A congruence automata modulo, 2, for the , AperyZeta2, sequence By Shalosh B. Ekhad Recall that the, AperyZeta2, sequence A(n) is defined as the n ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k = 0 The first 20 terms are 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009, 1313106114867738897, 13844511065506477501 We found an automata with, 2, states for the fast (linear-log) computation of A(n) mod, 2 given below Using this automata, it emerges that the following residue classes, modulo, 2, are missing {0} Just for fun, A(10000000000), modulo , 2, equals 1 Finally, the auotomata itself, with, 2, states where the i-th item of the following list, lists the , 2 children of state i (in order) [[2, 2], [2, 2]] The initial conditions are [1, 1] the whole thing took, 0.306, seconds. --------------------------------------------------------------- --------------------------------------------------------------- A congruence automata modulo, 4, for the , AperyZeta2, sequence By Shalosh B. Ekhad Recall that the, AperyZeta2, sequence A(n) is defined as the n ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k = 0 The first 20 terms are 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009, 1313106114867738897, 13844511065506477501 We found an automata with, 7, states for the fast (linear-log) computation of A(n) mod, 4 given below Using this automata, it emerges that the following residue classes, modulo, 4, are missing {0, 2} Just for fun, A(10000000000), modulo , 4, equals 3 Finally, the auotomata itself, with, 7, states where the i-th item of the following list, lists the , 2 children of state i (in order) [[2, 3], [2, 3], [4, 5], [4, 6], [4, 5], [2, 7], [2, 7]] The initial conditions are [1, 1, 3, 3, 3, 1, 1] the whole thing took, 0.448, seconds. --------------------------------------------------------------- --------------------------------------------------------------- A congruence automata modulo, 8, for the , AperyZeta2, sequence By Shalosh B. Ekhad Recall that the, AperyZeta2, sequence A(n) is defined as the n ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k = 0 The first 20 terms are 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009, 1313106114867738897, 13844511065506477501 We found an automata with, 50, states for the fast (linear-log) computation of A(n) mod, 8 given below Using this automata, it emerges that the following residue classes, modulo, 8, are missing {0, 2, 4, 6} Just for fun, A(10000000000), modulo , 8, equals 3 Finally, the auotomata itself, with, 50, states where the i-th item of the following list, lists the , 2 children of state i (in order) [[2, 3], [4, 5], [6, 7], [4, 8], [9, 10], [11, 12], [13, 14], [9, 10], [11, 12] , [15, 16], [11, 17], [18, 19], [11, 12], [13, 14], [20, 21], [22, 23], [24, 25 ], [26, 27], [28, 29], [20, 30], [31, 32], [20, 21], [22, 23], [4, 33], [34, 35 ], [26, 36], [37, 38], [4, 33], [39, 40], [41, 42], [4, 33], [34, 35], [43, 44] , [26, 27], [45, 46], [47, 48], [11, 12], [15, 16], [4, 33], [39, 40], [26, 27] , [28, 29], [20, 21], [49, 50], [26, 27], [45, 46], [20, 21], [49, 50], [11, 12 ], [13, 14]] The initial conditions are [1, 1, 3, 1, 3, 3, 3, 3, 3, 7, 3, 5, 3, 3, 7, 7, 1, 5, 1, 7, 1, 7, 7, 1, 5, 5, 3, 1, 1, 5, 1, 5, 7, 5, 5, 7, 3, 7, 1, 1, 5, 1, 7, 3, 5, 5, 7, 3, 3, 3] the whole thing took, 0.762, seconds. --------------------------------------------------------------- --------------------------------------------------------------- A congruence automata modulo, 3, for the , AperyZeta2, sequence By Shalosh B. Ekhad Recall that the, AperyZeta2, sequence A(n) is defined as the n ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k = 0 The first 20 terms are 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009, 1313106114867738897, 13844511065506477501 We found an automata with, 2, states for the fast (linear-log) computation of A(n) mod, 3 given below Using this automata, it emerges that the following residue classes, modulo, 3, are missing {2} Just for fun, A(10000000000), modulo , 3, equals 0 Finally, the auotomata itself, with, 2, states where the i-th item of the following list, lists the , 3 children of state i (in order) [[2, 0, 2], [2, 0, 2]] The initial conditions are [1, 1] the whole thing took, 0.444, seconds. --------------------------------------------------------------- --------------------------------------------------------------- A congruence automata modulo, 9, for the , AperyZeta2, sequence By Shalosh B. Ekhad Recall that the, AperyZeta2, sequence A(n) is defined as the n ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k = 0 The first 20 terms are 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009, 1313106114867738897, 13844511065506477501 We found an automata with, 9, states for the fast (linear-log) computation of A(n) mod, 9 given below Using this automata, it emerges that the following residue classes, modulo, 9, are missing {2, 4, 5, 7, 8} Just for fun, A(10000000000), modulo , 9, equals 0 Finally, the auotomata itself, with, 9, states where the i-th item of the following list, lists the , 3 children of state i (in order) [[2, 3, 4], [2, 3, 4], [5, 0, 6], [2, 3, 4], [5, 0, 7], [8, 0, 9], [5, 0, 7], [ 8, 0, 9], [8, 0, 9]] The initial conditions are [1, 1, 3, 1, 3, 6, 3, 6, 6] the whole thing took, 2.222, seconds. --------------------------------------------------------------- --------------------------------------------------------------- A congruence automata modulo, 5, for the , AperyZeta2, sequence By Shalosh B. Ekhad Recall that the, AperyZeta2, sequence A(n) is defined as the n ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k = 0 The first 20 terms are 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009, 1313106114867738897, 13844511065506477501 We found an automata with, 4, states for the fast (linear-log) computation of A(n) mod, 5 given below Using this automata, it emerges that the following residue classes, modulo, 5, are missing {0} Just for fun, A(10000000000), modulo , 5, equals 1 Finally, the auotomata itself, with, 4, states where the i-th item of the following list, lists the , 5 children of state i (in order) [[1, 3, 4, 2, 1], [2, 1, 3, 4, 2], [3, 4, 2, 1, 3], [4, 2, 1, 3, 4]] The initial conditions are [1, 2, 3, 4] the whole thing took, 0.304, seconds. --------------------------------------------------------------- --------------------------------------------------------------- A congruence automata modulo, 25, for the , AperyZeta2, sequence By Shalosh B. Ekhad Recall that the, AperyZeta2, sequence A(n) is defined as the n ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k = 0 The first 20 terms are 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009, 1313106114867738897, 13844511065506477501 We found an automata with, 261, states for the fast (linear-log) computation of A(n) mod, 25 given below Using this automata, it emerges that the following residue classes, modulo, 25, are missing {0, 5, 10, 15, 20} Just for fun, A(10000000000), modulo , 25, equals 16 Finally, the auotomata itself, with, 261, states where the i-th item of the following list, lists the , 5 children of state i (in order) [[2, 3, 4, 5, 6], [2, 3, 4, 5, 6], [7, 8, 9, 10, 11], [12, 13, 14, 15, 16], [17 , 18, 19, 20, 21], [2, 3, 4, 5, 6], [7, 22, 23, 24, 25], [26, 27, 28, 