On the Ellenberg-Gijswijt sequence that is an upper bound for the size of Su\ n bsets of, F[7] , with No Three-Term Arithemtical Progressions By Shalosh B. Ekhad In the brilliant paper "On large subsets of F_q^n with no Three-Term Arithmetic Progressions" by Jordan S. Ellenberg and Dion Gijswijt arxiv: 1605.09223v1 the authors prove that if M[d,n] is the sum of the coefficients from x^0 to \ x^d of the polynomial 6 5 4 3 2 n (x + x + x + x + x + x + 1) n then, for any d, the size of any subset of, F[7] , with no three-term arithmetical progressions is <= n 2 M[d/2, n] + 7 - M[d, n] Taking d to be , 4 n, it is easy to see that the above is bounded by a const\ ant time the sequence (2 m) A(m):=Coefficient of, x , in the polynomial , 6 5 4 3 2 m (x + x + x + x + x + x + 1) Thanks to the Almkvist-Zeilberger algorithm, the sequence A(m) satisfies the\ linear recurrence equation with polynomial coefficients of order, 6 9 8 16807 (m + 5) (m + 4) (m + 3) (m + 2) (m + 1) (61954364 m + 2471526088 m 7 6 5 4 + 43555740711 m + 444948892199 m + 2902982148157 m + 12540539858273 m 3 2 + 35857751442160 m + 65416513062912 m + 69062568528240 m + 32131293198480) A(m) + 9604 (m + 5) (m + 4) (m + 3) (m + 2) ( 10 9 8 7 30977182 m + 1282228817 m + 23627702154 m + 255061241113 m 6 5 4 + 1784889468362 m + 8453435152050 m + 27416944756665 m 3 2 + 60072473953266 m + 85022651512365 m + 70134050237202 m 11 + 25589561044320) A(m + 1) + 343 (m + 5) (m + 4) (m + 3) (185863092 m 10 9 8 7 + 8158030632 m + 160947693147 m + 1882908943634 m + 14504393413080 m 6 5 4 + 77187876135194 m + 289288623873389 m + 762611280489828 m 3 2 + 1383465761658076 m + 1641154823467544 m + 1142082651295488 m 12 + 351568925497440) A(m + 2) - 98 (m + 5) (m + 4) (10718104972 m 11 10 9 + 507959800514 m + 10934496950976 m + 141316686716919 m 8 7 6 + 1220714960216955 m + 7421376049848614 m + 32542638033017622 m 5 4 3 + 103638195397364651 m + 237726185475113219 m + 382683210588763518 m 2 + 409909109624601888 m + 261952516243353096 m + 75390889660919280) 13 12 A(m + 3) - 7 (m + 5) (43058282980 m + 2234410026920 m 11 10 9 + 53105261413509 m + 765190460932343 m + 7455452412642412 m 8 7 6 + 51852621691301606 m + 264750782726548049 m + 1004382245544805119 m 5 4 3 + 2829902277376160178 m + 5845988992848138780 m + 8602097011427568968 m 2 + 8533040089794697344 m + 5111388957232668096 m + 1395391936001172480) 13 12 A(m + 4) - 2 (2 m + 9) (78620087916 m + 4158427748580 m 11 10 9 + 100712713685615 m + 1478283586300892 m + 14666643125703819 m 8 7 6 + 103820625997407676 m + 539196676131670593 m + 2079167947173061500 m 5 4 + 5949142453801767133 m + 12466745810231823208 m 3 2 + 18582798791790828156 m + 18640677110717662112 m + 11266033095950322816 m + 3094059186889534080) A(m + 5) + 16 (2 m + 11) 9 8 (2 m + 9) (4 m + 21) (m + 6) (4 m + 23) (61954364 m + 1913936812 m 7 6 5 4 + 26013889111 m + 204057271110 m + 1017360138830 m + 3340612151884 m 3 2 + 7217685277191 m + 9883211605914 m + 7771530248944 m + 2668846724320) A(m + 6) = 0 and in Maple notation: 16807*(m+5)*(m+4)*(m+3)*(m+2)*(m+1)*(61954364*m^9+2471526088*m^8+43555740711*m^ 7+444948892199*m^6+2902982148157*m^5+12540539858273*m^4+35857751442160*m^3+ 65416513062912*m^2+69062568528240*m+32131293198480)*A(m)+9604*(m+5)*(m+4)*(m+3) *(m+2)*(30977182*m^10+1282228817*m^9+23627702154*m^8+255061241113*m^7+ 1784889468362*m^6+8453435152050*m^5+27416944756665*m^4+60072473953266*m^3+ 85022651512365*m^2+70134050237202*m+25589561044320)*A(m+1)+343*(m+5)*(m+4)*(m+3 )*(185863092*m^11+8158030632*m^10+160947693147*m^9+1882908943634*m^8+ 14504393413080*m^7+77187876135194*m^6+289288623873389*m^5+762611280489828*m^4+ 1383465761658076*m^3+1641154823467544*m^2+1142082651295488*m+351568925497440)*A (m+2)-98*(m+5)*(m+4)*(10718104972*m^12+507959800514*m^11+10934496950976*m^10+ 141316686716919*m^9+1220714960216955*m^8+7421376049848614*m^7+32542638033017622 *m^6+103638195397364651*m^5+237726185475113219*m^4+382683210588763518*m^3+ 409909109624601888*m^2+261952516243353096*m+75390889660919280)*A(m+3)-7*(m+5)*( 43058282980*m^13+2234410026920*m^12+53105261413509*m^11+765190460932343*m^10+ 7455452412642412*m^9+51852621691301606*m^8+264750782726548049*m^7+ 1004382245544805119*m^6+2829902277376160178*m^5+5845988992848138780*m^4+ 8602097011427568968*m^3+8533040089794697344*m^2+5111388957232668096*m+ 1395391936001172480)*A(m+4)-2*(2*m+9)*(78620087916*m^13+4158427748580*m^12+ 100712713685615*m^11+1478283586300892*m^10+14666643125703819*m^9+ 103820625997407676*m^8+539196676131670593*m^7+2079167947173061500*m^6+ 5949142453801767133*m^5+12466745810231823208*m^4+18582798791790828156*m^3+ 18640677110717662112*m^2+11266033095950322816*m+3094059186889534080)*A(m+5)+16* (2*m+11)*(2*m+9)*(4*m+21)*(m+6)*(4*m+23)*(61954364*m^9+1913936812*m^8+ 26013889111*m^7+204057271110*m^6+1017360138830*m^5+3340612151884*m^4+ 7217685277191*m^3+9883211605914*m^2+7771530248944*m+2668846724320)*A(m+6) = 0 The growth-constant is the cubic root of the largest root of the algebraic e\ quation in, M 6 5 4 3 2 1024 M - 5076 M - 4865 M - 16954 M + 1029 M + 4802 M + 16807 = 0 and in Maple notation 1024*M^6-5076*M^5-4865*M^4-16954*M^3+1029*M^2+4802*M+16807 = 0 that happens to be equal to 