On computing the Mod, 2, of Many Interesting sequences by Shalosh B. Ekhad Theorem Number, 1, : Let C(n) be the constant term, in x, of n (1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 2, : Let C(n) be the constant term, in x, of n (1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 3, : Let C(n) be the constant term, in x, of n (2 + x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 4, : Let C(n) be the constant term, in x, of n (2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 5, : Let C(n) be the constant term, in x, of n (3 + x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 6, : Let C(n) be the constant term, in x, of n (3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 7, : Let C(n) be the constant term, in x, of n (1/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 8, : Let C(n) be the constant term, in x, of n (1/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 9, : Let C(n) be the constant term, in x, of n (1/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 10, : Let C(n) be the constant term, in x, of n (1/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 11, : Let C(n) be the constant term, in x, of n (1/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 12, : Let C(n) be the constant term, in x, of n (1/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 13, : Let C(n) be the constant term, in x, of n (1/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 14, : Let C(n) be the constant term, in x, of n (1/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 15, : Let C(n) be the constant term, in x, of n (1/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 16, : Let C(n) be the constant term, in x, of n (1/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 17, : Let C(n) be the constant term, in x, of n (1/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 18, : Let C(n) be the constant term, in x, of n (1/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 19, : Let C(n) be the constant term, in x, of n (2/x + x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 20, : Let C(n) be the constant term, in x, of n (2/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 21, : Let C(n) be the constant term, in x, of n (2/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 22, : Let C(n) be the constant term, in x, of n (2/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 23, : Let C(n) be the constant term, in x, of n (2/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 24, : Let C(n) be the constant term, in x, of n (2/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 25, : Let C(n) be the constant term, in x, of n (2/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 26, : Let C(n) be the constant term, in x, of n (2/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 27, : Let C(n) be the constant term, in x, of n (2/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 28, : Let C(n) be the constant term, in x, of n (2/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 29, : Let C(n) be the constant term, in x, of n (2/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 30, : Let C(n) be the constant term, in x, of n (2/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 31, : Let C(n) be the constant term, in x, of n (3/x + x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 32, : Let C(n) be the constant term, in x, of n (3/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 33, : Let C(n) be the constant term, in x, of n (3/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 34, : Let C(n) be the constant term, in x, of n (3/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 35, : Let C(n) be the constant term, in x, of n (3/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 36, : Let C(n) be the constant term, in x, of n (3/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 19, 55, 595, 2611, 22141, 119449, 902035, 5420035, 38712169, 246360709, 1714206781, 11255897485, 77419522675, 517370395015 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 37, : Let C(n) be the constant term, in x, of n (3/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 38, : Let C(n) be the constant term, in x, of n (3/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 39, : Let C(n) be the constant term, in x, of n (3/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 40, : Let C(n) be the constant term, in x, of n (3/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 22, 116, 934, 6332, 48124, 352424, 2669062, 20107628, 153277972, 1170192344, 8981891164, 69111416792, 533463087928, 4126851588176 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0]], [1]] For example, C(100000), mudolo , 2, equals , 0 all the congruences classes mod, 2, show up Theorem Number, 41, : Let C(n) be the constant term, in x, of n (3/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! Theorem Number, 42, : Let C(n) be the constant term, in x, of n (3/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 2 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1]]], [1]] For example, C(100000), mudolo , 2, equals , 1 The congruence classes mod, 2, in the following set , {0}, never show up! ------------------------------------------ This ends this fascinating book that took, 0.177, to generate. ----------------------------------------- ----------------------------------------------------- On computing the Mod, 4, of Many Interesting sequences by Shalosh B. Ekhad Theorem Number, 1, : Let C(n) be the constant term, in x, of n (1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[2]], [A[2], A[2]], [(9 c[2] + 4) A[2] + (3 c[2] + 1) A[3], (9 c[2] + 4) A[2] + (3 c[2] + 1) A[3]]], [1, 1, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 2, : Let C(n) be the constant term, in x, of n (1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [A[2], A[3]], [A[2], A[3]]], [1, 1, 1, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 3, : Let C(n) be the constant term, in x, of n (2 + x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 0], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 4, : Let C(n) be the constant term, in x, of n (2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 0], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 5, : Let C(n) be the constant term, in x, of n (3 + x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [3 A[2], A[3]], [A[2], 3 A[3]]], [1, 1, 3, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 6, : Let C(n) be the constant term, in x, of n (3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], A[2]], [(c[2] + 4) A[2] + (c[2] + 1) A[3], (c[2] + 4) A[2] + (c[2] + 1) A[3]]], [1, 1, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 7, : Let C(n) be the constant term, in x, of n (1/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 2, states . Here it is: [[[A[2], 0], [A[2], 0]], [1, 1]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {2, 3}, never show up! Theorem Number, 8, : Let C(n) be the constant term, in x, of n (1/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 2, states . Here it is: [[[A[2], 0], [A[2], 2 A[2]]], [1, 1]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {2, 3}, never show up! Theorem Number, 9, : Let C(n) be the constant term, in x, of n (1/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [A[2], A[5]], [A[2], 2 A[2] + 2 A[3] + A[4]], [A[2], 2 A[2] + 2 A[3] + A[4]]], [1, 1, 1, 1, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 10, : Let C(n) be the constant term, in x, of n (1/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [A[2], A[5]], [3 A[2], 2 A[2] + A[4]], [A[2] + A[4] + A[5], 2 A[2] + A[4]]], [1, 1, 1, 3, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 11, : Let C(n) be the constant term, in x, of n (1/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 0], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 12, : Let C(n) be the constant term, in x, of n (1/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1]]], [1]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 13, : Let C(n) be the constant term, in x, of n (1/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 0], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 14, : Let C(n) be the constant term, in x, of n (1/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 2 A[2]], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 15, : Let C(n) be the constant term, in x, of n (1/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [3 A[2], A[3]], [A[2], 3 A[3]]], [1, 1, 3, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 16, : Let C(n) be the constant term, in x, of n (3 + 1/x + x) For the record, the first 15 terms of the sequence are: 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3]], [A[2], A[3]], [3 A[2], 2 A[2] + A[3]], [3 A[2], 2 A[2] + A[3]]] , [1, 1, 3, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 17, : Let C(n) be the constant term, in x, of n (1/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [3 A[2], A[5]], [A[2], 2 A[2] + 2 A[3] + A[4]], [A[2] + 2 A[3] + A[4] + A[5], A[5]]], [1, 1, 3, 1, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 18, : Let C(n) be the constant term, in x, of n (1/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [3 A[2], A[5]], [3 A[2], 2 A[2] + A[4]], [A[2], 2 A[3] + A[5]]], [1, 1, 3, 3, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 19, : Let C(n) be the constant term, in x, of n (2/x + x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 2, states . Here it is: [[[A[2], 0], [A[2], 0]], [1, 1]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {2, 3}, never show up! Theorem Number, 20, : Let C(n) be the constant term, in x, of n (2/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 2, states . Here it is: [[[A[2], 0], [A[2], 0]], [1, 1]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {2, 3}, never show up! Theorem Number, 21, : Let C(n) be the constant term, in x, of n (2/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[2]], [A[2], A[2]], [(9 c[2] + 4) A[2] + (3 c[2] + 1) A[3], (9 c[2] + 4) A[2] + (3 c[2] + 1) A[3]]], [1, 1, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 22, : Let C(n) be the constant term, in x, of n (2/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [A[2], A[5]], [A[2], 2 A[2] + 2 A[3] + A[4]], [A[2], 2 A[2] + 2 A[3] + A[4]]], [1, 1, 1, 1, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 23, : Let C(n) be the constant term, in x, of n (2/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[2]], [A[2], A[2]], [(9 c[2] + 4) A[2] + (3 c[2] + 1) A[3], (9 c[2] + 4) A[2] + (3 c[2] + 1) A[3]]], [1, 1, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 24, : Let C(n) be the constant term, in x, of n (2/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [A[2], A[5]], [A[2], 2 A[2] + 2 A[3] + A[4]], [A[2], 2 A[2] + 2 A[3] + A[4]]], [1, 1, 1, 1, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 25, : Let C(n) be the constant term, in x, of n (2/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 0], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 26, : Let C(n) be the constant term, in x, of n (2/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 0], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 27, : Let C(n) be the constant term, in x, of n (2/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], A[2]], [(c[2] + 4) A[2] + (c[2] + 1) A[3], (c[2] + 4) A[2] + (c[2] + 1) A[3]]], [1, 1, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 28, : Let C(n) be the constant term, in x, of n (3 + 2/x + x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [3 A[2], A[5]], [A[2], 2 A[2] + 2 A[3] + A[4]], [A[2] + 2 A[3] + A[4] + A[5], A[5]]], [1, 1, 3, 1, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 29, : Let C(n) be the constant term, in x, of n (2/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], A[2]], [(c[2] + 4) A[2] + (c[2] + 1) A[3], (c[2] + 4) A[2] + (c[2] + 1) A[3]]], [1, 1, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 30, : Let C(n) be the constant term, in x, of n (2/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [3 A[2], A[5]], [A[2], 2 A[2] + 2 A[3] + A[4]], [A[2] + 2 A[3] + A[4] + A[5], A[5]]], [1, 1, 3, 1, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 31, : Let C(n) be the constant term, in x, of n (3/x + x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 2, states . Here it is: [[[A[2], 0], [A[2], 2 A[2]]], [1, 1]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {2, 3}, never show up! Theorem Number, 32, : Let C(n) be the constant term, in x, of n (3/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 2, states . Here it is: [[[A[2], 0], [A[2], 0]], [1, 1]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {2, 3}, never show up! Theorem Number, 33, : Let C(n) be the constant term, in x, of n (1 + 3/x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [A[2], A[3]], [A[2], A[3]]], [1, 1, 1, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 34, : Let C(n) be the constant term, in x, of n (3/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [A[2], A[5]], [3 A[2], 2 A[2] + A[4]], [A[2] + A[4] + A[5], 2 A[2] + A[4]]], [1, 1, 1, 3, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 35, : Let C(n) be the constant term, in x, of n (3/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [A[2], A[5]], [A[2], 2 A[2] + 2 A[3] + A[4]], [A[2], 2 A[2] + 2 A[3] + A[4]]], [1, 1, 1, 1, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2, 3}, never show up! Theorem Number, 36, : Let C(n) be the constant term, in x, of n (3/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 19, 55, 595, 2611, 22141, 119449, 902035, 5420035, 38712169, 246360709, 1714206781, 11255897485, 77419522675, 517370395015 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [A[2], 2 A[2] + A[3]], [3 A[2], 2 A[2] + 3 A[3]]] , [1, 1, 1, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 37, : Let C(n) be the constant term, in x, of n (3/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 0], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 38, : Let C(n) be the constant term, in x, of n (3/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 2 A[2]], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 39, : Let C(n) be the constant term, in x, of n (3/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 0], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 40, : Let C(n) be the constant term, in x, of n (3/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 22, 116, 934, 6332, 48124, 352424, 2669062, 20107628, 153277972, 1170192344, 8981891164, 69111416792, 533463087928, 4126851588176 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3]], [A[2], 2 A[2]], [(2 c[3] + 2) A[2] + c[3] A[3], 0]], [1, 1, 2]] For example, C(100000), mudolo , 4, equals , 0 The congruence classes mod, 4, in the following set , {3}, never show up! Theorem Number, 41, : Let C(n) be the constant term, in x, of n (3/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [3 A[2], A[5]], [3 A[2], 2 A[2] + A[4]], [A[2], 2 A[3] + A[5]]], [1, 1, 3, 3, 1]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! Theorem Number, 42, : Let C(n) be the constant term, in x, of n (3/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 4 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [3 A[2], A[5]], [A[2], 2 A[2] + 2 A[3] + A[4]], [A[2] + 2 A[3] + A[4] + A[5], A[5]]], [1, 1, 3, 1, 3]] For example, C(100000), mudolo , 4, equals , 1 The congruence classes mod, 4, in the following set , {0, 2}, never show up! ------------------------------------------ This ends this fascinating book that took, 0.423, to generate. ----------------------------------------- ----------------------------------------------------- On computing the Mod, 8, of Many Interesting sequences by Shalosh B. Ekhad Theorem Number, 1, : Let C(n) be the constant term, in x, of n (1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[4], A[5]], [A[4], A[4]], [A[4], A[4]], [A[4], A[4]], [A[4], A[4]]], [1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 3, 4, 5, 6, 7}, never show up! Theorem Number, 2, : Let C(n) be the constant term, in x, of n (1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[5]], [A[9], A[10]], [A[4], 5 A[5] + 4 A[6]], [A[6], 4 A[5] + 5 A[7]], [6 A[4] + 3 A[6], 6 A[5] + 3 A[7]], [A[4], 5 A[5] + 4 A[6]], [4 A[4] + 5 A[6], A[7]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 3, 4, 5, 6, 7}, never show up! Theorem Number, 3, : Let C(n) be the constant term, in x, of n (2 + x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [4 A[4], 4 A[5]], [2 