29, 30], [31, 32, 33, 34, 35], [36, 37, 38, 39, 40], [7, 22, 23, 24, 25], [12, 13, 14, 15, 16], [31, 41, 42, 43, 44], [45, 46, 47, 48, 49], [50, 51, 52, 53, 54], [12, 13, 14, 15, 16], [17, 55, 56, 57, 58], [36, 59, 60, 61, 62], [50, 63, 64, 65, 66], [26, 67, 68, 69, 70], [17, 55, 56, 57, 58], [26, 27, 28, 29, 30], [31, 32, 33, 34, 35], [36, 37, 38, 39, 40], [7, 22, 23, 24, 25], [26, 71, 72, 73, 74], [ 75, 76, 77, 78, 79], [80, 81, 82, 83, 84], [85, 86, 87, 88, 89], [26, 71, 72, 73, 74], [31, 32, 33, 34, 35], [80, 90, 91, 92, 93], [94, 95, 96, 97, 98], [99, 100, 101, 102, 103], [31, 32, 33, 34, 35], [36, 104, 105, 106, 107], [85, 108, 109, 110, 111], [99, 112, 113, 114, 115], [75, 116, 117, 118, 119], [36, 104, 105, 106, 107], [80, 90, 91, 92, 93], [94, 95, 96, 97, 98], [99, 100, 101, 102, 103], [31, 32, 33, 34, 35], [45, 46, 47, 48, 49], [94, 120, 121, 122, 123], [26 , 71, 72, 73, 74], [124, 125, 126, 127, 128], [45, 46, 47, 48, 49], [50, 63, 64 , 65, 66], [99, 129, 130, 131, 132], [124, 133, 134, 135, 136], [80, 137, 138, 139, 140], [50, 63, 64, 65, 66], [36, 59, 60, 61, 62], [50, 63, 64, 65, 66], [ 26, 67, 68, 69, 70], [17, 55, 56, 57, 58], [85, 108, 109, 110, 111], [99, 112, 113, 114, 115], [75, 116, 117, 118, 119], [36, 104, 105, 106, 107], [99, 129, 130, 131, 132], [124, 133, 134, 135, 136], [80, 137, 138, 139, 140], [50, 63, 64, 65, 66], [75, 76, 77, 78, 79], [80, 81, 82, 83, 84], [85, 86, 87, 88, 89], [26, 71, 72, 73, 74], [75, 76, 77, 78, 79], [80, 81, 82, 83, 84], [85, 86, 87, 88, 89], [26, 71, 72, 73, 74], [75, 141, 142, 143, 144], [145, 146, 147, 148, 149], [150, 151, 152, 153, 154], [12, 155, 156, 157, 158], [75, 141, 142, 143, 144], [80, 81, 82, 83, 84], [150, 159, 160, 161, 162], [163, 164, 165, 166, 167 ], [168, 169, 170, 171, 172], [80, 81, 82, 83, 84], [85, 173, 174, 175, 176], [ 12, 177, 178, 179, 180], [168, 181, 182, 183, 184], [145, 185, 186, 187, 188], [85, 173, 174, 175, 176], [150, 159, 160, 161, 162], [163, 164, 165, 166, 167], [168, 169, 170, 171, 172], [80, 81, 82, 83, 84], [94, 95, 96, 97, 98], [163, 189, 190, 191, 192], [75, 141, 142, 143, 144], [2, 193, 194, 195, 196], [94, 95 , 96, 97, 98], [99, 112, 113, 114, 115], [168, 197, 198, 199, 200], [2, 3, 4, 5 , 6], [150, 201, 202, 203, 204], [99, 112, 113, 114, 115], [85, 108, 109, 110, 111], [99, 112, 113, 114, 115], [75, 116, 117, 118, 119], [36, 104, 105, 106, 107], [12, 177, 178, 179, 180], [168, 181, 182, 183, 184], [145, 185, 186, 187, 188], [85, 173, 174, 175, 176], [168, 197, 198, 199, 200], [2, 3, 4, 5, 6], [ 150, 201, 202, 203, 204], [99, 112, 113, 114, 115], [145, 146, 147, 148, 149], [150, 151, 152, 153, 154], [12, 155, 156, 157, 158], [75, 141, 142, 143, 144], [163, 189, 190, 191, 192], [75, 141, 142, 143, 144], [2, 193, 194, 195, 196], [ 94, 95, 96, 97, 98], [124, 133, 134, 135, 136], [2, 205, 206, 207, 208], [85, 173, 174, 175, 176], [163, 209, 210, 211, 212], [124, 133, 134, 135, 136], [168 , 197, 198, 199, 200], [2, 3, 4, 5, 6], [150, 201, 202, 203, 204], [99, 112, 113, 114, 115], [2, 205, 206, 207, 208], [85, 173, 174, 175, 176], [163, 209, 210, 211, 212], [124, 133, 134, 135, 136], [150, 159, 160, 161, 162], [163, 164 , 165, 166, 167], [168, 169, 170, 171, 172], [80, 81, 82, 83, 84], [145, 146, 147, 148, 149], [150, 151, 152, 153, 154], [12, 155, 156, 157, 158], [75, 141, 142, 143, 144], [145, 213, 214, 215, 216], [50, 217, 218, 219, 220], [221, 222, 223, 224, 225], [31, 226, 227, 228, 229], [145, 213, 214, 215, 216], [150, 151, 152, 153, 154], [221, 230, 231, 232, 233], [17, 55, 56, 57, 58], [45, 234, 235, 236, 237], [150, 151, 152, 153, 154], [31, 41, 42, 43, 44], [45, 46, 47, 48, 49 ], [50, 51, 52, 