6 5 4 3 2 RootOf(1024 _Z - 5076 _Z - 4865 _Z - 16954 _Z + 1029 _Z + 4802 _Z + 16807, index = 2) and in Maple notation RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2) that equals 6.1562048632167384164 as you can see, this is less than, 7, . n so indeed the size of the largest such set it is exponentially less than, 7 The asymptotic expansion to order, 2, of the sequence A(m) is / 5 5 m 1/2 |197675335144188928 %1 2830664220487221649152 %1 %2 %1 (1/m) |---------------------- + ---------------------- ---- \ %2 29279 m %2 4 4 1999569864878508572 %1 2876693522599598459691 %1 - ----------------------- - ---------------------- ---- %2 29279 m %2 3 3 4794803744468498565 %1 354275574729041694924489 %1 + ----------------------- - ------------------------ ---- %2 117116 m %2 2 2 1355876315770121894 %1 88431884441065664639763 %1 - ----------------------- + ----------------------- ---- %2 234232 m %2 3894301993629885376 %1 310482218856861967328 m + ---------------------- - ----------------------- %2 %2 3069414981250264599939 %1 86603286990130185373 + ---------------------- ---- + -------------------- 117116 m %2 %2 \ 7224962405705627039302937 1 | - ------------------------- ----|/m 468464 m %2/ %1 := RootOf( 6 5 4 3 2 1024 _Z - 5076 _Z - 4865 _Z - 16954 _Z + 1029 _Z + 4802 _Z + 16807, index = 2) 5 4 %2 := 197675335144188928 %1 - 1999569864878508572 %1 3 2 + 4794803744468498565 %1 - 1355876315770121894 %1 + 3894301993629885376 %1 + 86603286990130185373 and in Maple notation (197675335144188928*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^5-1999569864878508572*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-\ 1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^2+3894301993629885376*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+86603286990130185373)*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^m/ m*(1/m)^(1/2)*(197675335144188928/(197675335144188928*RootOf(1024*_Z^6-5076*_Z^ 5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^5-1999569864878508572 *RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029 *_Z^2+4802*_Z+16807,index = 2)^3-1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5 -4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+3894301993629885376* RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+86603286990130185373)*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029* _Z^2+4802*_Z+16807,index = 2)^5+2830664220487221649152/29279/m/( 197675335144188928*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^5-1999569864878508572*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-\ 1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^2+3894301993629885376*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+86603286990130185373)*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^5-\ 1999569864878508572/(197675335144188928*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-\ 16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^5-1999569864878508572*RootOf(1024 *_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+ 4794803744468498565*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^3-1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+3894301993629885376*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+ 86603286990130185373)*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2 +4802*_Z+16807,index = 2)^4-2876693522599598459691/29279/m/(197675335144188928* RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^5-1999569864878508572*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029* _Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf(1024*_Z^6-5076*_Z^5-\ 4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-1355876315770121894* RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+3894301993629885376*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029* _Z^2+4802*_Z+16807,index = 2)+86603286990130185373)*RootOf(1024*_Z^6-5076*_Z^5-\ 4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565/( 197675335144188928*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^5-1999569864878508572*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-\ 1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^2+3894301993629885376*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+86603286990130185373)*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-\ 354275574729041694924489/117116/m/(197675335144188928*RootOf(1024*_Z^6-5076*_Z^ 5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^5-1999569864878508572 *RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029 *_Z^2+4802*_Z+16807,index = 2)^3-1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5 -4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+3894301993629885376* RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+86603286990130185373)*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029* _Z^2+4802*_Z+16807,index = 2)^3-1355876315770121894/(197675335144188928*RootOf( 1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^5-\ 1999569864878508572*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-1355876315770121894*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+ 