A[4], 2 A[5]], 0, 0], [1, 1, 2, 1, 4, 2, 0, 0]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {3, 5, 6, 7}, never show up! Theorem Number, 4, : Let C(n) be the constant term, in x, of n (2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [4 A[4], 4 A[5]], [2 A[4], 2 A[5]], 0, 0], [1, 1, 2, 1, 4, 2, 0, 0]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {3, 5, 6, 7}, never show up! Theorem Number, 5, : Let C(n) be the constant term, in x, of n (3 + x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[5]], [A[9], A[10]], [3 A[4], 7 A[5] + 4 A[6]], [A[6], 4 A[5] + 5 A[7]], [6 A[4] + A[6], 6 A[5] + A[7]], [A[4], 5 A[5] + 4 A[6]], [4 A[4] + 7 A[6], 3 A[7]]], [1, 1, 3, 1, 1, 3, 3, 1, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 5, 6, 7}, never show up! Theorem Number, 6, : Let C(n) be the constant term, in x, of n (3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[4]], [A[4], A[4]], [6 A[4] + 2 A[5] + A[6], 6 A[4] + 2 A[5] + A[6]], [4 A[4] + A[5] + A[6] + A[7], 4 A[4] + A[5] + A[6] + A[7]]], [1, 1, 3, 1, 1, 3, 3]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 5, 6, 7}, never show up! Theorem Number, 7, : Let C(n) be the constant term, in x, of n (1/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[3], 0], [(4 c[4] + 4) A[3] + c[4] A[4], 0]], [1, 1, 1, 4]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {2, 3, 5, 6, 7}, never show up! Theorem Number, 8, : Let C(n) be the constant term, in x, of n (1/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[3], A[4]], [6 A[3], 2 A[4]], [6 A[3], 2 A[4]]], [1, 1, 1, 6, 6]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {2, 3, 4, 5, 7}, never show up! Theorem Number, 9, : Let C(n) be the constant term, in x, of n (1/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [5 A[4], 4 A[6] + A[8]], [ 6 A[4] + 6 A[6] + 4 A[7] + 5 A[9] + 4 A[10], 2 A[4] + A[5] + 5 A[6] + 3 A[7] + A[8] + A[9] + 4 A[10]], [ 2 A[5] + 6 A[6] + A[8] + 3 A[9] + A[11], 4 A[4] + 2 A[6] + 4 A[7] + 2 A[9] + A[10]], [ 5 A[4] + 2 A[5] + 4 A[6] + A[8] + 4 A[9] + A[11] + 4 A[12], 4 A[8] + A[11]] , [3 A[5] + A[6] + 3 A[7] + 5 A[8] + A[10] + 4 A[13], 2 A[5] + 2 A[7] + 4 A[8] + A[10]], [A[4], 3 A[5] + 3 A[6] + 3 A[7] + 2 A[8] + A[10] + A[12] + 4 A[13]], [ 2 A[5] + 6 A[6] + 2 A[7] + 2 A[8] + A[9] + A[12] + 2 A[13] + A[14], 2 A[5] + 2 A[8] + 3 A[10] + A[13] + A[15]]], [1, 1, 1, 1, 5, 1, 5, 1, 5, 5, 1, 5, 5, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 3, 4, 6, 7}, never show up! Theorem Number, 10, : Let C(n) be the constant term, in x, of n (1/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [7 A[4], 2 A[5] + 4 A[6] + A[8]], [ 2 A[4] + 7 A[5] + 6 A[6] + A[8] + 3 A[9], 4 A[5] + 4 A[7] + 2 A[8] + A[10]] , [4 A[6] + A[9], 4 A[7] + A[10]], [3 A[4], 2 A[4] + 5 A[5] + 2 A[6] + 3 A[8] + 3 A[9] + 3 A[11] + A[12]], [ 2 A[4] + 2 A[5] + 4 A[6] + A[8] + 3 A[9] + A[11] + 2 A[12], 2 A[4] + A[5] + 2 A[6] + 4 A[7] + A[8] + 3 A[9] + A[10] + A[12]], [3 A[4], 2 A[4] + 7 A[5] + 2 A[6] + 3 A[8] + 3 A[9] + A[11] + A[12]], [ 2 A[5] + 4 A[6] + A[8] + A[9] + A[11] + A[12] + A[14], 2 A[5] + 2 A[8] + 2 A[10] + A[13]]], [1, 1, 1, 1, 7, 1, 3, 3, 7, 1, 3, 3, 5, 3, 5]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6}, never show up! Theorem Number, 11, : Let C(n) be the constant term, in x, of n (1/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [4 A[4], 4 A[5]], [2 A[4], 2 A[5]], 0, 0], [1, 1, 2, 1, 4, 2, 0, 0]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {3, 5, 6, 7}, never show up! Theorem Number, 12, : Let C(n) be the constant term, in x, of n (1/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [2 A[2], 6 A[3]], [6 A[2], 2 A[3]]], [1, 1, 2, 6] ] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {0, 3, 4, 5, 7}, never show up! Theorem Number, 13, : Let C(n) be the constant term, in x, of n (1/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], 0, [2 A[4], 2 A[5]], 0, 0], [1, 1, 2, 1, 0, 2, 0, 0]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {3, 4, 5, 6, 7}, never show up! Theorem Number, 14, : Let C(n) be the constant term, in x, of n (1/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [2 A[4], 6 A[5]], [2 A[4], 2 A[5]], [4 A[4], 4 A[5]], [6 A[4], 2 A[5]]], [1, 1, 2, 1, 2, 2, 4, 6]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {0, 3, 5, 7}, never show up! Theorem Number, 15, : Let C(n) be the constant term, in x, of n (1/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[5]], [A[9], A[10]], [3 A[4], 7 A[5] + 4 A[6]], [A[6], 4 A[5] + 5 A[7]], [6 A[4] + A[6], 6 A[5] + A[7]], [A[4], 5 A[5] + 4 A[6]], [4 A[4] + 7 A[6], 3 A[7]]], [1, 1, 3, 1, 1, 3, 3, 1, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 5, 6, 7}, never show up! Theorem Number, 16, : Let C(n) be the constant term, in x, of n (3 + 1/x + x) For the record, the first 15 terms of the sequence are: 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[6], A[10]], [3 A[4], 4 A[4] + 7 A[5]], [2 A[4] + A[6], 2 A[5] + 4 A[6] + 5 A[7]], [A[6], 4 A[5] + 4 A[6] + A[7]], [3 A[4], 4 A[4] + 7 A[5]], [6 A[4] + 5 A[6], 6 A[5] + 4 A[6] + A[7]]], [1, 1, 3, 1, 3, 3, 5, 3, 3, 5]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6, 7}, never show up! Theorem Number, 17, : Let C(n) be the constant term, in x, of n (1/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [3 A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [5 A[4], 4 A[6] + A[8]], [ 2 A[4] + 2 A[6] + A[9], 4 A[4] + 5 A[5] + 5 A[6] + 3 A[7] + 3 A[8] + A[9] + 4 A[10]], [ 4 A[4] + 2 A[5] + 2 A[6] + A[8] + A[9] + 3 A[11], 4 A[4] + 5 A[5] + A[6] + 3 A[7] + 2 A[8] + A[9] + 4 A[10] + A[11]], [A[4] + 4 A[6] + A[9] + 3 A[12], 4 A[5] + 2 A[8] + 3 A[11]], [ 6 A[5] + 6 A[6] + 4 A[7] + 3 A[8] + 2 A[9] + 4 A[10] + A[11] + A[12], 3 A[5] + 7 A[6] + 5 A[7] + 2 A[8] + 5 A[9] + 4 A[10] + A[11] + 2 A[12]], [A[4], 4 A[5] + 2 A[6] + A[8] + A[9] + A[14]], [4 A[6] + A[9] + A[12] + A[14], 4 A[5] + 2 A[7] + 2 A[8] + A[10]]], [1, 1, 3, 1, 5, 3, 7, 1, 5, 5, 3, 7, 7, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6}, never show up! Theorem Number, 18, : Let C(n) be the constant term, in x, of n (1/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [3 A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [7 A[4], 2 A[5] + 4 A[6] + A[8]], [2 A[4] + A[9], 4 A[5] + 4 A[7] + 2 A[8] + A[10]], [A[5] + 2 A[6] + A[8] + A[9] + 2 A[11], 4 A[7] + 3 A[10]], [A[4], 5 A[5] + 2 A[6] + A[8] + 2 A[9] + A[11]], [A[5] + 2 A[6] + A[11] + A[12], 4 A[5] + 4 A[7] + 2 A[8] + 3 A[10]], [A[4] + 4 A[6] + 3 A[9] + A[12], 6 A[5] + 2 A[8] + A[11]], [4 A[6] + A[12], 2 A[5] + 2 A[8] + 3 A[10] + A[13] + A[15]]], [1, 1, 3, 1, 7, 3, 1, 3, 7, 1, 1, 1, 7, 3, 5]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6}, never show up! Theorem Number, 19, : Let C(n) be the constant term, in x, of n (2/x + x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[3], 0], [(4 c[4] + 4) A[3] + c[4] A[4], 0]], [1, 1, 1, 4]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {2, 3, 5, 6, 7}, never show up! Theorem Number, 20, : Let C(n) be the constant term, in x, of n (2/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[3], 0], [(4 c[4] + 4) A[3] + c[4] A[4], 0]], [1, 1, 1, 4]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {2, 3, 5, 6, 7}, never show up! Theorem Number, 21, : Let C(n) be the constant term, in x, of n (2/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[4], A[5]], [A[4], A[4]], [A[4], A[4]], [A[4], A[4]], [A[4], A[4]]], [1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 3, 4, 5, 6, 7}, never show up! Theorem Number, 22, : Let C(n) be the constant term, in x, of n (2/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [5 A[4], 4 A[6] + A[8]], [ 6 A[4] + 6 A[6] + 4 A[7] + 5 A[9] + 4 A[10], 2 A[4] + A[5] + 5 A[6] + 3 A[7] + A[8] + A[9] + 4 A[10]], [ 2 A[5] + 6 A[6] + A[8] + 3 A[9] + A[11], 4 A[4] + 2 A[6] + 4 A[7] + 2 A[9] + A[10]], [ 5 A[4] + 2 A[5] + 4 A[6] + A[8] + 4 A[9] + A[11] + 4 A[12], 4 A[8] + A[11]] , [3 A[5] + A[6] + 3 A[7] + 5 A[8] + A[10] + 4 A[13], 2 A[5] + 2 A[7] + 4 A[8] + A[10]], [A[4], 3 A[5] + 3 A[6] + 3 A[7] + 2 A[8] + A[10] + A[12] + 4 A[13]], [ 2 A[5] + 6 A[6] + 2 A[7] + 2 A[8] + A[9] + A[12] + 2 A[13] + A[14], 2 A[5] + 2 A[8] + 3 A[10] + A[13] + A[15]]], [1, 1, 1, 1, 5, 1, 5, 1, 5, 5, 1, 5, 5, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 3, 4, 6, 7}, never show up! Theorem Number, 23, : Let C(n) be the constant term, in x, of n (2/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[4], A[5]], [A[4], A[4]], [A[4], A[4]], [A[4], A[4]], [A[4], A[4]]], [1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 3, 4, 5, 6, 7}, never show up! Theorem Number, 24, : Let C(n) be the constant term, in x, of n (2/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [5 A[4], 4 A[6] + A[8]], [2 A[4] + 6 A[6] + A[9], 4 A[5] + 2 A[7] + 2 A[8] + A[10]], [ 4 A[4] + 4 A[5] + 4 A[6] + 2 A[7] + 3 A[8] + A[9] + 2 A[10] + A[11], 2 A[5] + 4 A[6] + 6 A[7] + 2 A[8] + 4 A[9] + 3 A[10]], [A[4] + 4 A[5] + 2 A[6] + 2 A[7] + 3 A[8] + 4 A[9] + 2 A[10] + A[11] + 2 A[12], 6 A[5] + 4 A[6] + 4 A[7] + 6 A[8] + 4 A[9] + 4 A[10] + A[11]], [ 2 A[5] + 4 A[6] + 2 A[7] + 2 A[8] + 4 A[9] + 2 A[10] + A[12], 3 A[5] + A[6] + 3 A[7] + 5 A[8] + 3 A[9] + 2 A[10]], [A[4], 6 A[5] + 2 A[8] + A[11]], [ 4 A[5] + 4 A[6] + 2 A[7] + 3 A[8] + 2 A[9] + 2 A[10] + A[11] + 3 A[12], 2 A[5] + 6 A[7] + 2 A[8] + 2 A[10] + A[13]]], [1, 1, 1, 1, 5, 1, 5, 1, 5, 5, 1, 5, 5, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 3, 4, 6, 7}, never show up! Theorem Number, 25, : Let C(n) be the constant term, in x, of n (2/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], 0, [2 A[4], 2 A[5]], 0, 0], [1, 1, 2, 1, 0, 2, 0, 0]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {3, 4, 5, 6, 7}, never show up! Theorem Number, 26, : Let C(n) be the constant term, in x, of n (2/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], 0, [2 A[4], 2 A[5]], 0, 0], [1, 1, 2, 1, 0, 2, 0, 0]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {3, 4, 5, 6, 7}, never show up! Theorem Number, 27, : Let C(n) be the constant term, in x, of n (2/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[4]], [A[4], A[4]], [6 A[4] + 2 A[5] + A[6], 6 A[4] + 2 A[5] + A[6]], [4 A[4] + A[5] + A[6] + A[7], 4 A[4] + A[5] + A[6] + A[7]]], [1, 1, 3, 1, 1, 3, 3]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 5, 6, 7}, never show up! Theorem Number, 28, : Let C(n) be the constant term, in x, of n (2/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [3 A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [5 A[4], 4 A[6] + A[8]], [ 2 A[4] + 2 A[6] + A[9], 4 A[4] + 5 A[5] + 5 A[6] + 3 A[7] + 3 A[8] + A[9] + 4 A[10]], [ 4 A[4] + 2 A[5] + 2 A[6] + A[8] + A[9] + 3 A[11], 4 A[4] + 5 A[5] + A[6] + 3 A[7] + 2 A[8] + A[9] + 4 A[10] + A[11]], [A[4] + 4 A[6] + A[9] + 3 A[12], 4 A[5] + 2 A[8] + 3 A[11]], [ 6 A[5] + 6 A[6] + 4 A[7] + 3 A[8] + 2 A[9] + 4 A[10] + A[11] + A[12], 3 A[5] + 7 A[6] + 5 A[7] + 2 A[8] + 5 A[9] + 4 A[10] + A[11] + 2 A[12]], [A[4], 4 A[5] + 2 A[6] + A[8] + A[9] + A[14]], [4 A[6] + A[9] + A[12] + A[14], 4 A[5] + 2 A[7] + 2 A[8] + A[10]]], [1, 1, 3, 1, 5, 3, 7, 1, 5, 5, 3, 7, 7, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6}, never show up! Theorem Number, 29, : Let C(n) be the constant term, in x, of n (2/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[4]], [A[4], A[4]], [6 A[4] + 2 A[5] + A[6], 6 A[4] + 2 A[5] + A[6]], [4 A[4] + A[5] + A[6] + A[7], 4 A[4] + A[5] + A[6] + A[7]]], [1, 1, 3, 1, 1, 3, 3]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 5, 6, 7}, never show up! Theorem Number, 30, : Let C(n) be the constant term, in x, of n (2/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [3 A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [5 A[4], 4 A[6] + A[8]], [6 A[4] + 2 A[6] + 4 A[7] + 5 A[9] + 4 A[10], 2 A[7] + 2 A[8] + A[10]], [2 A[6] + A[9], 2 A[7] + A[10]], [ A[4] + A[5] + 3 A[6] + 7 A[7] + 3 A[9] + 3 A[10] + A[11] + 2 A[12], 4 A[5] + 2 A[8] + 3 A[11]], [ 2 A[5] + 4 A[6] + 2 A[7] + A[8] + 2 A[10] + 3 A[11] + A[12], A[5] + A[6] + 7 A[7] + A[8] + 2 A[9] + 4 A[10] + A[12]], [ A[4], 2 A[4] + 4 A[5] + 2 A[7] + A[8] + 4 A[9] + 2 A[10] + 3 A[12] + A[14]] , [4 A[6] + A[9] + A[12] + A[14], A[5] + A[6] + 3 A[7] + A[8] + 2 A[9] + 2 A[10] + A[12]]], [1, 1, 3, 1, 5, 3, 7, 1, 5, 5, 3, 7, 7, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6}, never show up! Theorem Number, 31, : Let C(n) be the constant term, in x, of n (3/x + x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 5, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[3], A[4]], [6 A[3], 2 A[4]], [6 A[3], 2 A[4]]], [1, 1, 1, 6, 6]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {2, 3, 4, 5, 7}, never show up! Theorem Number, 32, : Let C(n) be the constant term, in x, of n (3/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[3], 0], [(4 c[4] + 4) A[3] + c[4] A[4], 0]], [1, 1, 1, 4]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {2, 3, 5, 6, 7}, never show up! Theorem Number, 33, : Let C(n) be the constant term, in x, of n (3/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[5]], [A[9], A[10]], [A[4], 5 A[5] + 4 A[6]], [A[6], 4 A[5] + 5 A[7]], [6 A[4] + 3 A[6], 6 A[5] + 3 A[7]], [A[4], 5 A[5] + 4 A[6]], [4 A[4] + 5 A[6], A[7]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 3, 4, 5, 6, 7}, never show up! Theorem Number, 34, : Let C(n) be the constant term, in x, of n (3/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [7 A[4], 2 A[5] + 4 A[6] + A[8]], [ 2 A[4] + 7 A[5] + 6 A[6] + A[8] + 3 A[9], 4 A[5] + 4 A[7] + 2 A[8] + A[10]] , [4 A[6] + A[9], 4 A[7] + A[10]], [3 A[4], 2 A[4] + 5 A[5] + 2 A[6] + 3 A[8] + 3 A[9] + 3 A[11] + A[12]], [ 2 A[4] + 2 A[5] + 4 A[6] + A[8] + 3 A[9] + A[11] + 2 A[12], 2 A[4] + A[5] + 2 A[6] + 4 A[7] + A[8] + 3 A[9] + A[10] + A[12]], [3 A[4], 2 A[4] + 7 A[5] + 2 A[6] + 3 A[8] + 3 A[9] + A[11] + A[12]], [ 2 A[5] + 4 A[6] + A[8] + A[9] + A[11] + A[12] + A[14], 2 A[5] + 2 A[8] + 2 A[10] + A[13]]], [1, 1, 1, 1, 7, 1, 3, 3, 7, 1, 3, 3, 5, 3, 5]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6}, never show up! Theorem Number, 35, : Let C(n) be the constant term, in x, of n (3/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [5 A[4], 4 A[6] + A[8]], [2 A[4] + 6 A[6] + A[9], 4 A[5] + 2 A[7] + 2 A[8] + A[10]], [ 4 A[4] + 4 A[5] + 4 A[6] + 2 A[7] + 3 A[8] + A[9] + 2 A[10] + A[11], 2 A[5] + 4 A[6] + 6 A[7] + 2 A[8] + 4 A[9] + 3 A[10]], [A[4] + 4 A[5] + 2 A[6] + 2 A[7] + 3 A[8] + 4 A[9] + 2 A[10] + A[11] + 2 A[12], 6 A[5] + 4 A[6] + 4 A[7] + 6 A[8] + 4 A[9] + 4 A[10] + A[11]], [ 2 A[5] + 4 A[6] + 2 A[7] + 2 A[8] + 4 A[9] + 2 A[10] + A[12], 3 A[5] + A[6] + 3 A[7] + 5 A[8] + 3 A[9] + 2 A[10]], [A[4], 6 A[5] + 2 A[8] + A[11]], [ 4 A[5] + 4 A[6] + 2 A[7] + 3 A[8] + 2 A[9] + 2 A[10] + A[11] + 3 A[12], 2 A[5] + 6 A[7] + 2 A[8] + 2 A[10] + A[13]]], [1, 1, 1, 1, 5, 1, 5, 1, 5, 5, 1, 5, 5, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 3, 4, 6, 7}, never show up! Theorem Number, 36, : Let C(n) be the constant term, in x, of n (3/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 19, 55, 595, 2611, 22141, 119449, 902035, 5420035, 38712169, 246360709, 1714206781, 11255897485, 77419522675, 517370395015 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [A[4], 4 A[4] + 5 A[5]], [6 A[4] + A[6], 6 A[5] + 4 A[6] + 5 A[7]], [3 A[6], 4 A[5] + 4 A[6] + 3 A[7]], [3 A[4], 4 A[4] + 7 A[5]], [6 A[4] + 7 A[6], 6 A[5] + 4 A[6] + 3 A[7]]], [1, 1, 1, 1, 3, 1, 7, 3, 3, 5]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6}, never show up! Theorem Number, 37, : Let C(n) be the constant term, in x, of n (3/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [4 A[4], 4 A[5]], [2 A[4], 2 A[5]], 0, 0], [1, 1, 2, 1, 4, 2, 0, 0]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {3, 5, 6, 7}, never show up! Theorem Number, 38, : Let C(n) be the constant term, in x, of n (3/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [2 A[4], 6 A[5]], [2 A[4], 2 A[5]], [4 A[4], 4 A[5]], [6 A[4], 2 A[5]]], [1, 1, 2, 1, 2, 2, 4, 6]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {0, 3, 5, 7}, never show up! Theorem Number, 39, : Let C(n) be the constant term, in x, of n (3/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], 0, [2 A[4], 2 A[5]], 0, 0], [1, 1, 2, 1, 0, 2, 0, 0]] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {3, 4, 5, 6, 7}, never show up! Theorem Number, 40, : Let C(n) be the constant term, in x, of n (3/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 22, 116, 934, 6332, 48124, 352424, 2669062, 20107628, 153277972, 1170192344, 8981891164, 69111416792, 533463087928, 4126851588176 