 680407328584630272, 17198182075392791838413294700370558181971889950882254055022\ 8550353886133541617680407328584630272] The first , 40, terms of the sequence, number of occurrences, 1, i shows up at the, 3 - 1, places are [1, 8, 72, 648, 5832, 52488, 472392, 4251528, 38263752, 344373768, 3099363912, 27894275208, 251048476872, 2259436291848, 20334926626632, 183014339639688, 1647129056757192, 14824161510814728, 133417453597332552, 1200757082375992968, 10806813741383936712, 97261323672455430408, 875351913052098873672, 7878167217468889863048, 70903504957220008767432, 638131544614980078906888, 5743183901534820710161992, 51688655113813386391457928, 465197896024320477523121352, 4186781064218884297708092168, 37681029577969958679372829512, 339129266201729628114355465608, 3052163395815566653029199190472, 27469470562340099877262792714248, 247225235061060898895365134428232, 2225027115549548090058286209854088, 20025244039945932810524575888686792, 180227196359513395294721182998181128, 1622044767235620557652490646983630152, 14598402905120585018872415822852671368, 131385626146085265169851742405674042312] The first , 40, terms of the sequence, number of occurrences, 2, i shows up at the, 3 - 1, places are [0, 8, 72, 648, 5832, 52488, 472392, 4251528, 38263752, 344373768, 3099363912, 27894275208, 251048476872, 2259436291848, 20334926626632, 183014339639688, 1647129056757192, 14824161510814728, 133417453597332552, 1200757082375992968, 10806813741383936712, 97261323672455430408, 875351913052098873672, 7878167217468889863048, 70903504957220008767432, 638131544614980078906888, 5743183901534820710161992, 51688655113813386391457928, 465197896024320477523121352, 4186781064218884297708092168, 37681029577969958679372829512, 339129266201729628114355465608, 3052163395815566653029199190472, 27469470562340099877262792714248, 247225235061060898895365134428232, 2225027115549548090058286209854088, 20025244039945932810524575888686792, 180227196359513395294721182998181128, 1622044767235620557652490646983630152, 14598402905120585018872415822852671368, 131385626146085265169851742405674042312] Using the found enumerative automaton with, 4, states, that we omit, it follows that the (rigorously) PROVED rational generating functions for these sparse subse\ quence are as follows. The generating function for the sequence for the number of occurrences of, 1, i in the sparse, 3 - 1, subseqeunce is t - 1 ------- 9 t - 1 and in Maple notation (t-1)/(9*t-1) The generating function for the sequence for the number of occurrences of, 2, i in the sparse, 3 - 1, subseqeunce is 8 t - ------- 9 t - 1 and in Maple notation -8*t/(9*t-1) This theorem took, 0.039, seconds. to state and prove ------------------------------------------------------------------ This concludes this webbook, that took, 0.067, seconds. to generate.