3894301993629885376*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)+86603286990130185373)*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+88431884441065664639763/ 234232/m/(197675335144188928*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+ 1029*_Z^2+4802*_Z+16807,index = 2)^5-1999569864878508572*RootOf(1024*_Z^6-5076* _Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+ 4794803744468498565*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^3-1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+3894301993629885376*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+ 86603286990130185373)*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2 +4802*_Z+16807,index = 2)^2+3894301993629885376/(197675335144188928*RootOf(1024 *_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^5-\ 1999569864878508572*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-1355876315770121894*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+ 3894301993629885376*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)+86603286990130185373)*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)-310482218856861967328*m/( 197675335144188928*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^5-1999569864878508572*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-\ 1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^2+3894301993629885376*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+86603286990130185373)+ 3069414981250264599939/117116/m/(197675335144188928*RootOf(1024*_Z^6-5076*_Z^5-\ 4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^5-1999569864878508572* RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029* _Z^2+4802*_Z+16807,index = 2)^3-1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5-\ 4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+3894301993629885376* RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+86603286990130185373)*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029* _Z^2+4802*_Z+16807,index = 2)+86603286990130185373/(197675335144188928*RootOf( 1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^5-\ 1999569864878508572*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)^4+4794803744468498565*RootOf(1024*_Z^6-5076*_Z^5-4865* _Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-1355876315770121894*RootOf (1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^2+ 3894301993629885376*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+ 4802*_Z+16807,index = 2)+86603286990130185373)-7224962405705627039302937/468464 /m/(197675335144188928*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^ 2+4802*_Z+16807,index = 2)^5-1999569864878508572*RootOf(1024*_Z^6-5076*_Z^5-\ 4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^4+4794803744468498565* RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)^3-1355876315770121894*RootOf(1024*_Z^6-5076*_Z^5-4865*_Z^4-16954*_Z^3+1029* _Z^2+4802*_Z+16807,index = 2)^2+3894301993629885376*RootOf(1024*_Z^6-5076*_Z^5-\ 4865*_Z^4-16954*_Z^3+1029*_Z^2+4802*_Z+16807,index = 2)+86603286990130185373)) and in floating-point .5375773846e20*6.156204863^m/m*(1/m)^(1/2)*(1.000000003+.130583557/m-5.77558185\ 6*m) 1 Note that we even gained a factor of, ---- 1/2 n hence be have an improvement on Ellenberg-Gijswijt! For the sake our beloved OEIS, the first, 30, terms of the upper bound discussed at the beginning are [8, 40, 224, 1276, 7343, 42700, 250062, 1472108, 8702918, 51628115, 307151215, 1831780132, 10947162065, 65541856742, 393033127184, 2360228017180, 14191470782090, 85426781616778, 514762192377454, 3104735514831671, 18741864474909303, 113224530247689945, 684511726603189548, 4141046599275360740, 25067273624323683468, 151828402251129207345, 920091487157589396419, 5578608695109401728562, 33839503094426750465572, 205358566257718962966560] The first, 30, terms of the above-mentioned sequence, A(m) are [1, 5, 28, 149, 826, 4676, 26769, 154645, 899857, 5266030, 30960755, 182734916, 1082062723, 6425392741, 38247229753, 228150186773, 1363493882569, 8162183126741, 48933232903016, 293752173960574, 1765559479960794, 10623308389733319, 63984055792660899, 385728634392522820, 2327330803964964576, 14053101287375422251, 84917834645327806465, 513470956968593968213, 3106726248407135333639, 18807979000611546214501] Finally, just for fun,, A(1000), equals 1409366608564660988253547693761455756725868559333728077441365379151248193616\ 168591616487916851839375676082021902496450451492121336065427253832955418\ 811028506550145564144161522996019225660682366447254264034011212523838756\ 446857376614285028272159421962322348654506873278560934342466156029354549\ 224095336804090910038922512157128373909438660522627331298519955216773547\ 291820266459614862521051183390616891537216900615072742074530824474340174\ 636034653823048671756074633040554283014373846682966855990539105274174016\ 975732460323442916322506392110523719608769358688252813038669109914915115\ 020321917297972448777345685653815091521052335608403404598772945819608866\ 476403440099703898023490859739711934018655400958046768102438991097342496\ 6107751636207254941763303322890211625034361020420502940786668795 --------------------------------------------------- This ends this article that took, 12.601, seconds to generate.