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3]], [A[2], A[4]], [2 A[2], 2 A[3]], [6 A[2], 6 A[3]]], [1, 1, 2, 6] ] For example, C(100000), mudolo , 8, equals , 0 The congruence classes mod, 8, in the following set , {0, 3, 4, 5, 7}, never show up! Theorem Number, 41, : Let C(n) be the constant term, in x, of n (3/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [3 A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [7 A[4], 2 A[5] + 4 A[6] + A[8]], [2 A[4] + A[9], 4 A[5] + 4 A[7] + 2 A[8] + A[10]], [A[5] + 2 A[6] + A[8] + A[9] + 2 A[11], 4 A[7] + 3 A[10]], [A[4], 5 A[5] + 2 A[6] + A[8] + 2 A[9] + A[11]], [A[5] + 2 A[6] + A[11] + A[12], 4 A[5] + 4 A[7] + 2 A[8] + 3 A[10]], [A[4] + 4 A[6] + 3 A[9] + A[12], 6 A[5] + 2 A[8] + A[11]], [4 A[6] + A[12], 2 A[5] + 2 A[8] + 3 A[10] + A[13] + A[15]]], [1, 1, 3, 1, 7, 3, 1, 3, 7, 1, 1, 1, 7, 3, 5]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6}, never show up! Theorem Number, 42, : Let C(n) be the constant term, in x, of n (3/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 8 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [3 A[4], A[11]], [A[12], A[13]], [A[14], A[15]], [5 A[4], 4 A[6] + A[8]], [6 A[4] + 2 A[6] + 4 A[7] + 5 A[9] + 4 A[10], 2 A[7] + 2 A[8] + A[10]], [2 A[6] + A[9], 2 A[7] + A[10]], [ A[4] + 2 A[5] + 2 A[6] + 2 A[7] + A[8] + A[9] + 2 A[10] + 3 A[11] + A[12], 4 A[5] + 2 A[8] + 3 A[11]], [ 2 A[5] + 4 A[6] + 2 A[7] + A[8] + 2 A[10] + 3 A[11] + A[12], A[5] + A[6] + 7 A[7] + A[8] + 2 A[9] + 4 A[10] + A[12]], [ A[4], 2 A[4] + 4 A[5] + 2 A[7] + A[8] + 4 A[9] + 2 A[10] + 3 A[12] + A[14]] , [4 A[6] + A[9] + A[12] + A[14], A[5] + A[6] + 3 A[7] + A[8] + 2 A[9] + 2 A[10] + A[12]]], [1, 1, 3, 1, 5, 3, 7, 1, 5, 5, 3, 7, 7, 1, 1]] For example, C(100000), mudolo , 8, equals , 1 The congruence classes mod, 8, in the following set , {0, 2, 4, 6}, never show up! ------------------------------------------ This ends this fascinating book that took, 1.117, to generate. ----------------------------------------- ----------------------------------------------------- On computing the Mod, 16, of Many Interesting sequences by Shalosh B. Ekhad Theorem Number, 1, : Let C(n) be the constant term, in x, of n (1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[8], A[9]], [A[11], A[10]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 2, : Let C(n) be the constant term, in x, of n (1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 23, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[9]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [A[8], A[23]], [13 A[10] + 4 A[12], 5 A[11] + 12 A[13]], [8 A[8] + 9 A[12], A[13]], [4 A[10] + 13 A[14], 12 A[11] + 5 A[15]], [8 A[8] + 9 A[10], 8 A[9] + 9 A[11]], [6 A[8] + 11 A[12], 6 A[9] + 11 A[13]], [14 A[10] + 3 A[14], 14 A[11] + 3 A[15]], [A[8], 9 A[9] + 8 A[14]], [5 A[10] + 12 A[12], 13 A[11] + 4 A[13]], [12 A[8] + 5 A[12], 4 A[9] + 13 A[13]], [A[14], 8 A[11] + 9 A[15]], [8 A[8] + A[10] + 8 A[12], 8 A[9] + A[11] + 8 A[13]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 3, : Let C(n) be the constant term, in x, of n (2 + x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 18, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [4 A[8], 4 A[9]], [12 A[10], 12 A[11]], [2 A[8], 2 A[9]], [8 A[8] + 10 A[10], 8 A[9] + 10 A[11]], [8 A[8], 8 A[9]], [8 A[10], 8 A[11]], [12 A[8] + A[10], 4 A[9] + A[11]], 0, 0], [1, 1, 2, 1, 4, 2, 8, 1, 0, 4, 0, 2, 0, 8, 0, 0, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 4, : Let C(n) be the constant term, in x, of n (2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 18, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [4 A[8], 4 A[9]], [12 A[10], 12 A[11]], [2 A[8], 2 A[9]], [8 A[8] + 10 A[10], 8 A[9] + 10 A[11]], [8 A[8], 8 A[9]], [8 A[10], 8 A[11]], [12 A[8] + 9 A[10], 4 A[9] + 9 A[11]], 0, 0], [1, 1, 2, 1, 4, 2, 8, 1, 0, 4, 0, 2, 0, 8, 0, 0, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 5, : Let C(n) be the constant term, in x, of n (3 + x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 23, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[9]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [3 A[8], A[23]], [15 A[10] + 4 A[12], 7 A[11] + 12 A[13]], [9 A[12], 8 A[9] + A[13]], [4 A[10] + 5 A[14], 12 A[11] + 13 A[15]], [8 A[8] + A[10], 8 A[9] + A[11]], [14 A[8] + A[12], 14 A[9] + A[13]], [6 A[10] + A[14], 6 A[11] + A[15]], [9 A[8], A[9] + 8 A[14]], [5 A[10] + 4 A[12], 13 A[11] + 12 A[13]], [4 A[8] + 7 A[12], 12 A[9] + 15 A[13]], [8 A[10] + 3 A[14], 11 A[15]], [8 A[8] + 11 A[10] + 8 A[12], 8 A[9] + 11 A[11] + 8 A[13]]], [1, 1, 3, 1, 9, 3, 11, 1, 1, 9, 9, 3, 3, 11, 11, 1, 1, 1, 9, 9, 9, 9, 3]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15}, never show up! Theorem Number, 6, : Let C(n) be the constant term, in x, of n (3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[8]], [A[8], A[8]], [%2, %2], [%2, %2], [3 A[8], 3 A[8]], [10 A[8] + 7 A[9] + 2 A[10], 10 A[8] + 7 A[9] + 2 A[10]], [%1, %1], [%1, %1]], [1, 1, 3, 1, 9, 3, 11, 1, 1, 9, 9, 3, 3, 11, 11]] %1 := 9 A[8] + 2 A[9] %2 := 7 A[8] + 2 A[9] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15}, never show up! Theorem Number, 7, : Let C(n) be the constant term, in x, of n (1/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 9, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[5], A[6]], [A[7], A[8]], [A[5], A[9]], [8 A[5], 8 A[6]], [4 A[5], 4 A[6]], 0, 0], [1, 1, 1, 4, 1, 8, 4, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 8, : Let C(n) be the constant term, in x, of n (1/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 13, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[5], A[6]], [A[7], A[8]], [A[5], A[6]], [A[10], A[11]], [6 A[5], 14 A[6]], [2 A[7], 10 A[8]], [A[10], A[11]], [6 A[5], 14 A[6]], [12 A[7] + 6 A[10], 10 A[8]], [6 A[5], 8 A[6] + 6 A[9]], [8 A[7] + 5 A[10] + 5 A[12], 10 A[8]]], [1, 1, 1, 6, 1, 6, 6, 12, 6, 6, 12, 6, 12]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15}, never show up! Theorem Number, 9, : Let C(n) be the constant term, in x, of n (1/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [5 A[8], 8 A[8] + 11 A[9] + 7 A[10] + 9 A[11] + 4 A[13] + 4 A[14] + 2 A[16] + A[17] + 7 A[18]], [12 A[8] + 2 A[9] + 12 A[10] + 6 A[12] + 10 A[13] + 4 A[14] + 6 A[16] + 9 A[17] + 2 A[19] + 6 A[20], 8 A[8] + 4 A[9] + 2 A[10] + 6 A[11] + 4 A[12] + 8 A[13] + 4 A[16] + 6 A[17] + 3 A[18] + 4 A[19] + 4 A[20]], [6 A[8] + 2 A[9] + 6 A[10] + 6 A[12] + 6 A[13] + 6 A[14] + 6 A[16] + 6 A[17] + 5 A[19] + 10 A[20] + 6 A[21], 8 A[9] + 2 A[10] + 8 A[13] + 2 A[14] + 14 A[16] + 2 A[17] + 11 A[20] + 2 A[21]], [14 A[10] + 8 A[12] + 8 A[13] + 6 A[14] + 12 A[17] + 8 A[19] + 8 A[20] + A[21], 8 A[8] + 11 A[9] + 12 A[11] + 4 A[12] + 6 A[13] + 3 A[14] + 9 A[15] + 15 A[16] + 14 A[17] + 6 A[18] + 4 A[19] + 15 A[21] + 6 A[22]], [2 A[8] + 7 A[9] + 14 A[10] + 5 A[12] + 9 A[13] + 6 A[14] + 5 A[16] + 5 A[17] + A[19] + 7 A[20] + 4 A[21], 14 A[8] + 13 A[9] + 6 A[10] + 8 A[11] + 11 A[12] + A[13] + 5 A[14] + 11 A[15] + 9 A[16] + 14 A[17] + A[18] + 7 A[19] + 13 A[20] + 3 A[21] + 5 A[22]], [ 4 A[10] + 10 A[12] + 2 A[17] + 3 A[19] + 2 A[24], 2 A[8] + 8 A[9] + 15 A[10] + 15 A[11] + A[12] + 5 A[13] + 5 A[14] + 15 A[15] + 4 A[16] + 10 A[17] + A[18] + A[19] + 2 A[20] + 3 A[21] + A[22] + 2 A[23] + 3 A[24] ], [6 A[8] + 6 A[9] + 7 A[10] + 13 A[11] + 3 A[12] + 7 A[13] + 8 A[14] + 3 A[16] + 14 A[17] + 5 A[18] + 3 A[19] + 5 A[20] + 7 A[21] + 3 A[23] + 5 A[24] + 6 A[25], 11 A[9] + 2 A[10] + 10 A[11] + 8 A[13] + 2 A[14] + 8 A[15] + 7 A[16] + 2 A[17] + 3 A[18] + 8 A[20] + 2 A[21] + 5 A[22] + 2 A[23] + 7 A[25]], [9 A[8] + 6 A[9] + A[10] + 3 A[11] + 9 A[12] + 7 A[13] + 4 A[14] + 3 A[16] + 11 A[17] + 5 A[18] + 6 A[19] + 5 A[20] + 6 A[21] + 3 A[23] + 2 A[24] + 8 A[25] + 5 A[26], 4 A[8] + 5 A[9] + 10 A[10] + 8 A[11] + 3 A[12] + 3 A[13] + 2 A[14] + 2 A[16] + 5 A[17] + A[18] + 2 A[19] + A[20] + 4 A[21] + 2 A[23] + 3 A[24] + 3 A[25] + 3 A[26]], [4 A[8] + 5 A[9] + 4 A[10] + 4 A[11] + 5 A[12] + 7 A[13] + 4 A[14] + 5 A[16] + 3 A[17] + 2 A[18] + 2 A[19] + 4 A[20] + 2 A[21] + 2 A[25] + A[26] + 7 A[27], 7 A[9] + 2 A[10] + 10 A[11] + 6 A[13] + 2 A[14] + 5 A[16] + 2 A[17] + A[18] + 4 A[20] + 2 A[21] + 6 A[25] + 6 A[27]], [2 A[8] + A[9] + A[10] + 7 A[11] + 8 A[12] + 2 A[13] + 4 A[14] + A[16] + 12 A[17] + 2 A[18] + A[19] + 3 A[20] + 3 A[24] + 3 A[25] + 2 A[26] + A[27] + 4 A[28], 2 A[8] + 8 A[9] + 13 A[10] + 11 A[11] + 7 A[12] + 11 A[13] + 8 A[14] + 7 A[16] + 8 A[17] + 4 A[18] + A[19] + 5 A[20] + 2 A[21] + A[23] + A[24] + 5 A[25] + 2 A[26] + 5 A[27] + 4 A[28]], [4 A[8] + 9 A[9] + 6 A[10] + 14 A[11] + 11 A[12] + 11 A[13] + 14 A[14] + 2 A[15] + 8 A[16] + A[17] + 3 A[18] + 2 A[19] + 2 A[20] + 5 A[21] + A[23] + A[24] + 7 A[25] + 3 A[26] + 9 A[27] + 6 A[28] + 2 A[29], 2 A[8] + 6 A[9] + 7 A[10] + 15 A[11] + 4 A[12] + 6 A[13] + 4 A[14] + 4 A[15] + 4 A[16] + 5 A[17] + A[18] + A[19] + A[20] + 2 A[21] + A[22] + 10 A[25] + A[26] + 3 A[27] + 2 A[28] + 2 A[29]], [7 A[9] + 12 A[10] + 8 A[11] + 6 A[12] + 6 A[13] + 8 A[14] + 2 A[15] + 6 A[16] + A[17] + 3 A[18] + A[19] + 3 A[20] + 3 A[21] + A[23] + A[24] + 5 A[25] + 2 A[26] + 5 A[27] + 3 A[28] + 2 A[29] + 8 A[30], 4 A[9] + 2 A[10] + 2 A[11] + 6 A[13] + 2 A[14] + 2 A[15] + 4 A[16] + 3 A[20] + 2 A[21] + 2 A[25] + 4 A[27] + 2 A[29] + 2 A[30]], [3 A[9] + 6 A[10] + 4 A[11] + 6 A[12] + 6 A[13] + 9 A[14] + A[15] + 3 A[16] + 2 A[17] + 2 A[21] + 2 A[24] + 2 A[25] + 2 A[26] + 4 A[27] + 4 A[28] + A[29] + 12 A[30], 3 A[9] + 2 A[10] + 2 A[11] + 6 A[12] + 6 A[13] + 4 A[14] + 6 A[15] + 3 A[16] + 2 A[17] + 3 A[18] + A[20] + 2 A[21] + A[22] + 2 A[24] + 10 A[25] + 2 A[26] + 3 A[27] + 2 A[28] + 6 A[29] + 10 A[30] + A[31]], [ A[8], 8 A[9] + 2 A[14] + 2 A[21] + 2 A[31] + 4 A[11] + 2 A[25] + 2 A[30] + 4 A[16] + 2 A[27] + 2 A[10] + A[23] + 2 A[13]], [9 A[9] + 8 A[15] + 8 A[14] + 4 A[28] + 4 A[21] + 4 A[32] + A[26] + 9 A[31] + 2 A[22] + 6 A[12] + 3 A[11] + 11 A[25] + 2 A[24] + 2 A[30] + 7 A[16] + 4 A[27] + 4 A[17] + A[10] + 3 A[19] + 6 A[13] + 10 A[29] + 6 A[8] + 6 A[33] + A[18], 4 A[9] + 10 A[15] + 4 A[14] + 2 A[28] + 2 A[21] + 2 A[32] + 7 A[31] + 3 A[22] + 2 A[12] + 12 A[11] + 8 A[25] + 8 A[30] + 4 A[16] + 2 A[27] + 2 A[17] + 14 A[10] + 4 A[13] + 13 A[29] + 4 A[8] + 6 A[33]], [ 7 A[9] + 8 A[15] + 6 A[14] + A[28] + A[21] + A[32] + 4 A[26] + 2 A[31] + 2 A[22] + 3 A[12] + 11 A[11] + 11 A[25] + 2 A[24] + 4 A[34] + 14 A[30] + 7 A[16] + 5 A[27] + 3 A[17] + 9 A[10] + A[19] + 7 A[13] + 10 A[29] + 2 A[33], 9 A[9] + 11 A[15] + 5 A[14] + A[21] + 5 A[31] + 2 A[22] + 4 A[11] + A[25] + 8 A[34] + 11 A[30] + 9 A[16] + 5 A[27] + 3 A[17] + 4 A[10] + 10 A[13] + 13 A[29] + 8 A[33]], [4 A[9] + A[35] + 4 A[15] + 2 A[14] + 4 A[28] + 3 A[31] + A[22] + 14 A[11] + 6 A[25] + A[24] + A[34] + 4 A[30] + 4 A[16] + 2 A[27] + A[17] + 8 A[10] + 4 A[19] + 4 A[13] + 6 A[29] + 2 A[33] + A[18], 4 A[9] + A[35] + 3 A[15] + 3 A[14] + 5 A[21] + 4 A[31] + 11 A[25] + 2 A[34] + 5 A[30] + 4 A[16] + 3 A[27] + A[17] + 8 A[10] + 4 A[23] + 4 A[13] + A[29] + A[33] + A[18] + 4 A[20]]], [1, 1, 1, 1, 5, 1, 13, 1, 1, 5, 5, 1, 1, 13, 13, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 13, 13, 13, 13, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15}, never show up! Theorem Number, 10, : Let C(n) be the constant term, in x, of n (1/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [7 A[8], 12 A[9] + 8 A[14] + 3 A[16]], [11 A[17] + 4 A[19], 8 A[13] + 3 A[18] + 4 A[20]], [ 2 A[8] + 10 A[10] + 8 A[12] + 6 A[17] + 9 A[19], 6 A[9] + 12 A[11] + 4 A[13] + 4 A[16] + 4 A[18] + 5 A[20]], [ 12 A[10] + 8 A[12] + 4 A[14] + 10 A[17] + 8 A[19] + A[21], 8 A[11] + 12 A[15] + 6 A[18] + A[22]], [8 A[10] + A[17], 8 A[11] + A[18]], [4 A[12] + A[19], 4 A[13] + A[20]], [6 A[9] + 4 A[14] + A[16] + A[21] + A[23], 12 A[15] + A[22]], [3 A[8], 2 A[16] + A[23]], [ 4 A[8] + 2 A[10] + 4 A[12] + 8 A[17] + 6 A[19] + A[24] + 2 A[26], 12 A[11] + 3 A[18] + 4 A[20] + 8 A[25]], [ 2 A[8] + 4 A[10] + 4 A[12] + 2 A[17] + 3 A[19] + 2 A[24] + 6 A[26], 12 A[9] + 8 A[11] + 4 A[13] + 4 A[16] + A[18] + 3 A[20] + 2 A[23] + 3 A[25] + 2 A[27]], [ 6 A[9] + 8 A[10] + 4 A[14] + A[16] + 6 A[17] + 3 A[21] + A[23] + 2 A[28], 8 A[11] + 4 A[15] + 2 A[18] + 5 A[22] + 8 A[25] + 4 A[29]], [6 A[8] + 10 A[10] + 8 A[12] + 2 A[17] + 2 A[19] + 3 A[24] + 9 A[26] + 5 A[30], 10 A[9] + 6 A[11] + 8 A[13] + 6 A[16] + 3 A[18] + 4 A[20] + 3 A[22] + 2 A[23] + 5 A[25] + 3 A[27] + 3 A[29]], [11 A[9] + 12 A[10] + 8 A[14] + 3 A[16] + A[17] + 4 A[21] + 2 A[23] + 2 A[24] + 3 A[28] + 7 A[30], 8 A[11] + 4 A[15] + 3 A[22] + 8 A[25] + 6 A[29]], [2 A[32] + A[26] + 4 A[12] + A[24] + 3 A[30] + 2 A[17] + 6 A[10] + 3 A[19] + 5 A[8], 8 A[9] + 2 A[16] + A[23]], [3 A[24] + 8 A[30] + 4 A[17] + 4 A[10], 2 A[22] + 8 A[11] + 7 A[25] + 2 A[29]], [3 A[32] + 2 A[26] + 8 A[12] + 2 A[24] + 2 A[30] + 4 A[10] + 4 A[19] + 2 A[8], 5 A[9] + 8 A[14] + 4 A[28] + A[21] + 4 A[32] + A[31] + 3 A[22] + 8 A[11] + 5 A[25] + 3 A[24] + 3 A[34] + 5 A[30] + 2 A[16] + A[27] + 8 A[10] + A[23] + 8 A[13] + 3 A[29] + 3 A[33] + 3 A[20]], [5 A[9] + A[35] + 4 A[15] + 8 A[14] + 2 A[28] + A[21] + 5 A[31] + 3 A[22] + 6 A[11] + 5 A[25] + 2 A[34] + 10 A[30] + 2 A[16] + 4 A[27] + 4 A[17] + 12 A[10] + A[23] + 4 A[13] + 8 A[29] + 2 A[33] + 2 A[20], 6 A[35] + 4 A[15] + 4 A[31] + 4 A[22] + 4 A[11] + 4 A[25] + 5 A[29]]], [1, 1, 1, 1, 7, 1, 3, 1, 11, 7, 1, 1, 11, 3, 5, 3, 11, 5, 7, 5, 1, 15, 3, 11, 5, 3, 9, 5, 11, 3, 13, 11, 1, 5, 11]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 6, 8, 10, 12, 14}, never show up! Theorem Number, 11, : Let C(n) be the constant term, in x, of n (1/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 18, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [4 A[8], 4 A[9]], [12 A[10], 12 A[11]], [2 A[8], 2 A[9]], [8 A[8] + 10 A[10], 8 A[9] + 10 A[11]], [8 A[8], 8 A[9]], [8 A[10], 8 A[11]], [12 A[8] + A[10], 4 A[9] + A[11]], 0, 0], [1, 1, 2, 1, 4, 2, 8, 1, 0, 4, 0, 2, 0, 8, 0, 0, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 12, : Let C(n) be the constant term, in x, of n (1/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [2 A[4], 10 A[5]], [8 A[4] + 6 A[6], 8 A[5] + 14 A[7]], [4 A[4] + A[6], 12 A[5] + A[7]], [6 A[4], 14 A[5]], [8 A[4] + 2 A[6], 8 A[5] + 10 A[7]]], [1, 1, 2, 1, 6, 2, 4, 6, 6, 12]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {0, 3, 5, 7, 8, 9, 10, 11, 13, 14, 15}, never show up! Theorem Number, 13, : Let C(n) be the constant term, in x, of n (1/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 20, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [8 A[8], 8 A[9]], [8 A[10], 8 A[11]], [2 A[8], 2 A[9]], [A[19], 8 A[10] + 10 A[11]], 0, 0, [A[20], 4 A[9] + 2 A[10] + A[11] + 2 A[13] + 3 A[14]], [5 A[9] + 4 A[13] + 3 A[16], 2 A[9] + 2 A[13] + 2 A[16]], [2 A[10] + A[14], 4 A[15]], 0, 0], [1, 1, 2, 1, 8, 2, 0, 1, 8, 8, 0, 2, 0, 0, 0, 0, 8, 0, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 14, : Let C(n) be the constant term, in x, of n (1/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 20, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [10 A[8], 2 A[9]], [8 A[8] + 14 A[10], 8 A[9] + 6 A[11]], [2 A[8], 2 A[9]], [8 A[8] + 10 A[10], 8 A[9] + 10 A[11]], [12 A[8], 12 A[9]], [4 A[10], 4 A[11]], [A[17], A[18]], [4 A[8] + 14 A[10] + 5 A[12] + 2 A[17], 14 A[9]], [2 A[8] + 3 A[12] + 3 A[14], 2 A[9] + 10 A[11] + 2 A[13] + 2 A[16]], [ 2 A[8] + 4 A[10] + 2 A[12] + 4 A[14] + 2 A[17] + 2 A[19], 2 A[9] + 14 A[11] + 2 A[13] + 2 A[18]], [2 A[10] + 5 A[14] + A[17] + A[19], 8 A[11] + A[18] + A[20]]], [1, 1, 2, 1, 10, 2, 12, 1, 6, 10, 4, 2, 12, 12, 8, 6, 6, 12, 6, 12]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {0, 3, 5, 7, 9, 11, 13, 14, 15}, never show up! Theorem Number, 15, : Let C(n) be the constant term, in x, of n (1/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 23, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[9]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [3 A[8], A[23]], [15 A[10] + 4 A[12], 7 A[11] + 12 A[13]], [9 A[12], 8 A[9] + A[13]], [4 A[10] + 5 A[14], 12 A[11] + 13 A[15]], [8 A[8] + A[10], 8 A[9] + A[11]], [14 A[8] + A[12], 14 A[9] + A[13]], [6 A[10] + A[14], 6 A[11] + A[15]], [9 A[8], A[9] + 8 A[14]], [5 A[10] + 4 A[12], 13 A[11] + 12 A[13]], [4 A[8] + 7 A[12], 12 A[9] + 15 A[13]], [8 A[10] + 3 A[14], 11 A[15]], [8 A[8] + 11 A[10] + 8 A[12], 8 A[9] + 11 A[11] + 8 A[13]]], [1, 1, 3, 1, 9, 3, 11, 1, 1, 9, 9, 3, 3, 11, 11, 1, 1, 1, 9, 9, 9, 9, 3]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15}, never show up! Theorem Number, 16, : Let C(n) be the constant term, in x, of n (1/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 22, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [3 A[8], 11 A[9] + 8 A[10]], [4 