53, 54], [12, 13, 14, 15, 16], [221, 230, 231, 232, 233], [17, 55, 56, 57, 58], [45, 234, 235, 236, 237], [150, 151, 152, 153, 154], [163, 164 , 165, 166, 167], [17, 238, 239, 240, 241], [145, 213, 214, 215, 216], [7, 242, 243, 244, 245], [163, 164, 165, 166, 167], [168, 181, 182, 183, 184], [45, 246, 247, 248, 249], [7, 22, 23, 24, 25], [221, 250, 251, 252, 253], [168, 181, 182, 183, 184], [12, 177, 178, 179, 180], [168, 181, 182, 183, 184], [145, 185, 186, 187, 188], [85, 173, 174, 175, 176], [31, 41, 42, 43, 44], [45, 46, 47, 48, 49] , [50, 51, 52, 53, 54], [12, 13, 14, 15, 16], [45, 246, 247, 248, 249], [7, 22, 23, 24, 25], [221, 250, 251, 252, 253], [168, 181, 182, 183, 184], [50, 217, 218, 219, 220], [221, 222, 223, 224, 225], [31, 226, 227, 228, 229], [145, 213, 214, 215, 216], [17, 238, 239, 240, 241], [145, 213, 214, 215, 216], [7, 242, 243, 244, 245], [163, 164, 165, 166, 167], [7, 8, 9, 10, 11], [12, 13, 14, 15, 16], [17, 18, 19, 20, 21], [2, 3, 4, 5, 6], [45, 246, 247, 248, 249], [7, 22, 23, 24, 25], [221, 250, 251, 252, 253], [168, 181, 182, 183, 184], [221, 230, 231, 232, 233], [17, 55, 56, 57, 58], [45, 234, 235, 236, 237], [150, 151, 152, 153, 154], [7, 8, 9, 10, 11], [12, 13, 14, 15, 16], [17, 18, 19, 20, 21], [2, 3 , 4, 5, 6], [17, 238, 239, 240, 241], [145, 213, 214, 215, 216], [7, 242, 243, 244, 245], [163, 164, 165, 166, 167], [50, 217, 218, 219, 220], [221, 222, 223, 224, 225], [31, 226, 227, 228, 229], [145, 213, 214, 215, 216], [99, 129, 130, 131, 132], [124, 133, 134, 135, 136], [80, 137, 138, 139, 140], [50, 63, 64, 65 , 66], [221, 222, 223, 224, 225], [124, 254, 255, 256, 257], [36, 104, 105, 106 , 107], [94, 258, 259, 260, 261], [221, 222, 223, 224, 225], [80, 90, 91, 92, 93], [94, 95, 96, 97, 98], [99, 100, 101, 102, 103], [31, 32, 33, 34, 35], [124 , 254, 255, 256, 257], [36, 104, 105, 106, 107], [94, 258, 259, 260, 261], [221 , 222, 223, 224, 225], [94, 120, 121, 122, 123], [26, 71, 72, 73, 74], [124, 125, 126, 127, 128], [45, 46, 47, 48, 49], [36, 59, 60, 61, 62], [50, 63, 64, 65, 66], [26, 67, 68, 69, 70], [17, 55, 56, 57, 58], [26, 27, 28, 29, 30], [31, 32, 33, 34, 35], [36, 37, 38, 39, 40], [7, 22, 23, 24, 25], [94, 120, 121, 122, 123], [26, 71, 72, 73, 74], [124, 125, 126, 127, 128], [45, 46, 47, 48, 49], [ 124, 254, 255, 256, 257], [36, 104, 105, 106, 107], [94, 258, 259, 260, 261], [ 221, 222, 223, 224, 225], [2, 205, 206, 207, 208], [85, 173, 174, 175, 176], [ 163, 209, 210, 211, 212], [124, 133, 134, 135, 136], [163, 189, 190, 191, 192], [75, 141, 142, 143, 144], [2, 193, 194, 195, 196], [94, 95, 96, 97, 98]] The initial conditions are [1, 1, 3, 19, 22, 1, 3, 9, 7, 16, 3, 19, 7, 11, 18, 19, 22, 16, 18, 9, 22, 9, 7, 16, 3, 9, 2, 21, 23, 9, 7, 21, 8, 4, 7, 16, 23, 4, 2, 16, 21, 8, 4, 7, 11, 8, 9, 17, 11, 18, 4, 17, 21, 18, 16, 18, 9, 22, 23, 4, 2, 16, 4, 17, 21, 18, 2, 21, 23, 9, 2, 21, 23, 9, 2, 6, 13, 19, 2, 21, 13, 24, 12, 21, 23, 19, 12, 6, 23, 13, 24, 12, 21, 8, 24, 2, 1, 8, 4, 12, 1, 13, 4, 23, 4, 2, 16, 19, 12, 6, 23, 12, 1, 13, 4, 6, 13, 19, 2, 24, 2, 1, 8, 17, 1, 23, 24, 17, 12, 1, 13, 4, 1, 23, 24, 17, 13, 24, 12, 21, 6, 13, 19, 2, 6, 18, 14, 7, 6, 13, 14, 22, 11, 13, 7, 11, 18, 19, 14, 22, 11, 13, 24, 22, 6, 3, 24, 12, 11, 3, 14, 12, 19, 12, 6, 23, 7, 11, 18, 19, 11, 3, 14, 12, 18, 14, 7, 6, 22, 6, 3, 24, 3, 19, 22, 1, 11, 3, 14, 12, 14, 22, 11, 13, 3, 19, 22, 1, 22, 6, 3, 24, 18, 14, 7, 6, 4, 17, 21, 18, 14, 17, 16, 8, 14, 21, 8, 4, 7, 17, 16, 8, 14, 8, 9, 17, 11, 16, 18, 9, 22, 9, 7, 16, 3, 8, 9, 17, 11, 17, 16, 8, 14, 1, 23, 24, 17, 24, 2, 1, 8] the whole thing took, 13.138, seconds. ---------------------------------------------------------------