A[8] + 15 A[10], 12 A[9] + 7 A[11]], [2 A[8] + 9 A[12], 2 A[9] + A[13] + 8 A[14]], [ 8 A[8] + 2 A[10] + 12 A[12] + 5 A[14], 8 A[9] + 2 A[11] + 4 A[13] + 13 A[15]], [12 A[8] + A[10] + 4 A[12], 4 A[9] + A[11] + 12 A[13]], [A[12], 8 A[9] + A[13] + 8 A[14]], [12 A[10] + 4 A[12] + A[14], 4 A[11] + 12 A[13] + A[15]], [11 A[8], 3 A[9] + 8 A[10]], [4 A[8] + 15 A[10] + 8 A[12], 12 A[9] + 7 A[11] + 8 A[13]], [14 A[8] + 13 A[12], 14 A[9] + 5 A[13] + 8 A[14]], [8 A[8] + 6 A[10] + 12 A[12] + A[14], 8 A[9] + 6 A[11] + 4 A[13] + 9 A[15]] ], [1, 1, 3, 1, 11, 3, 13, 1, 3, 11, 5, 3, 9, 13, 3, 3, 3, 13, 11, 1, 5, 11]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 6, 7, 8, 10, 12, 14, 15}, never show up! Theorem Number, 17, : Let C(n) be the constant term, in x, of n (1/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [3 A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [13 A[8], 4 A[9] + 8 A[14] + A[16]], [2 A[9] + 12 A[10] + 6 A[12] + 10 A[13] + 4 A[14] + 2 A[16] + 9 A[17] + 2 A[19] + 6 A[20], 3 A[9] + 8 A[10] + 10 A[11] + 7 A[12] + 15 A[13] + 6 A[14] + 3 A[16] + 10 A[17] + 7 A[18] + 11 A[19] + 5 A[20]], [10 A[8] + 13 A[9] + 6 A[10] + 7 A[12] + 9 A[13] + 6 A[14] + 13 A[16] + 8 A[17] + 2 A[19] + 7 A[20] + 4 A[21], 12 A[8] + 14 A[9] + 13 A[10] + 9 A[11] + 14 A[12] + 8 A[14] + 4 A[16] + 15 A[17] + 7 A[18] + 10 A[19] + 3 A[20]], [4 A[8] + 7 A[9] + 11 A[10] + 15 A[11] + 5 A[12] + 9 A[13] + 15 A[16] + A[17] + A[18] + 5 A[19] + 15 A[20] + 5 A[21], 15 A[9] + 4 A[10] + 8 A[12] + 14 A[13] + 11 A[14] + 13 A[15] + 15 A[16] + 8 A[17] + 12 A[18] + 12 A[19] + 4 A[20] + 7 A[21] + 10 A[22]], [6 A[9] + 12 A[10] + 10 A[11] + 12 A[12] + 6 A[13] + 6 A[14] + 8 A[15] + 3 A[16] + 7 A[17] + 6 A[18] + 12 A[19] + 10 A[20] + 6 A[21] + 8 A[22] + A[23], 4 A[8] + 5 A[9] + 15 A[10] + 3 A[11] + 13 A[12] + 11 A[13] + 8 A[15] + 12 A[16] + 7 A[17] + 8 A[18] + 13 A[19] + 13 A[20] + 10 A[21] + 8 A[22] + 3 A[23]], [4 A[8] + 15 A[9] + 2 A[10] + 14 A[12] + 8 A[13] + 5 A[16] + 9 A[17] + 13 A[19] + 8 A[20] + 4 A[23] + 3 A[24], 8 A[9] + 4 A[10] + 6 A[12] + 12 A[13] + 4 A[14] + 4 A[16] + 2 A[17] + 6 A[19] + 3 A[20] + 8 A[21] + 4 A[23] + 2 A[24]], [4 A[8] + 12 A[9] + 15 A[10] + 7 A[11] + 9 A[12] + 11 A[13] + 14 A[14] + 7 A[16] + 5 A[17] + 4 A[18] + 9 A[19] + 9 A[20] + 11 A[21] + 5 A[23] + 2 A[24] + 3 A[25], 6 A[9] + 4 A[11] + 7 A[12] + 11 A[13] + 11 A[14] + 5 A[15] + 4 A[16] + 10 A[17] + 2 A[18] + 7 A[19] + 7 A[20] + 9 A[21] + 2 A[22] + 2 A[23] + 4 A[24] + 4 A[25]], [3 A[8] + 11 A[9] + 6 A[10] + 2 A[11] + 9 A[12] + 7 A[13] + 4 A[14] + 2 A[16] + 3 A[17] + 7 A[19] + A[20] + 2 A[21] + 3 A[23] + 3 A[24] + 2 A[25] + 2 A[26], 4 A[9] + 2 A[16] + 3 A[23]], [8 A[8] + 6 A[9] + 12 A[10] + 8 A[11] + 10 A[12] + 10 A[13] + 8 A[14] + 3 A[16] + 6 A[17] + 5 A[18] + 6 A[19] + 5 A[20] + 4 A[21] + 3 A[23] + 3 A[24] + 5 A[25] + 4 A[26] + 5 A[27], 4 A[8] + 4 A[9] + 6 A[10] + 4 A[11] + 5 A[12] + 7 A[13] + 4 A[14] + 3 A[16] + 3 A[17] + A[18] + 3 A[19] + 2 A[20] + 2 A[21] + 3 A[23] + 3 A[24] + 2 A[25] + 2 A[26] + 3 A[27]], [4 A[8] + 4 A[9] + 14 A[10] + 2 A[11] + 10 A[12] + 4 A[13] + 4 A[14] + 2 A[16] + 9 A[17] + 4 A[19] + 2 A[20] + 2 A[21] + 2 A[23] + 3 A[24] + 2 A[25] + 3 A[26] + 2 A[27] + 2 A[28], 4 A[8] + 10 A[9] + 14 A[10] + 2 A[11] + 10 A[12] + 8 A[13] + 8 A[14] + 4 A[16] + 9 A[17] + 4 A[19] + 2 A[20] + 3 A[21] + 6 A[23] + A[24] + 2 A[25] + 2 A[26] + 3 A[27] + 5 A[28]], [4 A[8] + 4 A[9] + 6 A[10] + 2 A[11] + 6 A[12] + 4 A[13] + 4 A[14] + 2 A[16] + 2 A[17] + 4 A[19] + 2 A[20] + 2 A[23] + 4 A[24] + 2 A[25] + 2 A[26] + 2 A[27] + A[28], 2 A[8] + 6 A[9] + A[10] + 7 A[11] + 2 A[12] + 8 A[13] + A[14] + 7 A[15] + 3 A[16] + A[17] + 7 A[18] + A[19] + 2 A[20] + 3 A[22] + 3 A[23] + A[26] + 4 A[27] + A[28] + 3 A[29]], [2 A[8] + 11 A[9] + 11 A[10] + 9 A[11] + 7 A[12] + 11 A[13] + 5 A[14] + 7 A[15] + 6 A[16] + A[17] + 6 A[18] + 3 A[19] + 5 A[20] + 4 A[21] + 2 A[22] + 5 A[23] + A[24] + 3 A[25] + A[26] + 6 A[27] + 3 A[28] + 3 A[29] + 5 A[30], 2 A[8] + 12 A[9] + A[10] + 7 A[11] + 5 A[12] + 9 A[13] + 6 A[14] + 2 A[15] + 6 A[16] + 5 A[17] + 2 A[18] + 4 A[19] + 4 A[20] + A[21] + A[22] + 4 A[23] + 2 A[24] + 3 A[25] + A[26] + 4 A[27] + 3 A[28] + A[29] + 8 A[30]], [2 A[8] + 11 A[9] + 13 A[10] + 9 A[11] + 5 A[12] + 11 A[13] + 3 A[14] + 7 A[15] + 6 A[16] + 2 A[17] + 2 A[18] + 4 A[19] + 5 A[20] + A[21] + 2 A[22] + 5 A[23] + A[24] + 3 A[25] + A[26] + 6 A[27] + A[28] + 3 A[29] + 6 A[30] + 4 A[31], 6 A[9] + 4 A[10] + 6 A[11] + 2 A[12] + 6 A[13] + 2 A[14] + 8 A[15] + 4 A[16] + 2 A[18] + 2 A[19] + 2 A[20] + A[21] + 2 A[22] + 2 A[23] + 2 A[25] + 4 A[27] + A[28] + 3 A[29] + 2 A[30] + 4 A[31]], [ A[8], 14 A[9] + 6 A[15] + 10 A[14] + 5 A[28] + 3 A[21] + 3 A[32] + A[26] + 3 A[31] + 3 A[22] + 7 A[12] + 9 A[11] + 4 A[25] + 3 A[24] + 6 A[30] + 6 A[16] + 6 A[27] + 2 A[17] + 11 A[10] + 5 A[23] + 3 A[19] + 11 A[13] + 3 A[29] + 2 A[8] + 2 A[18] + 3 A[20]], [4 A[9] + 2 A[14] + A[28] + A[21] + 7 A[32] + A[26] + 7 A[31] + 6 A[12] + 9 A[11] + 2 A[25] + 2 A[24] + 4 A[30] + 2 A[16] + 2 A[27] + 2 A[17] + 11 A[10] + 2 A[23] + 2 A[19] + 4 A[13] + 2 A[8] + A[33] + 2 A[18] + A[20], 9 A[9] + 2 A[15] + 6 A[14] + 3 A[28] + A[21] + 2 A[32] + 8 A[31] + A[22] + 3 A[12] + 12 A[11] + 4 A[25] + 3 A[24] + 5 A[30] + 6 A[16] + 7 A[27] + 2 A[17] + 12 A[10] + 3 A[23] + A[19] + 15 A[13] + A[29] + 7 A[33] + 3 A[18] + 3 A[20]], [ 7 A[32] + 5 A[26] + 8 A[12] + 3 A[19], 6 A[9] + 2 A[14] + A[28] + 2 A[32] + 4 A[26] + 4 A[12] + 5 A[24] + A[34] + 7 A[30] + 3 A[16] + 8 A[27] + 2 A[17] + 6 A[10] + 3 A[23] + 2 A[19] + 8 A[13] + 7 A[33]], [4 A[9] + 4 A[35] + 8 A[15] + 14 A[14] + 11 A[28] + 7 A[21] + A[32] + 4 A[31] + 2 A[12] + 8 A[11] + 4 A[25] + 3 A[34] + 6 A[30] + 4 A[16] + 4 A[27] + 6 A[10] + 4 A[23] + 5 A[19] + 8 A[13] + 4 A[29] + A[33] + 3 A[20], A[9] + 3 A[35] + 12 A[15] + A[32] + 4 A[26] + 3 A[31] + 2 A[12] + 6 A[11] + 3 A[25] + A[16] + 5 A[27] + 4 A[23] + A[19] + 2 A[13] + 4 A[29] + A[33] + 4 A[20]]], [1, 1, 3, 1, 13, 3, 15, 1, 1, 13, 13, 3, 3, 15, 15, 1, 1, 1, 13, 13, 13, 13, 3, 3, 3, 15, 15, 15, 15, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14}, never show up! Theorem Number, 18, : Let C(n) be the constant term, in x, of n (1/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [3 A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [15 A[8], 6 A[9] + 8 A[14] + A[16]], [2 A[10] + 12 A[12] + A[17], 8 A[11] + 8 A[13] + 3 A[18] + 4 A[20]], [ 2 A[8] + 4 A[12] + 5 A[19], 6 A[9] + 14 A[11] + 8 A[13] + 4 A[16] + 2 A[18] + A[20]], [ 2 A[10] + 4 A[12] + 12 A[14] + 4 A[17] + 4 A[19] + A[21], 8 A[11] + 4 A[15] + 6 A[18] + A[22]], [ 6 A[10] + 4 A[12] + 5 A[17] + 4 A[19], 6 A[11] + 4 A[13] + 5 A[18] + 4 A[20]], [12 A[12] + 3 A[19], 12 A[13] + 3 A[20]], [4 A[14] + 3 A[21], 4 A[15] + 3 A[22]], [A[8], 6 A[9] + 2 A[16] + 3 A[23]], [6 A[10] + 12 A[12] + 2 A[17] + 4 A[19] + 3 A[24], 10 A[9] + 8 A[13] + 4 A[16] + 6 A[18] + 5 A[20] + 4 A[23] + A[25] + 3 A[27] ], [2 A[8] + 4 A[10] + 12 A[12] + 8 A[14] + 2 A[17] + 7 A[19] + 3 A[21] + 4 A[24] + 8 A[26] + A[28], 2 A[8] + 3 A[9] + 4 A[10] + 8 A[11] + 8 A[12] + 4 A[13] + 4 A[14] + 2 A[16] + 2 A[17] + A[18] + 5 A[19] + A[20] + 2 A[21] + 3 A[23] + 2 A[24] + A[25] + 3 A[26] + 2 A[27] + 2 A[28]], [ 6 A[10] + 4 A[12] + 12 A[14] + 6 A[17] + 4 A[19] + 4 A[21] + 3 A[28], 4 A[9] + 6 A[11] + 4 A[13] + 12 A[15] + 2 A[16] + 6 A[18] + 2 A[20] + 4 A[22] + 2 A[23] + 2 A[27] + 3 A[29]], [4 A[8] + 12 A[10] + 12 A[12] + 2 A[17] + 5 A[19] + 2 A[24] + 8 A[26] + 4 A[30], 2 A[8] + 9 A[9] + 2 A[10] + 2 A[11] + 4 A[12] + 8 A[13] + 4 A[14] + 4 A[16] + A[17] + A[18] + A[19] + 2 A[20] + 2 A[21] + 3 A[22] + 3 A[23] + A[25] + 3 A[26] + 3 A[27] + 2 A[28] + A[29] + A[30]], [ 4 A[9] + 8 A[10] + 4 A[14] + A[16] + 2 A[17] + A[23] + A[28] + 4 A[30], 8 A[9] + 12 A[11] + 8 A[13] + 4 A[15] + 4 A[16] + 5 A[18] + 4 A[20] + 4 A[23] + 2 A[25] + 4 A[27] + A[29] + 7 A[31]], [ 6 A[32] + 6 A[26] + 4 A[30] + 4 A[17] + 8 A[10] + 4 A[19] + 7 A[8], 6 A[9] + 2 A[16] + A[23]], [ 2 A[32] + 2 A[26] + 4 A[12] + A[24] + 4 A[30] + 2 A[17] + 10 A[10] + 4 A[8] , 2 A[9] + 4 A[31] + 10 A[11] + A[25] + 2 A[16] + 2 A[27] + 4 A[13] + 2 A[33] + 2 A[18]], [A[32] + 2 A[26] + 8 A[12] + 2 A[19] + 2 A[8], 6 A[9] + 4 A[16] + 4 A[27] + 12 A[13] + 2 A[33] + 3 A[20]], [10 A[9] + 8 A[14] + 6 A[28] + 2 A[21] + 6 A[32] + 2 A[26] + 4 A[12] + 2 A[11] + 2 A[25] + A[24] + A[34] + 7 A[30] + 7 A[16] + 10 A[27] + 2 A[17] + 6 A[10] + A[23] + 12 A[13] + 4 A[8] + 8 A[33] + 2 A[20], 4 A[35] + 4 A[15] + 5 A[29]]], [1, 1, 3, 1, 15, 3, 1, 1, 11, 15, 9, 3, 1, 1, 7, 3, 11, 5, 15, 13, 9, 7, 9, 1, 15, 1, 3, 7, 9, 3, 13, 11, 1, 5, 11]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 6, 8, 10, 12, 14}, never show up! Theorem Number, 19, : Let C(n) be the constant term, in x, of n (2/x + x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 9, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[5], A[6]], [A[7], A[8]], [A[5], A[9]], [8 A[5], 8 A[6]], [4 A[5], 4 A[6]], 0, 0], [1, 1, 1, 4, 1, 8, 4, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 20, : Let C(n) be the constant term, in x, of n (2/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 9, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[5], A[6]], [A[7], A[8]], [A[5], A[9]], [8 A[5], 8 A[6]], [12 A[5], 12 A[6]], 0, 0], [1, 1, 1, 12, 1, 8, 12, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15}, never show up! Theorem Number, 21, : Let C(n) be the constant term, in x, of n (2/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[8], A[9]], [A[11], A[10]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 22, : Let C(n) be the constant term, in x, of n (2/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [5 A[8], 8 A[8] + 11 A[9] + 7 A[10] + 9 A[11] + 4 A[13] + 4 A[14] + 2 A[16] + A[17] + 7 A[18]], [12 A[8] + 2 A[9] + 12 A[10] + 6 A[12] + 10 A[13] + 4 A[14] + 6 A[16] + 9 A[17] + 2 A[19] + 6 A[20], 8 A[8] + 4 A[9] + 2 A[10] + 6 A[11] + 4 A[12] + 8 A[13] + 4 A[16] + 6 A[17] + 3 A[18] + 4 A[19] + 4 A[20]], [6 A[8] + 2 A[9] + 6 A[10] + 6 A[12] + 6 A[13] + 6 A[14] + 6 A[16] + 6 A[17] + 5 A[19] + 10 A[20] + 6 A[21], 8 A[9] + 2 A[10] + 8 A[13] + 2 A[14] + 14 A[16] + 2 A[17] + 11 A[20] + 2 A[21]], [14 A[10] + 8 A[12] + 8 A[13] + 6 A[14] + 12 A[17] + 8 A[19] + 8 A[20] + A[21], 8 A[8] + 11 A[9] + 12 A[11] + 4 A[12] + 6 A[13] + 3 A[14] + 9 A[15] + 15 A[16] + 14 A[17] + 6 A[18] + 4 A[19] + 15 A[21] + 6 A[22]], [2 A[8] + 7 A[9] + 14 A[10] + 5 A[12] + 9 A[13] + 6 A[14] + 5 A[16] + 5 A[17] + A[19] + 7 A[20] + 4 A[21], 7 A[9] + 8 A[10] + 8 A[11] + 8 A[12] + 11 A[14] + 13 A[15] + 4 A[16] + 14 A[17] + A[18] + 8 A[19] + 2 A[20] + 7 A[21] + 3 A[22] + 3 A[23]], [ 4 A[10] + 10 A[12] + 2 A[17] + 3 A[19] + 2 A[24], 14 A[8] + 2 A[9] + 14 A[10] + 11 A[12] + 15 A[13] + 4 A[14] + A[16] + 7 A[19] + 6 A[20] + 14 A[21] + A[23] + 4 A[24]], [6 A[8] + 6 A[9] + 7 A[10] + 13 A[11] + 3 A[12] + 7 A[13] + 8 A[14] + 3 A[16] + 14 A[17] + 5 A[18] + 3 A[19] + 5 A[20] + 7 A[21] + 3 A[23] + 5 A[24] + 6 A[25], 11 A[9] + 2 A[10] + 10 A[11] + 8 A[13] + 2 A[14] + 8 A[15] + 7 A[16] + 2 A[17] + 3 A[18] + 8 A[20] + 2 A[21] + 5 A[22] + 2 A[23] + 7 A[25]], [9 A[8] + 6 A[9] + A[10] + 3 A[11] + 9 A[12] + 7 A[13] + 4 A[14] + 3 A[16] + 11 A[17] + 5 A[18] + 6 A[19] + 5 A[20] + 6 A[21] + 3 A[23] + 2 A[24] + 8 A[25] + 5 A[26], 4 A[8] + 5 A[9] + 10 A[10] + 8 A[11] + 3 A[12] + 3 A[13] + 2 A[14] + 2 A[16] + 5 A[17] + A[18] + 2 A[19] + A[20] + 4 A[21] + 2 A[23] + 3 A[24] + 3 A[25] + 3 A[26]], [4 A[8] + 5 A[9] + 4 A[10] + 4 A[11] + 5 A[12] + 7 A[13] + 4 A[14] + 5 A[16] + 3 A[17] + 2 A[18] + 2 A[19] + 4 A[20] + 2 A[21] + 2 A[25] + A[26] + 7 A[27], 7 A[9] + 2 A[10] + 10 A[11] + 6 A[13] + 2 A[14] + 5 A[16] + 2 A[17] + A[18] + 4 A[20] + 2 A[21] + 6 A[25] + 6 A[27]], [2 A[8] + A[9] + A[10] + 7 A[11] + 8 A[12] + 2 A[13] + 4 A[14] + A[16] + 12 A[17] + 2 A[18] + A[19] + 3 A[20] + 3 A[24] + 3 A[25] + 2 A[26] + A[27] + 4 A[28], 4 A[8] + 8 A[9] + 2 A[10] + 8 A[11] + 5 A[12] + 7 A[13] + 7 A[14] + 15 A[15] + 7 A[16] + 11 A[17] + 4 A[18] + 2 A[19] + 2 A[20] + 5 A[21] + A[22] + A[23] + 3 A[24] + 4 A[25] + A[26] + 4 A[27] + 4 A[28]], [4 A[8] + 9 A[9] + 6 A[10] + 14 A[11] + 11 A[12] + 11 A[13] + 14 A[14] + 2 A[15] + 8 A[16] + A[17] + 3 A[18] + 2 A[19] + 2 A[20] + 5 A[21] + A[23] + A[24] + 7 A[25] + 3 A[26] + 9 A[27] + 6 A[28] + 2 A[29], 2 A[8] + 6 A[9] + 7 A[10] + 15 A[11] + 4 A[12] + 6 A[13] + 4 A[14] + 4 A[15] + 4 A[16] + 5 A[17] + A[18] + A[19] + A[20] + 2 A[21] + A[22] + 10 A[25] + A[26] + 3 A[27] + 2 A[28] + 2 A[29]], [7 A[9] + 12 A[10] + 8 A[11] + 6 A[12] + 6 A[13] + 8 A[14] + 2 A[15] + 6 A[16] + A[17] + 3 A[18] + A[19] + 3 A[20] + 3 A[21] + A[23] + A[24] + 5 A[25] + 2 A[26] + 5 A[27] + 3 A[28] + 2 A[29] + 8 A[30], 4 A[9] + 2 A[10] + 2 A[11] + 6 A[13] + 2 A[14] + 2 A[15] + 4 A[16] + 3 A[20] + 2 A[21] + 2 A[25] + 4 A[27] + 2 A[29] + 2 A[30]], [3 A[9] + 6 A[10] + 4 A[11] + 6 A[12] + 6 A[13] + 9 A[14] + A[15] + 3 A[16] + 2 A[17] + 2 A[21] + 2 A[24] + 2 A[25] + 2 A[26] + 4 A[27] + 4 A[28] + A[29] + 12 A[30], 3 A[9] + 2 A[10] + 2 A[11] + 6 A[12] + 6 A[13] + 4 A[14] + 6 A[15] + 3 A[16] + 2 A[17] + 3 A[18] + A[20] + 2 A[21] + A[22] + 2 A[24] + 10 A[25] + 2 A[26] + 3 A[27] + 2 A[28] + 6 A[29] + 10 A[30] + A[31]], [A[8], 8 A[9] + 2 A[14] + 2 A[21] + 2 A[31] + 4 A[11] + 2 A[25] + 2 A[30] + 4 A[16] + 2 A[27] + 2 A[10] + A[23] + 2 A[13]], [9 A[9] + 8 A[15] + 8 A[14] + 4 A[28] + 4 A[21] + 4 A[32] + A[26] + 9 A[31] + 2 A[22] + 6 A[12] + 3 A[11] + 11 A[25] + 2 A[24] + 2 A[30] + 7 A[16] + 4 A[27] + 4 A[17] + A[10] + 3 A[19] + 6 A[13] + 10 A[29] + 6 A[8] + 6 A[33] + A[18], 4 A[9] + 10 A[15] + 4 A[14] + 2 A[28] + 2 A[21] + 2 A[32] + 7 A[31] + 3 A[22] + 2 A[12] + 12 A[11] + 8 A[25] + 8 A[30] + 4 A[16] + 2 A[27] + 2 A[17] + 14 A[10] + 4 A[13] + 13 A[29] + 4 A[8] + 6 A[33]], [7 A[9] + 8 A[15] + 6 A[14] + A[28] + A[21] + A[32] + 4 A[26] + 2 A[31] + 2 A[22] + 3 A[12] + 11 A[11] + 11 A[25] + 2 A[24] + 4 A[34] + 14 A[30] + 7 A[16] + 5 A[27] + 3 A[17] + 9 A[10] + A[19] + 7 A[13] + 10 A[29] + 2 A[33], 9 A[9] + 11 A[15] + 5 A[14] + A[21] + 5 A[31] + 2 A[22] + 4 A[11] + A[25] + 8 A[34] + 11 A[30] + 9 A[16] + 5 A[27] + 3 A[17] + 4 A[10] + 10 A[13] + 13 A[29] + 8 A[33]], [4 A[9] + A[35] + 4 A[15] + 2 A[14] + 4 A[28] + 3 A[31] + A[22] + 14 A[11] + 6 A[25] + A[24] + A[34] + 4 A[30] + 4 A[16] + 2 A[27] + A[17] + 8 A[10] + 4 A[19] + 4 A[13] + 6 A[29] + 2 A[33] + A[18], 4 A[9] + A[35] + 3 A[15] + 3 A[14] + 5 A[21] + 4 A[31] + 11 A[25] + 2 A[34] + 5 A[30] + 4 A[16] + 3 A[27] + A[17] + 8 A[10] + 4 A[23] + 4 A[13] + A[29] + A[33] + A[18] + 4 A[20]]], [1, 1, 1, 1, 5, 1, 13, 1, 1, 5, 5, 1, 1, 13, 13, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 13, 13, 13, 13, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15}, never show up! Theorem Number, 23, : Let C(n) be the constant term, in x, of n (2/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[8], A[9]], [A[11], A[10]], [A[8], A[8]], [A[8], A[8]], [%1, %1], [%1, %1], [A[8], A[8]], [A[8], A[8]], [%1, %1], [%1, %1]], [1, 1, 1, 1, 9, 1, 9, 1, 1, 9, 9, 1, 1, 9, 9]] %1 := 7 A[8] + 2 A[9] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 24, : Let C(n) be the constant term, in x, of n (2/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [13 A[8], 4 A[9] + 8 A[14] + A[16]], [12 A[8] + 7 A[9] + 7 A[10] + 5 A[11] + 6 A[12] + 14 A[13] + 8 A[14] + 9 A[16] + 14 A[17] + 3 A[18] + 10 A[19] + 14 A[20], 12 A[8] + 8 A[9] + 6 A[10] + 10 A[11] + 3 A[12] + 15 A[13] + 10 A[14] + 10 A[16] + 12 A[17] + 7 A[18] + 9 A[19] + 15 A[20]], [10 A[8] + 2 A[9] + 4 A[10] + 10 A[11] + 8 A[12] + 2 A[14] + 6 A[16] + 8 A[17] + 6 A[18] + 7 A[19] + 2 A[21], 12 A[8] + 8 A[9] + 9 A[10] + 5 A[11] + 6 A[13] + 14 A[16] + 15 A[17] + 3 A[18] + 12 A[19] + 9 A[20] + 12 A[21]], [4 A[8] + 8 A[9] + 13 A[10] + 15 A[11] + 6 A[12] + 10 A[13] + 14 A[14] + 8 A[15] + 8 A[16] + 13 A[17] + 9 A[18] + 6 A[19] + 10 A[20] + 5 A[21] + 8 A[22], 3 A[9] + 6 A[11] + 8 A[12] + 10 A[13] + 12 A[14] + 4 A[15] + 5 A[16] + 8 A[17] + 4 A[18] + 8 A[19] + 6 A[20] + 12 A[21] + 3 A[22]], [4 A[8] + 9 A[9] + A[10] + 15 A[11] + 6 A[12] + 10 A[13] + 12 A[14] + 8 A[15] + 7 A[16] + 4 A[17] + 9 A[18] + 6 A[19] + 10 A[20] + 4 A[21] + 8 A[22], A[10] + 11 A[11] + 7 A[12] + 13 A[13] + 4 A[14] + 8 A[15] + 9 A[16] + A[17] + 6 A[18] + A[19] + 9 A[20] + 2 A[21] + 8 A[22] + A[23]], [4 A[8] + 7 A[9] + 4 A[10] + 4 A[11] + 6 A[12] + 10 A[13] + 10 A[14] + 4 A[15] + 4 A[16] + 13 A[17] + 4 A[18] + 7 A[19] + 10 A[20] + 6 A[21] + 4 A[22] + A[23] + 3 A[24], 8 A[9] + 2 A[10] + 4 A[11] + 6 A[12] + 10 A[13] + 2 A[14] + 4 A[15] + 2 A[16] + 2 A[17] + 4 A[18] + 2 A[19] + 7 A[20] + 2 A[21] + 4 A[22] + 2 A[23]], [ 9 A[9] + 13 A[10] + A[11] + 7 A[12] + 3 A[13] + 8 A[14] + 4 A[16] + 8 A[17] + 2 A[18] + 5 A[19] + 3 A[20] + 7 A[21] + A[23] + 5 A[24] + 9 A[25], 9 A[9] + 6 A[10] + 12 A[11] + 6 A[12] + 2 A[14] + 6 A[15] + 4 A[16] + 6 A[17] + 2 A[18] + 2 A[19] + 2 A[21] + 3 A[22] + 3 A[23] + 6 A[25]], [3 A[8] + 7 A[9] + 14 A[10] + 6 A[11] + 7 A[12] + 5 A[13] + 4 A[14] + 4 A[15] + 4 A[16] + 10 A[17] + 3 A[18] + 2 A[19] + 5 A[20] + 6 A[21] + 4 A[22] + 3 A[23] + 2 A[24] + 11 A[25] + A[26], 2 A[8] + 8 A[9] + 6 A[10] + 2 A[12] + 2 A[14] + 3 A[16] + 6 A[17] + 3 A[19] + 2 A[21] + 2 A[23] + A[26]], [10 A[9] + 13 A[10] + 13 A[11] + 3 A[12] + 7 A[13] + 2 A[14] + 6 A[16] + 4 A[17] + 3 A[18] + A[19] + 5 A[20] + 4 A[21] + 2 A[23] + 2 A[24] + 8 A[25] + 2 A[27], 5 A[9] + 6 A[10] + 6 A[11] + 6 A[12] + 8 A[13] + 2 A[14] + 4 A[16] + 6 A[17] + A[18] + 2 A[19] + 5 A[20] + 2 A[21] + A[23] + 2 A[25] + 5 A[27]], [4 A[8] + 6 A[9] + 9 A[10] + 11 A[11] + 9 A[12] + A[13] + 8 A[14] + 3 A[16] + 8 A[17] + 2 A[18] + 5 A[19] + A[20] + 8 A[21] + A[23] + A[24] + 7 A[25] + 3 A[26] + 2 A[28], 2 A[8] + 4 A[9] + 12 A[10] + 8 A[12] + 6 A[13] + 8 A[14] + 3 A[16] + 12 A[17] + 5 A[19] + 2 A[20] + 4 A[21] + A[23] + A[26] + A[27] + 4 A[28] ], [2 A[8] + A[9] + 8 A[10] + 4 A[11] + 4 A[12] + 4 A[13] + 6 A[14] + A[16] + 6 A[17] + 2 A[18] + A[19] + A[20] + 3 A[21] + 2 A[25] + A[26] + A[27] + 2 A[28], 13 A[9] + 6 A[10] + 8 A[11] + 6 A[12] + 4 A[13] + 2 A[14] + 6 A[15] + 8 A[16] + 6 A[17] + 2 A[19] + 2 A[21] + 3 A[22] + 3 A[23] + 6 A[25] + 4 A[27] + 2 A[29]], [6 A[8] + 3 A[9] + A[10] + 5 A[11] + 7 A[12] + 3 A[13] + 8 A[14] + 3 A[16] + 5 A[17] + 2 A[18] + A[19] + 3 A[20] + 5 A[21] + 3 A[24] + 5 A[25] + 3 A[26] + 3 A[28] + 3 A[30], 10 A[8] + 10 A[9] + 13 A[10] + 5 A[11] + 11 A[12] + 9 A[13] + 2 A[14] + 7 A[16] + 5 A[17] + 2 A[18] + 2 A[19] + 4 A[20] + 11 A[21] + A[23] + 3 A[24] + 5 A[25] + 5 A[26] + 2 A[27] + 7 A[28] + 11 A[30]], [6 A[8] + 5 A[9] + 6 A[10] + 6 A[12] + 12 A[14] + 2 A[16] + 3 A[17] + A[19] + 8 A[21] + A[23] + 3 A[26] + 3 A[28] + 13 A[30], 4 A[8] + 5 A[9] + 6 A[10] + 10 A[11] + 2 A[12] + 4 A[13] + 9 A[14] + 3 A[15] + 4 A[16] + 4 A[17] + A[18] + 4 A[20] + 3 A[21] + A[22] + A[23] + A[24] + 6 A[25] + 2 A[26] + 2 A[27] + 4 A[28] + A[29] + 13 A[30] + 3 A[31]], [A[8], 3 A[9] + 4 A[14] + 2 A[28] + 2 A[21] + 2 A[26] + 2 A[12] + 4 A[11] + 2 A[25] + 2 A[24] + 12 A[30] + 2 A[16] + 4 A[17] + 6 A[10] + 2 A[13] + 4 A[8] + 2 A[18] + 2 A[20]], [5 A[9] + 2 A[14] + 2 A[21] + A[32] + A[26] + 4 A[12] + 2 A[24] + 10 A[30] + 2 A[16] + A[17] + 12 A[10] + A[23] + 2 A[8], 5 A[9] + 5 A[15] + 11 A[14] + 3 A[28] + 6 A[21] + 2 A[32] + 4 A[26] + 3 A[22] + 9 A[12] + 5 A[11] + 2 A[25] + 3 A[24] + 6 A[30] + 5 A[16] + 3 A[27] + 4 A[17] + 15 A[10] + A[19] + 5 A[13] + 2 A[29] + 8 A[8] + A[33] + A[20]], [8 A[9] + A[15] + 5 A[14] + 3 A[28] + A[32] + 2 A[26] + 4 A[31] + A[22] + 4 A[12] + A[11] + 8 A[25] + A[24] + 6 A[34] + 10 A[30] + 6 A[16] + 5 A[27] + 2 A[17] + A[10] + 2 A[23] + 4 A[13] + 4 A[29] + 2 A[8] + A[33] + A[18] + 2 A[20], 3 A[9] + 4 A[14] + 2 A[28] + A[21] + 4 A[26] + 5 A[24] + A[34] + 4 A[30] + 2 A[16] + A[17] + 6 A[10] + A[23] + 2 A[13] + A[33]], [8 A[9] + 6 A[14] + 8 A[28] + 4 A[21] + 6 A[11] + 2 A[25] + 3 A[24] + A[34] + 11 A[30] + 6 A[16] + 4 A[10] + 6 A[23] + 4 A[19] + 4 A[13] + A[33] + 3 A[20], 6 A[9] + 3 A[15] + 3 A[14] + 3 A[28] + 4 A[21] + 4 A[31] + 6 A[22] + 8 A[25] + A[24] + 5 A[30] + 6 A[16] + 7 A[27] + 8 A[10] + 6 A[13] + 4 A[29] + A[33] + 6 A[20]]], [1, 1, 1, 1, 13, 1, 5, 1, 1, 13, 13, 1, 1, 5, 5, 1, 1, 1, 13, 13, 13, 13, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15}, never show up! Theorem Number, 25, : Let C(n) be the constant term, in x, of n (2/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 20, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [8 A[8], 8 A[9]], [8 A[10], 8 A[11]], [2 A[8], 2 A[9]], [A[19], 8 A[10] + 10 A[11]], 0, 0, [A[20], 4 A[9] + 2 A[10] + A[11] + 2 A[13] + 3 A[14]], [5 A[9] + 4 A[13] + 3 A[16], 2 A[9] + 2 A[13] + 2 A[16]], [2 A[10] + A[14], 4 A[15]], 0, 0], [1, 1, 2, 1, 8, 2, 0, 1, 8, 8, 0, 2, 0, 0, 0, 0, 8, 0, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 26, : Let C(n) be the constant term, in x, of n (2/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 20, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], 0, 0, [2 A[8], 2 A[9]], [A[19], 8 A[10] + 10 A[11]], 0, 0, [A[20], 2 A[10] + A[11] + 3 A[14] + 4 A[15]], [5 A[9] + 4 A[13] + 3 A[16], 2 A[9] + 2 A[13] + 2 A[16]], [2 A[10] + A[14], 2 A[9] + A[11] + 2 A[13] + 2 A[14] + A[15] + 2 A[16] + A[18]], 0, 0], [1, 1, 2, 1, 0, 2, 0, 1, 8, 0, 0, 2, 0, 0, 0, 0, 8, 0, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 27, : Let C(n) be the constant term, in x, of n (2/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[8]], [A[8], A[8]], [%2, %2], [%2, %2], [3 A[8], 3 A[8]], [10 A[8] + 7 A[9] + 2 A[10], 10 A[8] + 7 A[9] + 2 A[10]], [%1, %1], [%1, %1]], [1, 1, 3, 1, 9, 3, 11, 1, 1, 9, 9, 3, 3, 11, 11]] %1 := 9 A[8] + 2 A[9] %2 := 7 A[8] + 2 A[9] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15}, never show up! Theorem Number, 28, : Let C(n) be the constant term, in x, of n (2/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [3 A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [13 A[8], 4 A[9] + 8 A[14] + A[16]], [2 A[9] + 12 A[10] + 6 A[12] + 10 A[13] + 4 A[14] + 2 A[16] + 9 A[17] + 2 A[19] + 6 A[20], 3 A[9] + 8 A[10] + 10 A[11] + 7 A[12] + 15 A[13] + 6 A[14] + 3 A[16] + 10 A[17] + 7 A[18] + 11 A[19] + 5 A[20]], [10 A[8] + 13 A[9] + 6 A[10] + 7 A[12] + 9 A[13] + 6 A[14] + 13 A[16] + 8 A[17] + 2 A[19] + 7 A[20] + 4 A[21], 12 A[8] + 14 A[9] + 13 A[10] + 9 A[11] + 14 A[12] + 8 A[14] + 4 A[16] + 15 A[17] + 7 A[18] + 10 A[19] + 3 A[20]], [4 A[8] + 7 A[9] + 11 A[10] + 15 A[11] + 5 A[12] + 9 A[13] + 15 A[16] + A[17] + A[18] + 5 A[19] + 15 A[20] + 5 A[21], 15 A[9] + 4 A[10] + 8 A[12] + 14 A[13] + 11 A[14] + 13 A[15] + 15 A[16] + 8 A[17] + 12 A[18] + 12 A[19] + 4 A[20] + 7 A[21] + 10 A[22]], [6 A[9] + 12 A[10] + 10 A[11] + 12 A[12] + 6 A[13] + 6 A[14] + 8 A[15] + 3 A[16] + 7 A[17] + 6 A[18] + 12 A[19] + 10 A[20] + 6 A[21] + 8 A[22] + A[23], 4 A[8] + 5 A[9] + 15 A[10] + 3 A[11] + 13 A[12] + 11 A[13] + 8 A[15] + 12 A[16] + 7 A[17] + 8 A[18] + 13 A[19] + 13 A[20] + 10 A[21] + 8 A[22] + 3 A[23]], [4 A[8] + 15 A[9] + 2 A[10] + 14 A[12] + 8 A[13] + 5 A[16] + 9 A[17] + 13 A[19] + 8 A[20] + 4 A[23] + 3 A[24], 8 A[9] + 4 A[10] + 6 A[12] + 12 A[13] + 4 A[14] + 4 A[16] + 2 A[17] + 6 A[19] + 3 A[20] + 8 A[21] + 4 A[23] + 2 A[24]], [4 A[8] + 12 A[9] + 15 A[10] + 7 A[11] + 9 A[12] + 11 A[13] + 14 A[14] + 7 A[16] + 5 A[17] + 4 A[18] + 9 A[19] + 9 A[20] + 11 A[21] + 5 A[23] + 2 A[24] + 3 A[25], 6 A[9] + 4 A[11] + 7 A[12] + 11 A[13] + 11 A[14] + 5 A[15] + 4 A[16] + 10 A[17] + 2 A[18] + 7 A[19] + 7 A[20] + 9 A[21] + 2 A[22] + 2 A[23] + 4 A[24] + 4 A[25]], [3 A[8] + 11 A[9] + 6 A[10] + 2 A[11] + 9 A[12] + 7 A[13] + 4 A[14] + 2 A[16] + 3 A[17] + 7 A[19] + A[20] + 2 A[21] + 3 A[23] + 3 A[24] + 2 A[25] + 2 A[26], 4 A[9] + 2 A[16] + 3 A[23]], [8 A[8] + 6 A[9] + 12 A[10] + 8 A[11] + 10 A[12] + 10 A[13] + 8 A[14] + 3 A[16] + 6 A[17] + 5 A[18] + 6 A[19] + 5 A[20] + 4 A[21] + 3 A[23] + 3 A[24] + 5 A[25] + 4 A[26] + 5 A[27], 4 A[8] + 4 A[9] + 6 A[10] + 4 A[11] + 5 A[12] + 7 A[13] + 4 A[14] + 3 A[16] + 3 A[17] + A[18] + 3 A[19] + 2 A[20] + 2 A[21] + 3 A[23] + 3 A[24] + 2 A[25] + 2 A[26] + 3 A[27]], [4 A[8] + 4 A[9] + 14 A[10] + 2 A[11] + 10 A[12] + 4 A[13] + 4 A[14] + 2 A[16] + 9 A[17] + 4 A[19] + 2 A[20] + 2 A[21] + 2 A[23] + 3 A[24] + 2 A[25] + 3 A[26] + 2 A[27] + 2 A[28], 4 A[8] + 10 A[9] + 14 A[10] + 2 A[11] + 10 A[12] + 8 A[13] + 8 A[14] + 4 A[16] + 9 A[17] + 4 A[19] + 2 A[20] + 3 A[21] + 6 A[23] + A[24] + 2 A[25] + 2 A[26] + 3 A[27] + 5 A[28]], [4 A[8] + 4 A[9] + 6 A[10] + 2 A[11] + 6 A[12] + 4 A[13] + 4 A[14] + 2 A[16] + 2 A[17] + 4 A[19] + 2 A[20] + 2 A[23] + 4 A[24] + 2 A[25] + 2 A[26] + 2 A[27] + A[28], 2 A[8] + 6 A[9] + A[10] + 7 A[11] + 2 A[12] + 8 A[13] + A[14] + 7 A[15] + 3 A[16] + A[17] + 7 A[18] + A[19] + 2 A[20] + 3 A[22] + 3 A[23] + A[26] + 4 A[27] + A[28] + 3 A[29]], [2 A[8] + 11 A[9] + 11 A[10] + 9 A[11] + 7 A[12] + 11 A[13] + 5 A[14] + 7 A[15] + 6 A[16] + A[17] + 6 A[18] + 3 A[19] + 5 A[20] + 4 A[21] + 2 A[22] + 5 A[23] + A[24] + 3 A[25] + A[26] + 6 A[27] + 3 A[28] + 3 A[29] + 5 A[30], 2 A[8] + 12 A[9] + A[10] + 7 A[11] + 5 A[12] + 9 A[13] + 6 A[14] + 2 A[15] + 6 A[16] + 5 A[17] + 2 A[18] + 4 A[19] + 4 A[20] + A[21] + A[22] + 4 A[23] + 2 A[24] + 3 A[25] + A[26] + 4 A[27] + 3 A[28] + A[29] + 8 A[30]], [2 A[8] + 11 A[9] + 13 A[10] + 9 A[11] + 5 A[12] + 11 A[13] + 3 A[14] + 7 A[15] + 6 A[16] + 2 A[17] + 2 A[18] + 4 A[19] + 5 A[20] + A[21] + 2 A[22] + 5 A[23] + A[24] + 3 A[25] + A[26] + 6 A[27] + A[28] + 3 A[29] + 6 A[30] + 4 A[31], 6 A[9] + 4 A[10] + 6 A[11] + 2 A[12] + 6 A[13] + 2 A[14] + 8 A[15] + 4 A[16] + 2 A[18] + 2 A[19] + 2 A[20] + A[21] + 2 A[22] + 2 A[23] + 2 A[25] + 4 A[27] + A[28] + 3 A[29] + 2 A[30] + 4 A[31]], [ A[8], 14 A[9] + 6 A[15] + 10 A[14] + 5 A[28] + 3 A[21] + 3 A[32] + A[26] + 3 A[31] + 3 A[22] + 7 A[12] + 9 A[11] + 4 A[25] + 3 A[24] + 6 A[30] + 6 A[16] + 6 A[27] + 2 A[17] + 11 A[10] + 5 A[23] + 3 A[19] + 11 A[13] + 3 A[29] + 2 A[8] + 2 A[18] + 3 A[20]], [4 A[9] + 2 A[14] + A[28] + A[21] + 7 A[32] + A[26] + 7 A[31] + 6 A[12] + 9 A[11] + 2 A[25] + 2 A[24] + 4 A[30] + 2 A[16] + 2 A[27] + 2 A[17] + 11 A[10] + 2 A[23] + 2 A[19] + 4 A[13] + 2 A[8] + A[33] + 2 A[18] + A[20], 9 A[9] + 2 A[15] + 6 A[14] + 3 A[28] + A[21] + 2 A[32] + 8 A[31] + A[22] + 3 A[12] + 12 A[11] + 4 A[25] + 3 A[24] + 5 A[30] + 6 A[16] + 7 A[27] + 2 A[17] + 12 A[10] + 3 A[23] + A[19] + 15 A[13] + A[29] + 7 A[33] + 3 A[18] + 3 A[20]], [ 7 A[32] + 5 A[26] + 8 A[12] + 3 A[19], 6 A[9] + 2 A[14] + A[28] + 2 A[32] + 4 A[26] + 4 A[12] + 5 A[24] + A[34] + 7 A[30] + 3 A[16] + 8 A[27] + 2 A[17] + 6 A[10] + 3 A[23] + 2 A[19] + 8 A[13] + 7 A[33]], [4 A[9] + 4 A[35] + 8 A[15] + 14 A[14] + 11 A[28] + 7 A[21] + A[32] + 4 A[31] + 2 A[12] + 8 A[11] + 4 A[25] + 3 A[34] + 6 A[30] + 4 A[16] + 4 A[27] + 6 A[10] + 4 A[23] + 5 A[19] + 8 A[13] + 4 A[29] + A[33] + 3 A[20], A[9] + 3 A[35] + 12 A[15] + A[32] + 4 A[26] + 3 A[31] + 2 A[12] + 6 A[11] + 3 A[25] + A[16] + 5 A[27] + 4 A[23] + A[19] + 2 A[13] + 4 A[29] + A[33] + 4 A[20]]], [1, 1, 3, 1, 13, 3, 15, 1, 1, 13, 13, 3, 3, 15, 15, 1, 1, 1, 13, 13, 13, 13, 3, 3, 3, 15, 15, 15, 15, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14}, never show up! Theorem Number, 29, : Let C(n) be the constant term, in x, of n (2/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [A[8], A[8]], [3 A[8], 3 A[8]], [10 A[8] + 7 A[9] + 2 A[10], 10 A[8] + 7 A[9] + 2 A[10]], [3 A[8], 3 A[8]], [3 A[8], 3 A[8]]], [1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 3, 3]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 30, : Let C(n) be the constant term, in x, of n (2/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [3 A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [5 A[8], 8 A[8] + 15 A[9] + 7 A[10] + 5 A[11] + 8 A[12] + 12 A[13] + 4 A[14] + 6 A[16] + A[17] + 3 A[18]], [8 A[9] + 15 A[10] + 15 A[11] + 4 A[12] + 12 A[13] + 12 A[14] + 10 A[17] + 9 A[18] + 8 A[19] + 8 A[20], 2 A[8] + 8 A[9] + 6 A[10] + 6 A[11] + A[12] + 13 A[13] + 6 A[14] + 12 A[16] + 12 A[17] + 3 A[18] + 3 A[19] + 13 A[20]], [10 A[8] + 2 A[10] + 12 A[12] + 4 A[13] + 2 A[14] + 6 A[17] + 3 A[19] + 4 A[20] + 2 A[21], 4 A[8] + 6 A[9] + 4 A[10] + 2 A[12] + 12 A[13] + 8 A[14] + 8 A[16] + 8 A[17] + 6 A[19] + 7 A[20] + 4 A[21]], [14 A[8] + 2 A[9] + 2 A[10] + 15 A[12] + 11 A[13] + 4 A[14] + 8 A[15] + 2 A[16] + 6 A[17] + 13 A[19] + 7 A[20] + A[21] + 8 A[22], 14 A[8] + 6 A[9] + 2 A[10] + 12 A[11] + 15 A[12] + 13 A[13] + 7 A[14] + 11 A[15] + 12 A[16] + 6 A[17] + 6 A[18] + 13 A[19] + 11 A[20] + 15 A[21] + 6 A[22]], [6 A[8] + 9 A[9] + 4 A[11] + 11 A[12] + 7 A[13] + A[14] + 5 A[15] + 6 A[16] + 7 A[17] + 4 A[18] + 9 A[19] + 5 A[20] + 13 A[21] + A[22] + A[23], 2 A[8] + 11 A[9] + 8 A[10] + 2 A[11] + 9 A[12] + 11 A[13] + 15 A[14] + 9 A[15] + 9 A[16] + 12 A[17] + A[18] + 11 A[19] + 9 A[20] + 11 A[21] + 5 A[22] + 2 A[23]], [ 4 A[8] + 12 A[9] + 4 A[12] + 6 A[13] + 8 A[14] + 4 A[15] + 3 A[16] + 6 A[17] + 7 A[19] + 6 A[20] + 4 A[21] + 4 A[22] + 3 A[23] + 2 A[24], 4 A[8] + 8 A[9] + 8 A[10] + 2 A[12] + 8 A[13] + 8 A[14] + 4 A[15] + 4 A[16] + 4 A[17] + 6 A[19] + 7 A[20] + 4 A[21] + 4 A[22] + 4 A[23]], [3 A[9] + 12 A[10] + 2 A[11] + 6 A[13] + 8 A[14] + 4 A[15] + 2 A[16] + 8 A[17] + 6 A[20] + 3 A[21] + 4 A[22] + A[23] + 2 A[25], 6 A[9] + 2 A[10] + 6 A[11] + 2 A[13] + 4 A[14] + 4 A[15] + 10 A[17] + 5 A[18] + 2 A[20] + 4 A[21] + 3 A[22] + 2 A[23] + 4 A[24] + 3 A[25]], [5 A[8] + 4 A[9] + 2 A[10] + 10 A[12] + 2 A[14] + A[16] + A[17] + 3 A[19] + 2 A[21] + A[23] + 3 A[24] + 7 A[26], 2 A[8] + 3 A[9] + 3 A[10] + A[11] + 2 A[12] + 2 A[13] + 4 A[14] + A[16] + 2 A[17] + A[19] + 2 A[20] + 4 A[21] + A[23] + A[24] + A[25] + A[26]], [6 A[8] + 9 A[9] + 9 A[10] + 5 A[11] + 6 A[12] + 12 A[13] + 10 A[14] + 4 A[15] + 6 A[16] + 14 A[17] + 3 A[19] + 4 A[20] + 6 A[21] + 4 A[22] + 3 A[23] + 4 A[24] + 5 A[25] + 3 A[26] + 8 A[27], 2 A[8] + 9 A[9] + 12 A[10] + 8 A[11] + 3 A[12] + 5 A[13] + 8 A[14] + 4 A[16] + 7 A[17] + 5 A[18] + 3 A[19] + 2 A[20] + 2 A[21] + 3 A[23] + A[24] + 2 A[25] + 2 A[26] + 3 A[27]], [8 A[8] + 15 A[9] + 8 A[11] + 13 A[12] + 13 A[13] + 8 A[14] + 4 A[15] + 8 A[16] + 13 A[17] + 5 A[18] + 6 A[19] + 5 A[20] + 6 A[21] + 4 A[22] + 7 A[23] + 3 A[24] + 5 A[25] + 6 A[26] + 8 A[27] + 12 A[28], 4 A[8] + 9 A[9] + 10 A[10] + 2 A[11] + 4 A[12] + 14 A[14] + 4 A[15] + 6 A[16] + 8 A[17] + 2 A[19] + 5 A[20] + 4 A[21] + 4 A[22] + 5 A[23] + 2 A[24] + 2 A[25] + 2 A[26] + 8 A[27] + 6 A[28]], [3 A[9] + 2 A[10] + 6 A[11] + 6 A[13] + 3 A[14] + A[15] + 2 A[16] + 4 A[17] + 5 A[18] + A[20] + A[21] + A[23] + A[25] + 3 A[27] + A[28] + A[29], 4 A[9] + 4 A[10] + 10 A[11] + 8 A[13] + 2 A[14] + 8 A[15] + 2 A[16] + 2 A[17] + 7 A[18] + 4 A[20] + A[21] + 2 A[22] + 2 A[23] + A[25] + 4 A[27] + A[28] + 3 A[29]], [11 A[9] + 12 A[10] + 6 A[11] + 6 A[12] + 10 A[13] + 9 A[14] + A[15] + 5 A[16] + 3 A[17] + 5 A[18] + A[19] + 3 A[20] + 2 A[21] + 4 A[23] + 3 A[24] + A[25] + 2 A[26] + 5 A[27] + 5 A[28] + A[29] + 8 A[30], 5 A[9] + 4 A[10] + 2 A[11] + 6 A[13] + 2 A[14] + 2 A[15] + 3 A[16] + 2 A[18] + A[20] + A[21] + 4 A[23] + 2 A[27] + A[28] + 2 A[29] + 2 A[30]], [2 A[8] + 10 A[9] + 7 A[10] + A[11] + 2 A[12] + 8 A[13] + 3 A[14] + A[15] + 5 A[16] + 2 A[17] + 4 A[18] + A[19] + 2 A[20] + A[21] + 5 A[23] + A[24] + 2 A[25] + A[26] + 4 A[27] + A[28] + A[29] + 4 A[30] + 7 A[31], 2 A[8] + 6 A[9] + 7 A[10] + 9 A[11] + 2 A[12] + 6 A[13] + 2 A[14] + 4 A[15] + 4 A[16] + A[17] + 3 A[18] + A[19] + 3 A[20] + A[21] + 4 A[23] + A[24] + 2 A[25] + A[26] + 3 A[27] + A[28] + A[29] + 3 A[30] + 4 A[31]], [A[8], 4 A[9] + 10 A[14] + 5 A[28] + 3 A[21] + 2 A[32] + 2 A[26] + 2 A[31] + 7 A[12] + 7 A[11] + 2 A[25] + 6 A[30] + 2 A[16] + 2 A[27] + 3 A[17] + 7 A[10] + A[23] + A[19] + 3 A[13] + 2 A[8] + A[18] + A[20]], [4 A[9] + 4 A[14] + 2 A[28] + 2 A[21] + A[32] + A[26] + 2 A[12] + A[24] + 10 A[30] + 2 A[16] + 2 A[27] + 4 A[17] + 12 A[10] + 2 A[23] + 4 A[13] + 2 A[8] + 2 A[20], 6 A[9] + 2 A[14] + A[28] + A[21] + 8 A[31] + 12 A[11] + 2 A[25] + 7 A[30] + 2 A[16] + A[27] + A[17] + 6 A[10] + 2 A[23] + 2 A[13] + 4 A[33] + 3 A[18] + A[20]], [8 A[9] + 10 A[14] + 5 A[28] + 2 A[21] + 5 A[32] + 5 A[26] + 2 A[31] + 4 A[12] + 9 A[11] + 4 A[25] + 2 A[24] + 3 A[34] + 9 A[30] + 3 A[16] + 6 A[27] + 7 A[10] + 3 A[23] + A[19] + 4 A[13] + A[18] + 2 A[20], 5 A[9] + 12 A[14] + 10 A[28] + 2 A[21] + 4 A[26] + A[31] + 5 A[11] + 7 A[25] + 5 A[24] + 4 A[34] + 10 A[30] + 2 A[16] + 3 A[27] + 7 A[10] + 3 A[23] + 6 A[13] + A[18] + 2 A[20]], [2 A[9] + A[35] + 4 A[15] + 6 A[14] + 3 A[28] + 4 A[31] + A[22] + 8 A[11] + 6 A[25] + 4 A[24] + 2 A[34] + 2 A[30] + A[16] + A[27] + 2 A[10] + 5 A[23] + 2 A[13] + 2 A[29] + 2 A[18] + 5 A[20], A[9] + 4 A[15] + 3 A[31] + A[22] + 6 A[11] + 5 A[25] + A[16] + A[27] + 2 A[13] + 6 A[29] + 2 A[18] + A[20]]], [1, 1, 3, 1, 5, 3, 7, 1, 1, 5, 5, 3, 3, 7, 7, 1, 1, 1, 5, 5, 5, 5, 3, 3, 3, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 31, : Let C(n) be the constant term, in x, of n (3/x + x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 13, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[5], A[6]], [A[7], A[8]], [A[5], A[6]], [A[10], A[11]], [6 A[5], 14 A[6]], [2 A[7], 10 A[8]], [A[10], A[11]], [6 A[5], 14 A[6]], [12 A[7] + 6 A[10], 10 A[8]], [6 A[5], 8 A[6] + 6 A[9]], [8 A[7] + 5 A[10] + 5 A[12], 10 A[8]]], [1, 1, 1, 6, 1, 6, 6, 12, 6, 6, 12, 6, 12]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15}, never show up! Theorem Number, 32, : Let C(n) be the constant term, in x, of n (3/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 9, states . Here it is: [[[A[2], 0], [A[3], A[4]], [A[5], A[6]], [A[7], A[8]], [A[5], A[9]], [8 A[5], 8 A[6]], [12 A[5], 12 A[6]], 0, 0], [1, 1, 1, 12, 1, 8, 12, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15}, never show up! Theorem Number, 33, : Let C(n) be the constant term, in x, of n (3/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 23, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[9]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [A[8], A[23]], [13 A[10] + 4 A[12], 5 A[11] + 12 A[13]], [8 A[8] + 9 A[12], A[13]], [4 A[10] + 13 A[14], 12 A[11] + 5 A[15]], [8 A[8] + 9 A[10], 8 A[9] + 9 A[11]], [6 A[8] + 11 A[12], 6 A[9] + 11 A[13]], [14 A[10] + 3 A[14], 14 A[11] + 3 A[15]], [A[8], 9 A[9] + 8 A[14]], [5 A[10] + 12 A[12], 13 A[11] + 4 A[13]], [12 A[8] + 5 A[12], 4 A[9] + 13 A[13]], [A[14], 8 A[11] + 9 A[15]], [8 A[8] + A[10] + 8 A[12], 8 A[9] + A[11] + 8 A[13]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 34, : Let C(n) be the constant term, in x, of n (3/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [7 A[8], 12 A[9] + 8 A[14] + 3 A[16]], [11 A[17] + 4 A[19], 8 A[13] + 3 A[18] + 4 A[20]], [ 2 A[8] + 10 A[10] + 8 A[12] + 6 A[17] + 9 A[19], 6 A[9] + 12 A[11] + 4 A[13] + 4 A[16] + 4 A[18] + 5 A[20]], [ 12 A[10] + 8 A[12] + 4 A[14] + 10 A[17] + 8 A[19] + A[21], 8 A[11] + 12 A[15] + 6 A[18] + A[22]], [8 A[10] + A[17], 8 A[11] + A[18]], [4 A[12] + A[19], 4 A[13] + A[20]], [6 A[9] + 4 A[14] + A[16] + A[21] + A[23], 12 A[15] + A[22]], [3 A[8], 2 A[16] + A[23]], [ 4 A[8] + 2 A[10] + 4 A[12] + 8 A[17] + 6 A[19] + A[24] + 2 A[26], 12 A[11] + 3 A[18] + 4 A[20] + 8 A[25]], [ 2 A[8] + 4 A[10] + 4 A[12] + 2 A[17] + 3 A[19] + 2 A[24] + 6 A[26], 12 A[9] + 8 A[11] + 4 A[13] + 4 A[16] + A[18] + 3 A[20] + 2 A[23] + 3 A[25] + 2 A[27]], [ 6 A[9] + 8 A[10] + 4 A[14] + A[16] + 6 A[17] + 3 A[21] + A[23] + 2 A[28], 8 A[11] + 4 A[15] + 2 A[18] + 5 A[22] + 8 A[25] + 4 A[29]], [6 A[8] + 10 A[10] + 8 A[12] + 2 A[17] + 2 A[19] + 3 A[24] + 9 A[26] + 5 A[30], 10 A[9] + 6 A[11] + 8 A[13] + 6 A[16] + 3 A[18] + 4 A[20] + 3 A[22] + 2 A[23] + 5 A[25] + 3 A[27] + 3 A[29]], [11 A[9] + 12 A[10] + 8 A[14] + 3 A[16] + A[17] + 4 A[21] + 2 A[23] + 2 A[24] + 3 A[28] + 7 A[30], 8 A[11] + 4 A[15] + 3 A[22] + 8 A[25] + 6 A[29]], [2 A[32] + A[26] + 4 A[12] + A[24] + 3 A[30] + 2 A[17] + 6 A[10] + 3 A[19] + 5 A[8], 8 A[9] + 2 A[16] + A[23]], [3 A[24] + 8 A[30] + 4 A[17] + 4 A[10], 2 A[22] + 8 A[11] + 7 A[25] + 2 A[29]], [3 A[32] + 2 A[26] + 8 A[12] + 2 A[24] + 2 A[30] + 4 A[10] + 4 A[19] + 2 A[8], 5 A[9] + 8 A[14] + 4 A[28] + A[21] + 4 A[32] + A[31] + 3 A[22] + 8 A[11] + 5 A[25] + 3 A[24] + 3 A[34] + 5 A[30] + 2 A[16] + A[27] + 8 A[10] + A[23] + 8 A[13] + 3 A[29] + 3 A[33] + 3 A[20]], [5 A[9] + A[35] + 4 A[15] + 8 A[14] + 2 A[28] + A[21] + 5 A[31] + 3 A[22] + 6 A[11] + 5 A[25] + 2 A[34] + 10 A[30] + 2 A[16] + 4 A[27] + 4 A[17] + 12 A[10] + A[23] + 4 A[13] + 8 A[29] + 2 A[33] + 2 A[20], 6 A[35] + 4 A[15] + 4 A[31] + 4 A[22] + 4 A[11] + 4 A[25] + 5 A[29]]], [1, 1, 1, 1, 7, 1, 3, 1, 11, 7, 1, 1, 11, 3, 5, 3, 11, 5, 7, 5, 1, 15, 3, 11, 5, 3, 9, 5, 11, 3, 13, 11, 1, 5, 11]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 6, 8, 10, 12, 14}, never show up! Theorem Number, 35, : Let C(n) be the constant term, in x, of n (3/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [13 A[8], 4 A[9] + 8 A[14] + A[16]], [12 A[8] + 7 A[9] + 7 A[10] + 5 A[11] + 6 A[12] + 14 A[13] + 8 A[14] + 9 A[16] + 14 A[17] + 3 A[18] + 10 A[19] + 14 A[20], 12 A[8] + 8 A[9] + 6 A[10] + 10 A[11] + 3 A[12] + 15 A[13] + 10 A[14] + 10 A[16] + 12 A[17] + 7 A[18] + 9 A[19] + 15 A[20]], [10 A[8] + 2 A[9] + 4 A[10] + 10 A[11] + 8 A[12] + 2 A[14] + 6 A[16] + 8 A[17] + 6 A[18] + 7 A[19] + 2 A[21], 12 A[8] + 8 A[9] + 9 A[10] + 5 A[11] + 6 A[13] + 14 A[16] + 15 A[17] + 3 A[18] + 12 A[19] + 9 A[20] + 12 A[21]], [4 A[8] + 8 A[9] + 13 A[10] + 15 A[11] + 6 A[12] + 10 A[13] + 14 A[14] + 8 A[15] + 8 A[16] + 13 A[17] + 9 A[18] + 6 A[19] + 10 A[20] + 5 A[21] + 8 A[22], 3 A[9] + 6 A[11] + 8 A[12] + 10 A[13] + 12 A[14] + 4 A[15] + 5 A[16] + 8 A[17] + 4 A[18] + 8 A[19] + 6 A[20] + 12 A[21] + 3 A[22]], [4 A[8] + 9 A[9] + A[10] + 15 A[11] + 6 A[12] + 10 A[13] + 12 A[14] + 8 A[15] + 7 A[16] + 4 A[17] + 9 A[18] + 6 A[19] + 10 A[20] + 4 A[21] + 8 A[22], A[10] + 11 A[11] + 7 A[12] + 13 A[13] + 4 A[14] + 8 A[15] + 9 A[16] + A[17] + 6 A[18] + A[19] + 9 A[20] + 2 A[21] + 8 A[22] + A[23]], [4 A[8] + 7 A[9] + 4 A[10] + 4 A[11] + 6 A[12] + 10 A[13] + 10 A[14] + 4 A[15] + 4 A[16] + 13 A[17] + 4 A[18] + 7 A[19] + 10 A[20] + 6 A[21] + 4 A[22] + A[23] + 3 A[24], 8 A[9] + 2 A[10] + 4 A[11] + 6 A[12] + 10 A[13] + 2 A[14] + 4 A[15] + 2 A[16] + 2 A[17] + 4 A[18] + 2 A[19] + 7 A[20] + 2 A[21] + 4 A[22] + 2 A[23]], [ 9 A[9] + 13 A[10] + A[11] + 7 A[12] + 3 A[13] + 8 A[14] + 4 A[16] + 8 A[17] + 2 A[18] + 5 A[19] + 3 A[20] + 7 A[21] + A[23] + 5 A[24] + 9 A[25], 9 A[9] + 6 A[10] + 12 A[11] + 6 A[12] + 2 A[14] + 6 A[15] + 4 A[16] + 6 A[17] + 2 A[18] + 2 A[19] + 2 A[21] + 3 A[22] + 3 A[23] + 6 A[25]], [3 A[8] + 7 A[9] + 14 A[10] + 6 A[11] + 7 A[12] + 5 A[13] + 4 A[14] + 4 A[15] + 4 A[16] + 10 A[17] + 3 A[18] + 2 A[19] + 5 A[20] + 6 A[21] + 4 A[22] + 3 A[23] + 2 A[24] + 11 A[25] + A[26], 2 A[8] + 8 A[9] + 6 A[10] + 2 A[12] + 2 A[14] + 3 A[16] + 6 A[17] + 3 A[19] + 2 A[21] + 2 A[23] + A[26]], [10 A[9] + 13 A[10] + 13 A[11] + 3 A[12] + 7 A[13] + 2 A[14] + 6 A[16] + 4 A[17] + 3 A[18] + A[19] + 5 A[20] + 4 A[21] + 2 A[23] + 2 A[24] + 8 A[25] + 2 A[27], 5 A[9] + 6 A[10] + 6 A[11] + 6 A[12] + 8 A[13] + 2 A[14] + 4 A[16] + 6 A[17] + A[18] + 2 A[19] + 5 A[20] + 2 A[21] + A[23] + 2 A[25] + 5 A[27]], [4 A[8] + 6 A[9] + 9 A[10] + 11 A[11] + 9 A[12] + A[13] + 8 A[14] + 3 A[16] + 8 A[17] + 2 A[18] + 5 A[19] + A[20] + 8 A[21] + A[23] + A[24] + 7 A[25] + 3 A[26] + 2 A[28], 2 A[8] + 4 A[9] + 12 A[10] + 8 A[12] + 6 A[13] + 8 A[14] + 3 A[16] + 12 A[17] + 5 A[19] + 2 A[20] + 4 A[21] + A[23] + A[26] + A[27] + 4 A[28] ], [2 A[8] + A[9] + 8 A[10] + 4 A[11] + 4 A[12] + 4 A[13] + 6 A[14] + A[16] + 6 A[17] + 2 A[18] + A[19] + A[20] + 3 A[21] + 2 A[25] + A[26] + A[27] + 2 A[28], 13 A[9] + 6 A[10] + 8 A[11] + 6 A[12] + 4 A[13] + 2 A[14] + 6 A[15] + 8 A[16] + 6 A[17] + 2 A[19] + 2 A[21] + 3 A[22] + 3 A[23] + 6 A[25] + 4 A[27] + 2 A[29]], [6 A[8] + 3 A[9] + A[10] + 5 A[11] + 7 A[12] + 3 A[13] + 8 A[14] + 3 A[16] + 5 A[17] + 2 A[18] + A[19] + 3 A[20] + 5 A[21] + 3 A[24] + 5 A[25] + 3 A[26] + 3 A[28] + 3 A[30], 10 A[8] + 10 A[9] + 13 A[10] + 5 A[11] + 11 A[12] + 9 A[13] + 2 A[14] + 7 A[16] + 5 A[17] + 2 A[18] + 2 A[19] + 4 A[20] + 11 A[21] + A[23] + 3 A[24] + 5 A[25] + 5 A[26] + 2 A[27] + 7 A[28] + 11 A[30]], [6 A[8] + 5 A[9] + 6 A[10] + 6 A[12] + 12 A[14] + 2 A[16] + 3 A[17] + A[19] + 8 A[21] + A[23] + 3 A[26] + 3 A[28] + 13 A[30], 4 A[8] + 5 A[9] + 6 A[10] + 10 A[11] + 2 A[12] + 4 A[13] + 9 A[14] + 3 A[15] + 4 A[16] + 4 A[17] + A[18] + 4 A[20] + 3 A[21] + A[22] + A[23] + A[24] + 6 A[25] + 2 A[26] + 2 A[27] + 4 A[28] + A[29] + 13 A[30] + 3 A[31]], [A[8], 3 A[9] + 4 A[14] + 2 A[28] + 2 A[21] + 2 A[26] + 2 A[12] + 4 A[11] + 2 A[25] + 2 A[24] + 12 A[30] + 2 A[16] + 4 A[17] + 6 A[10] + 2 A[13] + 4 A[8] + 2 A[18] + 2 A[20]], [5 A[9] + 2 A[14] + 2 A[21] + A[32] + A[26] + 4 A[12] + 2 A[24] + 10 A[30] + 2 A[16] + A[17] + 12 A[10] + A[23] + 2 A[8], 5 A[9] + 5 A[15] + 11 A[14] + 3 A[28] + 6 A[21] + 2 A[32] + 4 A[26] + 3 A[22] + 9 A[12] + 5 A[11] + 2 A[25] + 3 A[24] + 6 A[30] + 5 A[16] + 3 A[27] + 4 A[17] + 15 A[10] + A[19] + 5 A[13] + 2 A[29] + 8 A[8] + A[33] + A[20]], [8 A[9] + A[15] + 5 A[14] + 3 A[28] + A[32] + 2 A[26] + 4 A[31] + A[22] + 4 A[12] + A[11] + 8 A[25] + A[24] + 6 A[34] + 10 A[30] + 6 A[16] + 5 A[27] + 2 A[17] + A[10] + 2 A[23] + 4 A[13] + 4 A[29] + 2 A[8] + A[33] + A[18] + 2 A[20], 3 A[9] + 4 A[14] + 2 A[28] + A[21] + 4 A[26] + 5 A[24] + A[34] + 4 A[30] + 2 A[16] + A[17] + 6 A[10] + A[23] + 2 A[13] + A[33]], [8 A[9] + 6 A[14] + 8 A[28] + 4 A[21] + 6 A[11] + 2 A[25] + 3 A[24] + A[34] + 11 A[30] + 6 A[16] + 4 A[10] + 6 A[23] + 4 A[19] + 4 A[13] + A[33] + 3 A[20], 6 A[9] + 3 A[15] + 3 A[14] + 3 A[28] + 4 A[21] + 4 A[31] + 6 A[22] + 8 A[25] + A[24] + 5 A[30] + 6 A[16] + 7 A[27] + 8 A[10] + 6 A[13] + 4 A[29] + A[33] + 6 A[20]]], [1, 1, 1, 1, 13, 1, 5, 1, 1, 13, 13, 1, 1, 5, 5, 1, 1, 1, 13, 13, 13, 13, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15}, never show up! Theorem Number, 36, : Let C(n) be the constant term, in x, of n (3/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 19, 55, 595, 2611, 22141, 119449, 902035, 5420035, 38712169, 246360709, 1714206781, 11255897485, 77419522675, 517370395015 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 22, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [A[8], 9 A[9] + 8 A[10]], [12 A[8] + 13 A[10], 4 A[9] + 5 A[11]], [14 A[8] + 9 A[12], 14 A[9] + A[13] + 8 A[14]], [ 8 A[8] + 14 A[10] + 12 A[12] + 13 A[14], 8 A[9] + 14 A[11] + 4 A[13] + 5 A[15]], [12 A[8] + 9 A[10] + 12 A[12], 4 A[9] + 9 A[11] + 4 A[13]], [8 A[8] + 11 A[12], 11 A[13] + 8 A[14]], [4 A[10] + 12 A[12] + 3 A[14], 12 A[11] + 4 A[13] + 3 A[15]], [3 A[8], 11 A[9] + 8 A[10]], [4 A[8] + 15 A[10] + 8 A[12], 12 A[9] + 7 A[11] + 8 A[13]], [6 A[8] + 7 A[12], 6 A[9] + 15 A[13] + 8 A[14]], [ 8 A[8] + 14 A[10] + 4 A[12] + 11 A[14], 8 A[9] + 14 A[11] + 12 A[13] + 3 A[15]]], [1, 1, 1, 1, 3, 1, 7, 1, 3, 3, 13, 1, 3, 7, 9, 3, 3, 13, 3, 9, 13, 3]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 5, 6, 8, 10, 11, 12, 14, 15}, never show up! Theorem Number, 37, : Let C(n) be the constant term, in x, of n (3/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 18, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [4 A[8], 4 A[9]], [12 A[10], 12 A[11]], [2 A[8], 2 A[9]], [8 A[8] + 10 A[10], 8 A[9] + 10 A[11]], [8 A[8], 8 A[9]], [8 A[10], 8 A[11]], [12 A[8] + 9 A[10], 4 A[9] + 9 A[11]], 0, 0], [1, 1, 2, 1, 4, 2, 8, 1, 0, 4, 0, 2, 0, 8, 0, 0, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 38, : Let C(n) be the constant term, in x, of n (3/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 20, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [10 A[8], 2 A[9]], [8 A[8] + 14 A[10], 8 A[9] + 6 A[11]], [2 A[8], 2 A[9]], [8 A[8] + 10 A[10], 8 A[9] + 10 A[11]], [12 A[8], 12 A[9]], [4 A[10], 4 A[11]], [A[17], A[18]], [4 A[8] + 14 A[10] + 5 A[12] + 2 A[17], 14 A[9]], [2 A[8] + 3 A[12] + 3 A[14], 2 A[9] + 10 A[11] + 2 A[13] + 2 A[16]], [ 2 A[8] + 4 A[10] + 2 A[12] + 4 A[14] + 2 A[17] + 2 A[19], 2 A[9] + 14 A[11] + 2 A[13] + 2 A[18]], [2 A[10] + 5 A[14] + A[17] + A[19], 8 A[11] + A[18] + A[20]]], [1, 1, 2, 1, 10, 2, 12, 1, 6, 10, 4, 2, 12, 12, 8, 6, 6, 12, 6, 12]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {0, 3, 5, 7, 9, 11, 13, 14, 15}, never show up! Theorem Number, 39, : Let C(n) be the constant term, in x, of n (3/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 20, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], 0, 0, [2 A[8], 2 A[9]], [A[19], 8 A[10] + 10 A[11]], 0, 0, [A[20], 2 A[10] + A[11] + 3 A[14] + 4 A[15]], [5 A[9] + 4 A[13] + 3 A[16], 2 A[9] + 2 A[13] + 2 A[16]], [2 A[10] + A[14], 2 A[9] + A[11] + 2 A[13] + 2 A[14] + A[15] + 2 A[16] + A[18]], 0, 0], [1, 1, 2, 1, 0, 2, 0, 1, 8, 0, 0, 2, 0, 0, 0, 0, 8, 0, 0, 0]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15}, never show up! Theorem Number, 40, : Let C(n) be the constant term, in x, of n (3/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 22, 116, 934, 6332, 48124, 352424, 2669062, 20107628, 153277972, 1170192344, 8981891164, 69111416792, 533463087928, 4126851588176 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [2 A[4], 10 A[5]], [2 A[6], 10 A[7]], [11 A[6], 3 A[7]], [6 A[4], 14 A[5]], [6 A[6], 14 A[7]]], [1, 1, 2, 1, 6, 2, 4, 6, 6, 12]] For example, C(100000), mudolo , 16, equals , 0 The congruence classes mod, 16, in the following set , {0, 3, 5, 7, 8, 9, 10, 11, 13, 14, 15}, never show up! Theorem Number, 41, : Let C(n) be the constant term, in x, of n (3/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [3 A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [15 A[8], 6 A[9] + 8 A[14] + A[16]], [2 A[10] + 12 A[12] + A[17], 8 A[11] + 8 A[13] + 3 A[18] + 4 A[20]], [ 2 A[8] + 4 A[12] + 5 A[19], 6 A[9] + 14 A[11] + 8 A[13] + 4 A[16] + 2 A[18] + A[20]], [ 2 A[10] + 4 A[12] + 12 A[14] + 4 A[17] + 4 A[19] + A[21], 8 A[11] + 4 A[15] + 6 A[18] + A[22]], [ 6 A[10] + 4 A[12] + 5 A[17] + 4 A[19], 6 A[11] + 4 A[13] + 5 A[18] + 4 A[20]], [12 A[12] + 3 A[19], 12 A[13] + 3 A[20]], [4 A[14] + 3 A[21], 4 A[15] + 3 A[22]], [A[8], 6 A[9] + 2 A[16] + 3 A[23]], [6 A[10] + 12 A[12] + 2 A[17] + 4 A[19] + 3 A[24], 10 A[9] + 8 A[13] + 4 A[16] + 6 A[18] + 5 A[20] + 4 A[23] + A[25] + 3 A[27] ], [2 A[8] + 4 A[10] + 12 A[12] + 8 A[14] + 2 A[17] + 7 A[19] + 3 A[21] + 4 A[24] + 8 A[26] + A[28], 2 A[8] + 3 A[9] + 4 A[10] + 8 A[11] + 8 A[12] + 4 A[13] + 4 A[14] + 2 A[16] + 2 A[17] + A[18] + 5 A[19] + A[20] + 2 A[21] + 3 A[23] + 2 A[24] + A[25] + 3 A[26] + 2 A[27] + 2 A[28]], [ 6 A[10] + 4 A[12] + 12 A[14] + 6 A[17] + 4 A[19] + 4 A[21] + 3 A[28], 4 A[9] + 6 A[11] + 4 A[13] + 12 A[15] + 2 A[16] + 6 A[18] + 2 A[20] + 4 A[22] + 2 A[23] + 2 A[27] + 3 A[29]], [4 A[8] + 12 A[10] + 12 A[12] + 2 A[17] + 5 A[19] + 2 A[24] + 8 A[26] + 4 A[30], 2 A[8] + 9 A[9] + 2 A[10] + 2 A[11] + 4 A[12] + 8 A[13] + 4 A[14] + 4 A[16] + A[17] + A[18] + A[19] + 2 A[20] + 2 A[21] + 3 A[22] + 3 A[23] + A[25] + 3 A[26] + 3 A[27] + 2 A[28] + A[29] + A[30]], [ 4 A[9] + 8 A[10] + 4 A[14] + A[16] + 2 A[17] + A[23] + A[28] + 4 A[30], 8 A[9] + 12 A[11] + 8 A[13] + 4 A[15] + 4 A[16] + 5 A[18] + 4 A[20] + 4 A[23] + 2 A[25] + 4 A[27] + A[29] + 7 A[31]], [ 6 A[32] + 6 A[26] + 4 A[30] + 4 A[17] + 8 A[10] + 4 A[19] + 7 A[8], 6 A[9] + 2 A[16] + A[23]], [ 2 A[32] + 2 A[26] + 4 A[12] + A[24] + 4 A[30] + 2 A[17] + 10 A[10] + 4 A[8] , 2 A[9] + 4 A[31] + 10 A[11] + A[25] + 2 A[16] + 2 A[27] + 4 A[13] + 2 A[33] + 2 A[18]], [A[32] + 2 A[26] + 8 A[12] + 2 A[19] + 2 A[8], 6 A[9] + 4 A[16] + 4 A[27] + 12 A[13] + 2 A[33] + 3 A[20]], [10 A[9] + 8 A[14] + 6 A[28] + 2 A[21] + 6 A[32] + 2 A[26] + 4 A[12] + 2 A[11] + 2 A[25] + A[24] + A[34] + 7 A[30] + 7 A[16] + 10 A[27] + 2 A[17] + 6 A[10] + A[23] + 12 A[13] + 4 A[8] + 8 A[33] + 2 A[20], 4 A[35] + 4 A[15] + 5 A[29]]], [1, 1, 3, 1, 15, 3, 1, 1, 11, 15, 9, 3, 1, 1, 7, 3, 11, 5, 15, 13, 9, 7, 9, 1, 15, 1, 3, 7, 9, 3, 13, 11, 1, 5, 11]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 6, 8, 10, 12, 14}, never show up! Theorem Number, 42, : Let C(n) be the constant term, in x, of n (3/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 16 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 35, states . Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12], A[13]], [A[14], A[15]], [A[8], A[16]], [A[17], A[18]], [A[19], A[20]], [A[21], A[22]], [3 A[8], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A[32], A[33]], [A[34], A[35]], [5 A[8], 8 A[8] + 15 A[9] + 7 A[10] + 5 A[11] + 8 A[12] + 12 A[13] + 4 A[14] + 6 A[16] + A[17] + 3 A[18]], [8 A[9] + 15 A[10] + 15 A[11] + 4 A[12] + 12 A[13] + 12 A[14] + 10 A[17] + 9 A[18] + 8 A[19] + 8 A[20], 2 A[8] + 8 A[9] + 6 A[10] + 6 A[11] + A[12] + 13 A[13] + 6 A[14] + 12 A[16] + 12 A[17] + 3 A[18] + 3 A[19] + 13 A[20]], [10 A[8] + 2 A[10] + 12 A[12] + 4 A[13] + 2 A[14] + 6 A[17] + 3 A[19] + 4 A[20] + 2 A[21], 4 A[8] + 6 A[9] + 4 A[10] + 2 A[12] + 12 A[13] + 8 A[14] + 8 A[16] + 8 A[17] + 6 A[19] + 7 A[20] + 4 A[21]], [14 A[8] + 2 A[9] + 2 A[10] + 15 A[12] + 11 A[13] + 4 A[14] + 8 A[15] + 2 A[16] + 6 A[17] + 13 A[19] + 7 A[20] + A[21] + 8 A[22], 14 A[8] + 6 A[9] + 2 A[10] + 12 A[11] + 15 A[12] + 13 A[13] + 7 A[14] + 11 A[15] + 12 A[16] + 6 A[17] + 6 A[18] + 13 A[19] + 11 A[20] + 15 A[21] + 6 A[22]], [6 A[8] + 9 A[9] + 4 A[11] + 11 A[12] + 7 A[13] + A[14] + 5 A[15] + 6 A[16] + 7 A[17] + 4 A[18] + 9 A[19] + 5 A[20] + 13 A[21] + A[22] + A[23], 2 A[8] + 11 A[9] + 8 A[10] + 2 A[11] + 9 A[12] + 11 A[13] + 15 A[14] + 9 A[15] + 9 A[16] + 12 A[17] + A[18] + 11 A[19] + 9 A[20] + 11 A[21] + 5 A[22] + 2 A[23]], [ 4 A[8] + 12 A[9] + 4 A[12] + 6 A[13] + 8 A[14] + 4 A[15] + 3 A[16] + 6 A[17] + 7 A[19] + 6 A[20] + 4 A[21] + 4 A[22] + 3 A[23] + 2 A[24], 4 A[8] + 8 A[9] + 8 A[10] + 2 A[12] + 8 A[13] + 8 A[14] + 4 A[15] + 4 A[16] + 4 A[17] + 6 A[19] + 7 A[20] + 4 A[21] + 4 A[22] + 4 A[23]], [3 A[9] + 12 A[10] + 2 A[11] + 6 A[13] + 8 A[14] + 4 A[15] + 2 A[16] + 8 A[17] + 6 A[20] + 3 A[21] + 4 A[22] + A[23] + 2 A[25], 6 A[9] + 2 A[10] + 6 A[11] + 2 A[13] + 4 A[14] + 4 A[15] + 10 A[17] + 5 A[18] + 2 A[20] + 4 A[21] + 3 A[22] + 2 A[23] + 4 A[24] + 3 A[25]], [5 A[8] + 4 A[9] + 2 A[10] + 10 A[12] + 2 A[14] + A[16] + A[17] + 3 A[19] + 2 A[21] + A[23] + 3 A[24] + 7 A[26], 2 A[8] + 3 A[9] + 3 A[10] + A[11] + 2 A[12] + 2 A[13] + 4 A[14] + A[16] + 2 A[17] + A[19] + 2 A[20] + 4 A[21] + A[23] + A[24] + A[25] + A[26]], [6 A[8] + 9 A[9] + 9 A[10] + 5 A[11] + 6 A[12] + 12 A[13] + 10 A[14] + 4 A[15] + 6 A[16] + 14 A[17] + 3 A[19] + 4 A[20] + 6 A[21] + 4 A[22] + 3 A[23] + 4 A[24] + 5 A[25] + 3 A[26] + 8 A[27], 2 A[8] + 9 A[9] + 12 A[10] + 8 A[11] + 3 A[12] + 5 A[13] + 8 A[14] + 4 A[16] + 7 A[17] + 5 A[18] + 3 A[19] + 2 A[20] + 2 A[21] + 3 A[23] + A[24] + 2 A[25] + 2 A[26] + 3 A[27]], [8 A[8] + 15 A[9] + 8 A[11] + 13 A[12] + 13 A[13] + 8 A[14] + 4 A[15] + 8 A[16] + 13 A[17] + 5 A[18] + 6 A[19] + 5 A[20] + 6 A[21] + 4 A[22] + 7 A[23] + 3 A[24] + 5 A[25] + 6 A[26] + 8 A[27] + 12 A[28], 4 A[8] + 9 A[9] + 10 A[10] + 2 A[11] + 4 A[12] + 14 A[14] + 4 A[15] + 6 A[16] + 8 A[17] + 2 A[19] + 5 A[20] + 4 A[21] + 4 A[22] + 5 A[23] + 2 A[24] + 2 A[25] + 2 A[26] + 8 A[27] + 6 A[28]], [3 A[9] + 2 A[10] + 6 A[11] + 6 A[13] + 3 A[14] + A[15] + 2 A[16] + 4 A[17] + 5 A[18] + A[20] + A[21] + A[23] + A[25] + 3 A[27] + A[28] + A[29], 4 A[9] + 4 A[10] + 10 A[11] + 8 A[13] + 2 A[14] + 8 A[15] + 2 A[16] + 2 A[17] + 7 A[18] + 4 A[20] + A[21] + 2 A[22] + 2 A[23] + A[25] + 4 A[27] + A[28] + 3 A[29]], [11 A[9] + 12 A[10] + 6 A[11] + 6 A[12] + 10 A[13] + 9 A[14] + A[15] + 5 A[16] + 3 A[17] + 5 A[18] + A[19] + 3 A[20] + 2 A[21] + 4 A[23] + 3 A[24] + A[25] + 2 A[26] + 5 A[27] + 5 A[28] + A[29] + 8 A[30], 5 A[9] + 4 A[10] + 2 A[11] + 6 A[13] + 2 A[14] + 2 A[15] + 3 A[16] + 2 A[18] + A[20] + A[21] + 4 A[23] + 2 A[27] + A[28] + 2 A[29] + 2 A[30]], [2 A[8] + 10 A[9] + 7 A[10] + A[11] + 2 A[12] + 8 A[13] + 3 A[14] + A[15] + 5 A[16] + 2 A[17] + 4 A[18] + A[19] + 2 A[20] + A[21] + 5 A[23] + A[24] + 2 A[25] + A[26] + 4 A[27] + A[28] + A[29] + 4 A[30] + 7 A[31], 2 A[8] + 6 A[9] + 7 A[10] + 9 A[11] + 2 A[12] + 6 A[13] + 2 A[14] + 4 A[15] + 4 A[16] + A[17] + 3 A[18] + A[19] + 3 A[20] + A[21] + 4 A[23] + A[24] + 2 A[25] + A[26] + 3 A[27] + A[28] + A[29] + 3 A[30] + 4 A[31]], [A[8], 4 A[9] + 10 A[14] + 5 A[28] + 3 A[21] + 2 A[32] + 2 A[26] + 2 A[31] + 7 A[12] + 7 A[11] + 2 A[25] + 6 A[30] + 2 A[16] + 2 A[27] + 3 A[17] + 7 A[10] + A[23] + A[19] + 3 A[13] + 2 A[8] + A[18] + A[20]], [4 A[9] + 4 A[14] + 2 A[28] + 2 A[21] + A[32] + A[26] + 2 A[12] + A[24] + 10 A[30] + 2 A[16] + 2 A[27] + 4 A[17] + 12 A[10] + 2 A[23] + 4 A[13] + 2 A[8] + 2 A[20], 6 A[9] + 2 A[14] + A[28] + A[21] + 8 A[31] + 12 A[11] + 2 A[25] + 7 A[30] + 2 A[16] + A[27] + A[17] + 6 A[10] + 2 A[23] + 2 A[13] + 4 A[33] + 3 A[18] + A[20]], [8 A[9] + 10 A[14] + 5 A[28] + 2 A[21] + 5 A[32] + 5 A[26] + 2 A[31] + 4 A[12] + 9 A[11] + 4 A[25] + 2 A[24] + 3 A[34] + 9 A[30] + 3 A[16] + 6 A[27] + 7 A[10] + 3 A[23] + A[19] + 4 A[13] + A[18] + 2 A[20], 5 A[9] + 12 A[14] + 10 A[28] + 2 A[21] + 4 A[26] + A[31] + 5 A[11] + 7 A[25] + 5 A[24] + 4 A[34] + 10 A[30] + 2 A[16] + 3 A[27] + 7 A[10] + 3 A[23] + 6 A[13] + A[18] + 2 A[20]], [2 A[9] + A[35] + 4 A[15] + 6 A[14] + 3 A[28] + 4 A[31] + A[22] + 8 A[11] + 6 A[25] + 4 A[24] + 2 A[34] + 2 A[30] + A[16] + A[27] + 2 A[10] + 5 A[23] + 2 A[13] + 2 A[29] + 2 A[18] + 5 A[20], A[9] + 4 A[15] + 3 A[31] + A[22] + 6 A[11] + 5 A[25] + A[16] + A[27] + 2 A[13] + 6 A[29] + 2 A[18] + A[20]]], [1, 1, 3, 1, 5, 3, 7, 1, 1, 5, 5, 3, 3, 7, 7, 1, 1, 1, 5, 5, 5, 5, 3, 3, 3, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 16, equals , 1 The congruence classes mod, 16, in the following set , {0, 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15}, never show up! ------------------------------------------ This ends this fascinating book that took, 5.150, to generate. ----------------------------------------- ----------------------------------------------------- On computing the Mod, 3, of Many Interesting sequences by Shalosh B. Ekhad Theorem Number, 1, : Let C(n) be the constant term, in x, of n (1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 2, : Let C(n) be the constant term, in x, of n (1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 3, : Let C(n) be the constant term, in x, of n (2 + x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 4, : Let C(n) be the constant term, in x, of n (2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 5, : Let C(n) be the constant term, in x, of n (3 + x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 6, : Let C(n) be the constant term, in x, of n (3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 7, : Let C(n) be the constant term, in x, of n (1/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 8, : Let C(n) be the constant term, in x, of n (1/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 9, : Let C(n) be the constant term, in x, of n (1/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 2 A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 2 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 10, : Let C(n) be the constant term, in x, of n (1/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 11, : Let C(n) be the constant term, in x, of n (1/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 12, : Let C(n) be the constant term, in x, of n (1/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 13, : Let C(n) be the constant term, in x, of n (1/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 2 A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 2 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 14, : Let C(n) be the constant term, in x, of n (1/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 15, : Let C(n) be the constant term, in x, of n (1/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 16, : Let C(n) be the constant term, in x, of n (1/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 2 A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 17, : Let C(n) be the constant term, in x, of n (1/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 18, : Let C(n) be the constant term, in x, of n (1/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 19, : Let C(n) be the constant term, in x, of n (2/x + x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 20, : Let C(n) be the constant term, in x, of n (2/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 21, : Let C(n) be the constant term, in x, of n (2/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 22, : Let C(n) be the constant term, in x, of n (2/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 2 A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 2 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 23, : Let C(n) be the constant term, in x, of n (2/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 24, : Let C(n) be the constant term, in x, of n (2/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 25, : Let C(n) be the constant term, in x, of n (2/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 2 A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 2 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 26, : Let C(n) be the constant term, in x, of n (2/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 27, : Let C(n) be the constant term, in x, of n (2/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 28, : Let C(n) be the constant term, in x, of n (2/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 29, : Let C(n) be the constant term, in x, of n (2/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 2 A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 30, : Let C(n) be the constant term, in x, of n (2/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 31, : Let C(n) be the constant term, in x, of n (3/x + x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 32, : Let C(n) be the constant term, in x, of n (3/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 33, : Let C(n) be the constant term, in x, of n (3/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 34, : Let C(n) be the constant term, in x, of n (3/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 35, : Let C(n) be the constant term, in x, of n (3/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 36, : Let C(n) be the constant term, in x, of n (3/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 19, 55, 595, 2611, 22141, 119449, 902035, 5420035, 38712169, 246360709, 1714206781, 11255897485, 77419522675, 517370395015 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 37, : Let C(n) be the constant term, in x, of n (3/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 38, : Let C(n) be the constant term, in x, of n (3/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 39, : Let C(n) be the constant term, in x, of n (3/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 40, : Let C(n) be the constant term, in x, of n (3/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 22, 116, 934, 6332, 48124, 352424, 2669062, 20107628, 153277972, 1170192344, 8981891164, 69111416792, 533463087928, 4126851588176 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 3, equals , 1 The congruence classes mod, 3, in the following set , {0, 2}, never show up! Theorem Number, 41, : Let C(n) be the constant term, in x, of n (3/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! Theorem Number, 42, : Let C(n) be the constant term, in x, of n (3/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 3 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 0]], [1]] For example, C(100000), mudolo , 3, equals , 0 The congruence classes mod, 3, in the following set , {2}, never show up! ------------------------------------------ This ends this fascinating book that took, 0.246, to generate. ----------------------------------------- ----------------------------------------------------- On computing the Mod, 9, of Many Interesting sequences by Shalosh B. Ekhad Theorem Number, 1, : Let C(n) be the constant term, in x, of n (1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [A[2], 3 A[2] + 7 A[3], A[7]], %1, [A[2], 3 A[2] + 7 A[3], 6 A[3] + 4 A[4]], %1, %1], [1, 1, 1, 1, 1, 1, 1]] %1 := [A[2], 6 A[2] + 4 A[3], 3 A[3] + 7 A[4]] For example, C(100000), mudolo , 9, equals , 1 The congruence classes mod, 9, in the following set , {0, 2, 3, 4, 5, 6, 7, 8}, never show up! Theorem Number, 2, : Let C(n) be the constant term, in x, of n (1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[2], A[2]], [A[2], A[2], A[2]], [ (64 c[2] + 9) A[2] + (8 c[2] + 1) A[3], (64 c[2] + 9) A[2] + (8 c[2] + 1) A[3], (64 c[2] + 9) A[2] + (8 c[2] + 1) A[3]], [(640 c[2] + 576 c[3] + 81) A[2] + (72 c[2] + 64 c[3] + 9) A[3] + (8 c[2] + 8 c[3] + 1) A[4], (640 c[2] + 576 c[3] + 81) A[2] + (72 c[2] + 64 c[3] + 9) A[3] + (8 c[2] + 8 c[3] + 1) A[4], (640 c[2] + 576 c[3] + 81) A[2] + (72 c[2] + 64 c[3] + 9) A[3] + (8 c[2] + 8 c[3] + 1) A[4]]], [1, 1, 1, 1]] For example, C(100000), mudolo , 9, equals , 1 The congruence classes mod, 9, in the following set , {0, 2, 3, 4, 5, 6, 7, 8}, never show up! Theorem Number, 3, : Let C(n) be the constant term, in x, of n (2 + x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [2 A[2], 8 A[3], A[7]], [4 A[2], 6 A[2] + 4 A[3], 3 A[3] + 4 A[4]], [8 A[2], 5 A[3], 2 A[4]], [A[2], 6 A[2] + A[3], 3 A[3] + A[4]], [2 A[2], 3 A[2] + 2 A[3], 6 A[3] + 2 A[4]]], [1, 1, 2, 4, 8, 1, 2]] For example, C(100000), mudolo , 9, equals , 7 The congruence classes mod, 9, in the following set , {0, 3, 5, 6, 7}, never show up! Theorem Number, 4, : Let C(n) be the constant term, in x, of n (2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3], A[4]], [A[2], 8 A[2], A[2]], [ (28 c[2] + 9) A[2] + (4 c[2] + 1) A[3], (28 c[2] + 63) A[2] + (4 c[2] + 8) A[3], (28 c[2] + 9) A[2] + (4 c[2] + 1) A[3]], [(136 c[2] + 216 c[3] + 72) A[2] + (18 c[2] + 28 c[3] + 9) A[3] + (2 c[2] + 4 c[3] + 1) A[4], (136 c[2] + 216 c[3] + 486) A[2] + (18 c[2] + 28 c[3] + 63) A[3] + (2 c[2] + 4 c[3] + 8) A[4], (136 c[2] + 216 c[3] + 72) A[2] + (18 c[2] + 28 c[3] + 9) A[3] + (2 c[2] + 4 c[3] + 1) A[4]]], [1, 1, 2, 4]] For example, C(100000), mudolo , 9, equals , 7 The congruence classes mod, 9, in the following set , {0, 3, 5, 6, 7, 8}, never show up! Theorem Number, 5, : Let C(n) be the constant term, in x, of n (3 + x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3], 0], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0]], [1, 1, 3]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 6, 7, 8}, never show up! Theorem Number, 6, : Let C(n) be the constant term, in x, of n (3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3], 0], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0]], [1, 1, 3]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 6, 7, 8}, never show up! Theorem Number, 7, : Let C(n) be the constant term, in x, of n (1/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [0, 6 A[2], 3 A[3]], [4 A[2], A[5], A[8]], [0, 6 A[2], 3 A[3]], [7 A[2], 3 A[3] + A[5], A[6]], [0, 6 A[2], 3 A[3]], [4 A[2], A[5], 2 A[6] + 2 A[8]]], [1, 1, 0, 4, 0, 7, 0, 4]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 3, 5, 6, 8}, never show up! Theorem Number, 8, : Let C(n) be the constant term, in x, of n (1/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], 0, A[3]], [A[2], 0, 0], [(3 c[3] + 6) A[2] + c[3] A[3], 0, 0]], [1, 1, 6]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 3, 4, 5, 7, 8}, never show up! Theorem Number, 9, : Let C(n) be the constant term, in x, of n (1/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 11, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [A[2], A[7], A[8]], [5 A[2], A[9], A[10]], [4 A[2], 6 A[2] + 6 A[3] + A[5], A[11]], [5 A[2], 2 A[5], 6 A[4] + 2 A[6]], [4 A[2], 3 A[3] + A[7], 3 A[2] + 6 A[3] + 2 A[5] + A[6] + A[7]], [2 A[2], 6 A[3] + 2 A[5], 3 A[4] + 2 A[6]], [ 2 A[2] + 6 A[3] + 3 A[4] + 2 A[5] + A[6] + A[7] + 2 A[8], 6 A[3] + 2 A[7], 6 A[4] + 2 A[8]], [A[2] + 6 A[3] + A[5] + 2 A[6] + 2 A[7] + A[8], 3 A[3] + A[5] + 2 A[6] + A[8], A[6]], [ 2 A[2] + 6 A[3] + 3 A[4] + 2 A[5] + A[6] + A[7] + 2 A[8], 6 A[3] + 3 A[4] + 2 A[5] + A[6] + 2 A[8], 2 A[6]]], [1, 1, 1, 5, 4, 5, 4, 2, 8, 4, 8]] For example, C(100000), mudolo , 9, equals , 2 The congruence classes mod, 9, in the following set , {0, 3, 6, 7}, never show up! Theorem Number, 10, : Let C(n) be the constant term, in x, of n (1/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [A[2], A[7], A[8]], [7 A[2], A[9], A[10]], [A[2], 6 A[2] + 3 A[3] + A[5], 3 A[2] + A[3] + 7 A[4] + 2 A[5]], [A[2], A[5], 3 A[2] + 7 A[3] + A[4] + 2 A[5]], [A[2], 6 A[2] + 3 A[3] + A[5], 6 A[2] + 2 A[3] + 5 A[4] + A[5] + 2 A[6]], [A[2], 6 A[2] + 2 A[5] + 2 A[7], 6 A[2] + 4 A[3] + A[4] + A[5] + A[7]], [ A[2] + 6 A[3] + A[5] + A[7] + A[9], A[5] + A[7] + 2 A[9], 6 A[3] + 3 A[4] + A[5] + A[6] + 2 A[9]], [ A[2] + 6 A[3] + A[5] + A[7] + A[9], 2 A[5] + 2 A[9], 8 A[3] + 5 A[4] + A[5] + 2 A[6] + A[7] + 2 A[9]]], [1, 1, 1, 7, 1, 1, 1, 1, 7, 7]] For example, C(100000), mudolo , 9, equals , 1 The congruence classes mod, 9, in the following set , {0, 2, 3, 4, 5, 6, 8}, never show up! Theorem Number, 11, : Let C(n) be the constant term, in x, of n (1/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [2 A[2], 8 A[3], A[7]], [4 A[2], 6 A[2] + 4 A[3], 3 A[3] + 4 A[4]], [8 A[2], 5 A[3], 2 A[4]], [A[2], 6 A[2] + A[3], 3 A[3] + A[4]], [2 A[2], 3 A[2] + 2 A[3], 6 A[3] + 2 A[4]]], [1, 1, 2, 4, 8, 1, 2]] For example, C(100000), mudolo , 9, equals , 7 The congruence classes mod, 9, in the following set , {0, 3, 5, 6, 7}, never show up! Theorem Number, 12, : Let C(n) be the constant term, in x, of n (1/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[3], A[4]], [2 A[2], 8 A[3], A[7]], [6 A[2], 0, 3 A[4]], [2 A[2], 8 A[3], 5 A[4]], [6 A[2], 0, 3 A[4]], [3 A[2], 0, 6 A[4]]], [1, 1, 2, 6, 2, 6, 3]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {4, 5, 7, 8}, never show up! Theorem Number, 13, : Let C(n) be the constant term, in x, of n (1/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 11, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [2 A[2], A[7], A[8]], [8 A[2], A[9], A[10]], [5 A[2], 6 A[2] + 2 A[5], A[11]], [5 A[2], 2 A[5], 6 A[4] + 2 A[6]], [A[2], 3 A[2] + 3 A[3] + A[5], 6 A[2] + 6 A[3] + 6 A[4] + 2 A[5] + A[6] + 2 A[7]], [ A[2] + 6 A[3] + 3 A[4] + 2 A[5] + A[6] + 2 A[7] + A[8], 3 A[3] + 3 A[4] + A[5] + A[6] + A[8], A[6]], [ A[2] + 3 A[3] + 6 A[4] + A[5] + 2 A[6] + A[7] + 2 A[8], 3 A[3] + 2 A[7], 3 A[4] + 2 A[8]], [A[2], 3 A[3] + A[5], 6 A[4] + A[6]], [A[2], 3 A[3] + A[5], 6 A[4] + A[6]]], [1, 1, 2, 8, 5, 5, 1, 4, 7, 1, 1]] For example, C(100000), mudolo , 9, equals , 5 The congruence classes mod, 9, in the following set , {0, 3, 6}, never show up! Theorem Number, 14, : Let C(n) be the constant term, in x, of n (1/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [2 A[2], A[7], A[8]], [A[2], A[5], A[6]], [8 A[2], 6 A[2] + 3 A[3] + 2 A[5], 3 A[2] + 7 A[3] + 2 A[4] + 2 A[5]], [A[2], A[5], 6 A[2] + 2 A[3] + A[4] + A[5]], [ 7 A[2], 3 A[3] + 2 A[7], 6 A[2] + 6 A[3] + 3 A[4] + 2 A[5] + A[6] + 2 A[7]] , [2 A[2], 6 A[3] + 3 A[4] + 2 A[5] + A[6] + A[8], 4 A[3] + 5 A[4] + A[5] + A[6] + 2 A[7] + A[8]], [ 2 A[2] + 3 A[3] + 6 A[4] + A[5] + 2 A[6] + A[7] + 2 A[8], 3 A[3] + 3 A[4] + A[6] + A[7] + A[8], 4 A[3] + 8 A[4] + A[5] + 2 A[6] + 2 A[7] + 2 A[8]], [A[2], 3 A[3] + 6 A[4] + A[5] + 2 A[6] + 2 A[8], 5 A[3] + 7 A[4] + 2 A[5] + 2 A[6] + A[7] + 2 A[8]]], [1, 1, 2, 1, 8, 1, 7, 2, 8, 1]] For example, C(100000), mudolo , 9, equals , 7 The congruence classes mod, 9, in the following set , {0, 3, 4, 5, 6}, never show up! Theorem Number, 15, : Let C(n) be the constant term, in x, of n (1/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3], 0], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0]], [1, 1, 3]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 6, 7, 8}, never show up! Theorem Number, 16, : Let C(n) be the constant term, in x, of n (1/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [3 A[2], 6 A[2] + 3 A[3], A[7]], [2 A[2], 3 A[2] + 5 A[3], 6 A[3] + 8 A[4]], [0, 6 A[2], 3 A[3]], [2 A[2], 3 A[2] + 5 A[3], 6 A[3] + 8 A[4]], [6 A[2], 6 A[3], 6 A[4]]], [1, 1, 3, 2, 0, 2, 6]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {4, 5, 7, 8}, never show up! Theorem Number, 17, : Let C(n) be the constant term, in x, of n (1/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 9, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [3 A[2], 6 A[2] + 3 A[3], A[7]], [4 A[2], A[5], A[9]], [0, 6 A[2], 3 A[3]], [7 A[2], 3 A[3] + A[5], A[6]], [3 A[2], 6 A[4] + A[7], 6 A[3] + A[7]], [0, 6 A[2], 6 A[4] + A[7]], [4 A[2], 3 A[3] + 3 A[4] + A[5] + 2 A[7], 6 A[3] + 3 A[4] + A[6] + A[7]]], [1, 1, 3, 4, 0, 7, 3, 0, 4]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 5, 6, 8}, never show up! Theorem Number, 18, : Let C(n) be the constant term, in x, of n (1/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3], A[4]], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0], [(6 c[3] + 3 c[4] + 6) A[2] + c[3] A[3] + c[4] A[4], 0, 0]], [1, 1, 3, 6]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 7, 8}, never show up! Theorem Number, 19, : Let C(n) be the constant term, in x, of n (2/x + x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 8, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [0, 6 A[2], 3 A[3]], [4 A[2], A[5], A[8]], [0, 6 A[2], 3 A[3]], [7 A[2], 3 A[3] + A[5], A[6]], [0, 6 A[2], 3 A[3]], [4 A[2], A[5], 2 A[6] + 2 A[8]]], [1, 1, 0, 4, 0, 7, 0, 4]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 3, 5, 6, 8}, never show up! Theorem Number, 20, : Let C(n) be the constant term, in x, of n (2/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], 0, A[3]], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0]], [1, 1, 3]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 6, 7, 8}, never show up! Theorem Number, 21, : Let C(n) be the constant term, in x, of n (2/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [A[2], 3 A[2] + 7 A[3], A[7]], %1, [A[2], 3 A[2] + 7 A[3], 6 A[3] + 4 A[4]], %1, %1], [1, 1, 1, 1, 1, 1, 1]] %1 := [A[2], 6 A[2] + 4 A[3], 3 A[3] + 7 A[4]] For example, C(100000), mudolo , 9, equals , 1 The congruence classes mod, 9, in the following set , {0, 2, 3, 4, 5, 6, 7, 8}, never show up! Theorem Number, 22, : Let C(n) be the constant term, in x, of n (2/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 11, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [A[2], A[7], A[8]], [5 A[2], A[9], A[10]], [4 A[2], 6 A[2] + 6 A[3] + A[5], A[11]], [5 A[2], 2 A[5], 6 A[4] + 2 A[6]], [4 A[2], 3 A[3] + A[7], 3 A[2] + 6 A[3] + 2 A[5] + A[6] + A[7]], [2 A[2], 6 A[3] + 2 A[5], 3 A[4] + 2 A[6]], [ 2 A[2] + 6 A[3] + 3 A[4] + 2 A[5] + A[6] + A[7] + 2 A[8], 6 A[3] + 2 A[7], 6 A[4] + 2 A[8]], [A[2] + 6 A[3] + A[5] + 2 A[6] + 2 A[7] + A[8], 3 A[3] + A[5] + 2 A[6] + A[8], A[6]], [ 2 A[2] + 6 A[3] + 3 A[4] + 2 A[5] + A[6] + A[7] + 2 A[8], 6 A[3] + 3 A[4] + 2 A[5] + A[6] + 2 A[8], 2 A[6]]], [1, 1, 1, 5, 4, 5, 4, 2, 8, 4, 8]] For example, C(100000), mudolo , 9, equals , 2 The congruence classes mod, 9, in the following set , {0, 3, 6, 7}, never show up! Theorem Number, 23, : Let C(n) be the constant term, in x, of n (2/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [A[2], 3 A[2] + 7 A[3], A[7]], [0, 3 A[3], 6 A[4]], [7 A[2], 3 A[2] + 4 A[3], 6 A[3] + A[4]], [6 A[2], 0, 3 A[4]], [6 A[2], 0, 3 A[4]]], [1, 1, 1, 0, 7, 6, 6]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 3, 4, 5, 8}, never show up! Theorem Number, 24, : Let C(n) be the constant term, in x, of n (2/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [A[2], A[7], A[8]], [4 A[2], A[9], A[10]], [A[2], 6 A[2] + 3 A[3] + A[5], 3 A[2] + 7 A[3] + 4 A[4] + 2 A[5]], [A[2], A[5], A[6]], [A[2], A[3] + 4 A[4] + 2 A[6], 2 A[3] + 8 A[4] + 2 A[6] + A[7]], [A[2], 7 A[3] + 4 A[4] + A[5] + 2 A[6] + 2 A[7], 4 A[3] + 7 A[4] + 2 A[6] + 2 A[7] + A[8]], [A[2] + 2 A[3] + 2 A[4] + A[5] + A[6], 3 A[3] + A[7], 3 A[4] + A[8]], [A[2] + 2 A[3] + 2 A[4] + A[5] + A[6], 3 A[3] + A[5], 3 A[4] + A[6]]], [1, 1, 1, 4, 1, 1, 1, 1, 4, 4]] For example, C(100000), mudolo , 9, equals , 1 The congruence classes mod, 9, in the following set , {0, 2, 3, 5, 6, 7, 8}, never show up! Theorem Number, 25, : Let C(n) be the constant term, in x, of n (2/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 11, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [2 A[2], A[7], A[8]], [8 A[2], A[9], A[10]], [5 A[2], 6 A[2] + 2 A[5], A[11]], [5 A[2], 2 A[5], 6 A[4] + 2 A[6]], [A[2], 3 A[2] + 3 A[3] + A[5], 6 A[2] + 6 A[3] + 6 A[4] + 2 A[5] + A[6] + 2 A[7]], [ A[2] + 6 A[3] + 3 A[4] + 2 A[5] + A[6] + 2 A[7] + A[8], 3 A[3] + 3 A[4] + A[5] + A[6] + A[8], A[6]], [ A[2] + 3 A[3] + 6 A[4] + A[5] + 2 A[6] + A[7] + 2 A[8], 3 A[3] + 2 A[7], 3 A[4] + 2 A[8]], [A[2], 3 A[3] + A[5], 6 A[4] + A[6]], [A[2], 3 A[3] + A[5], 6 A[4] + A[6]]], [1, 1, 2, 8, 5, 5, 1, 4, 7, 1, 1]] For example, C(100000), mudolo , 9, equals , 5 The congruence classes mod, 9, in the following set , {0, 3, 6}, never show up! Theorem Number, 26, : Let C(n) be the constant term, in x, of n (2/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [2 A[2], A[7], A[8]], [7 A[2], A[9], A[10]], [8 A[2], 6 A[2] + 3 A[3] + 2 A[5], 3 A[2] + 4 A[3] + 5 A[4] + 2 A[5]], [A[2], 6 A[2] + 5 A[3] + 4 A[4] + 2 A[5] + 2 A[6], A[6]], [7 A[2], 5 A[3] + 4 A[4] + 2 A[5] + 2 A[6], 6 A[2] + 2 A[3] + A[4] + A[5]], [2 A[2], 6 A[3] + 3 A[4] + 2 A[5] + A[6] + A[8], 2 A[6]], [2 A[2], 4 A[3] + 2 A[4] + A[5] + 2 A[8], A[8]], [ A[2] + 4 A[3] + 2 A[4] + 2 A[5] + 2 A[8], 3 A[3] + 3 A[4] + A[5] + A[6] + A[8], 2 A[6] + 2 A[10]]], [1, 1, 2, 7, 8, 1, 7, 2, 2, 7]] For example, C(100000), mudolo , 9, equals , 7 The congruence classes mod, 9, in the following set , {0, 3, 4, 5, 6}, never show up! Theorem Number, 27, : Let C(n) be the constant term, in x, of n (2/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], A[3], 0], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0]], [1, 1, 3]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 6, 7, 8}, never show up! Theorem Number, 28, : Let C(n) be the constant term, in x, of n (2/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 9, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [3 A[2], 6 A[2] + 3 A[3], A[7]], [4 A[2], A[5], A[9]], [0, 6 A[2], 3 A[3]], [7 A[2], 3 A[3] + A[5], A[6]], [3 A[2], 6 A[4] + A[7], 6 A[3] + A[7]], [0, 6 A[2], 6 A[4] + A[7]], [4 A[2], 3 A[3] + 3 A[4] + A[5] + 2 A[7], 6 A[4] + A[6]]], [1, 1, 3, 4, 0, 7, 3, 0, 4]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 5, 6, 8}, never show up! Theorem Number, 29, : Let C(n) be the constant term, in x, of n (2/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 7, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [3 A[2], 6 A[2] + 3 A[3], A[7]], [8 A[2], 3 A[2] + 8 A[3], 6 A[3] + 8 A[4]], [0, 6 A[2], 3 A[3]], [2 A[2], 3 A[2] + 2 A[3], 6 A[3] + 2 A[4]], [6 A[2], 6 A[3], 6 A[4]]], [1, 1, 3, 8, 0, 2, 6]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {4, 5, 7}, never show up! Theorem Number, 30, : Let C(n) be the constant term, in x, of n (2/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3], A[3]], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0], [(6 c[3] + 6 c[4] + 3) A[2] + c[3] A[3] + c[4] A[4], 0, 0]], [1, 1, 3, 3]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 6, 7, 8}, never show up! Theorem Number, 31, : Let C(n) be the constant term, in x, of n (3/x + x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], 0, A[3]], [A[2], 0, 0], [(3 c[3] + 6) A[2] + c[3] A[3], 0, 0]], [1, 1, 6]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 3, 4, 5, 7, 8}, never show up! Theorem Number, 32, : Let C(n) be the constant term, in x, of n (3/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 3, states . Here it is: [[[A[2], 0, A[3]], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0]], [1, 1, 3]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 6, 7, 8}, never show up! Theorem Number, 33, : Let C(n) be the constant term, in x, of n (3/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[2], A[2]], [A[2], A[2], A[2]], [ (64 c[2] + 9) A[2] + (8 c[2] + 1) A[3], (64 c[2] + 9) A[2] + (8 c[2] + 1) A[3], (64 c[2] + 9) A[2] + (8 c[2] + 1) A[3]], [(640 c[2] + 576 c[3] + 81) A[2] + (72 c[2] + 64 c[3] + 9) A[3] + (8 c[2] + 8 c[3] + 1) A[4], (640 c[2] + 576 c[3] + 81) A[2] + (72 c[2] + 64 c[3] + 9) A[3] + (8 c[2] + 8 c[3] + 1) A[4], (640 c[2] + 576 c[3] + 81) A[2] + (72 c[2] + 64 c[3] + 9) A[3] + (8 c[2] + 8 c[3] + 1) A[4]]], [1, 1, 1, 1]] For example, C(100000), mudolo , 9, equals , 1 The congruence classes mod, 9, in the following set , {0, 2, 3, 4, 5, 6, 7, 8}, never show up! Theorem Number, 34, : Let C(n) be the constant term, in x, of n (3/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [A[2], A[7], A[8]], [7 A[2], A[9], A[10]], [A[2], 6 A[2] + 3 A[3] + A[5], 3 A[2] + A[3] + 7 A[4] + 2 A[5]], [A[2], A[5], 3 A[2] + 7 A[3] + A[4] + 2 A[5]], [A[2], 6 A[2] + 3 A[3] + A[5], 6 A[2] + 2 A[3] + 5 A[4] + A[5] + 2 A[6]], [A[2], 6 A[2] + 2 A[5] + 2 A[7], 6 A[2] + 4 A[3] + A[4] + A[5] + A[7]], [ A[2] + 6 A[3] + A[5] + A[7] + A[9], A[5] + A[7] + 2 A[9], 6 A[3] + 3 A[4] + A[5] + A[6] + 2 A[9]], [ A[2] + 6 A[3] + A[5] + A[7] + A[9], 2 A[5] + 2 A[9], 8 A[3] + 5 A[4] + A[5] + 2 A[6] + A[7] + 2 A[9]]], [1, 1, 1, 7, 1, 1, 1, 1, 7, 7]] For example, C(100000), mudolo , 9, equals , 1 The congruence classes mod, 9, in the following set , {0, 2, 3, 4, 5, 6, 8}, never show up! Theorem Number, 35, : Let C(n) be the constant term, in x, of n (3/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [A[2], A[7], A[8]], [4 A[2], A[9], A[10]], [A[2], 6 A[2] + 3 A[3] + A[5], 3 A[2] + 7 A[3] + 4 A[4] + 2 A[5]], [A[2], A[5], A[6]], [A[2], A[3] + 4 A[4] + 2 A[6], 2 A[3] + 8 A[4] + 2 A[6] + A[7]], [A[2], 7 A[3] + 4 A[4] + A[5] + 2 A[6] + 2 A[7], 4 A[3] + 7 A[4] + 2 A[6] + 2 A[7] + A[8]], [A[2] + 2 A[3] + 2 A[4] + A[5] + A[6], 3 A[3] + A[7], 3 A[4] + A[8]], [A[2] + 2 A[3] + 2 A[4] + A[5] + A[6], 3 A[3] + A[5], 3 A[4] + A[6]]], [1, 1, 1, 4, 1, 1, 1, 1, 4, 4]] For example, C(100000), mudolo , 9, equals , 1 The congruence classes mod, 9, in the following set , {0, 2, 3, 5, 6, 7, 8}, never show up! Theorem Number, 36, : Let C(n) be the constant term, in x, of n (3/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 19, 55, 595, 2611, 22141, 119449, 902035, 5420035, 38712169, 246360709, 1714206781, 11255897485, 77419522675, 517370395015 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[2], A[2]], [A[2], A[2], A[2]], [ (64 c[2] + 9) A[2] + (8 c[2] + 1) A[3], (64 c[2] + 9) A[2] + (8 c[2] + 1) A[3], (64 c[2] + 9) A[2] + (8 c[2] + 1) A[3]], [(640 c[2] + 576 c[3] + 81) A[2] + (72 c[2] + 64 c[3] + 9) A[3] + (8 c[2] + 8 c[3] + 1) A[4], (640 c[2] + 576 c[3] + 81) A[2] + (72 c[2] + 64 c[3] + 9) A[3] + (8 c[2] + 8 c[3] + 1) A[4], (640 c[2] + 576 c[3] + 81) A[2] + (72 c[2] + 64 c[3] + 9) A[3] + (8 c[2] + 8 c[3] + 1) A[4]]], [1, 1, 1, 1]] For example, C(100000), mudolo , 9, equals , 1 The congruence classes mod, 9, in the following set , {0, 2, 3, 4, 5, 6, 7, 8}, never show up! Theorem Number, 37, : Let C(n) be the constant term, in x, of n (3/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3], A[4]], [A[2], 8 A[2], A[2]], [ (28 c[2] + 9) A[2] + (4 c[2] + 1) A[3], (28 c[2] + 63) A[2] + (4 c[2] + 8) A[3], (28 c[2] + 9) A[2] + (4 c[2] + 1) A[3]], [(136 c[2] + 216 c[3] + 72) A[2] + (18 c[2] + 28 c[3] + 9) A[3] + (2 c[2] + 4 c[3] + 1) A[4], (136 c[2] + 216 c[3] + 486) A[2] + (18 c[2] + 28 c[3] + 63) A[3] + (2 c[2] + 4 c[3] + 8) A[4], (136 c[2] + 216 c[3] + 72) A[2] + (18 c[2] + 28 c[3] + 9) A[3] + (2 c[2] + 4 c[3] + 1) A[4]]], [1, 1, 2, 4]] For example, C(100000), mudolo , 9, equals , 7 The congruence classes mod, 9, in the following set , {0, 3, 5, 6, 7, 8}, never show up! Theorem Number, 38, : Let C(n) be the constant term, in x, of n (3/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [2 A[2], A[7], A[8]], [A[2], A[5], A[6]], [8 A[2], 6 A[2] + 3 A[3] + 2 A[5], 3 A[2] + 7 A[3] + 2 A[4] + 2 A[5]], [A[2], A[5], 6 A[2] + 2 A[3] + A[4] + A[5]], [ 7 A[2], 3 A[3] + 2 A[7], 6 A[2] + 6 A[3] + 3 A[4] + 2 A[5] + A[6] + 2 A[7]] , [2 A[2], 6 A[3] + 3 A[4] + 2 A[5] + A[6] + A[8], 4 A[3] + 5 A[4] + A[5] + A[6] + 2 A[7] + A[8]], [ 2 A[2] + 3 A[3] + 6 A[4] + A[5] + 2 A[6] + A[7] + 2 A[8], 3 A[3] + 3 A[4] + A[6] + A[7] + A[8], 4 A[3] + 8 A[4] + A[5] + 2 A[6] + 2 A[7] + 2 A[8]], [A[2], 3 A[3] + 6 A[4] + A[5] + 2 A[6] + 2 A[8], 5 A[3] + 7 A[4] + 2 A[5] + 2 A[6] + A[7] + 2 A[8]]], [1, 1, 2, 1, 8, 1, 7, 2, 8, 1]] For example, C(100000), mudolo , 9, equals , 7 The congruence classes mod, 9, in the following set , {0, 3, 4, 5, 6}, never show up! Theorem Number, 39, : Let C(n) be the constant term, in x, of n (3/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 10, states . Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [2 A[2], A[7], A[8]], [7 A[2], A[9], A[10]], [8 A[2], 6 A[2] + 3 A[3] + 2 A[5], 3 A[2] + 4 A[3] + 5 A[4] + 2 A[5]], [A[2], 6 A[2] + 5 A[3] + 4 A[4] + 2 A[5] + 2 A[6], A[6]], [7 A[2], 5 A[3] + 4 A[4] + 2 A[5] + 2 A[6], 6 A[2] + 2 A[3] + A[4] + A[5]], [2 A[2], 6 A[3] + 3 A[4] + 2 A[5] + A[6] + A[8], 2 A[6]], [2 A[2], 4 A[3] + 2 A[4] + A[5] + 2 A[8], A[8]], [ A[2] + 4 A[3] + 2 A[4] + 2 A[5] + 2 A[8], 3 A[3] + 3 A[4] + A[5] + A[6] + A[8], 2 A[6] + 2 A[10]]], [1, 1, 2, 7, 8, 1, 7, 2, 2, 7]] For example, C(100000), mudolo , 9, equals , 7 The congruence classes mod, 9, in the following set , {0, 3, 4, 5, 6}, never show up! Theorem Number, 40, : Let C(n) be the constant term, in x, of n (3/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 22, 116, 934, 6332, 48124, 352424, 2669062, 20107628, 153277972, 1170192344, 8981891164, 69111416792, 533463087928, 4126851588176 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3], A[4]], [A[2], 8 A[2], A[2]], [ (28 c[2] + 9) A[2] + (4 c[2] + 1) A[3], (28 c[2] + 63) A[2] + (4 c[2] + 8) A[3], (28 c[2] + 9) A[2] + (4 c[2] + 1) A[3]], [(136 c[2] + 216 c[3] + 72) A[2] + (18 c[2] + 28 c[3] + 9) A[3] + (2 c[2] + 4 c[3] + 1) A[4], (136 c[2] + 216 c[3] + 486) A[2] + (18 c[2] + 28 c[3] + 63) A[3] + (2 c[2] + 4 c[3] + 8) A[4], (136 c[2] + 216 c[3] + 72) A[2] + (18 c[2] + 28 c[3] + 9) A[3] + (2 c[2] + 4 c[3] + 1) A[4]]], [1, 1, 2, 4]] For example, C(100000), mudolo , 9, equals , 7 The congruence classes mod, 9, in the following set , {0, 3, 5, 6, 7, 8}, never show up! Theorem Number, 41, : Let C(n) be the constant term, in x, of n (3/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3], A[4]], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0], [(6 c[3] + 3 c[4] + 6) A[2] + c[3] A[3] + c[4] A[4], 0, 0]], [1, 1, 3, 6]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 7, 8}, never show up! Theorem Number, 42, : Let C(n) be the constant term, in x, of n (3/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 9 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 4, states . Here it is: [[[A[2], A[3], A[3]], [A[2], 0, 0], [(6 c[3] + 3) A[2] + c[3] A[3], 0, 0], [(6 c[3] + 6 c[4] + 3) A[2] + c[3] A[3] + c[4] A[4], 0, 0]], [1, 1, 3, 3]] For example, C(100000), mudolo , 9, equals , 0 The congruence classes mod, 9, in the following set , {2, 4, 5, 6, 7, 8}, never show up! ------------------------------------------ This ends this fascinating book that took, 1.330, to generate. ----------------------------------------- ----------------------------------------------------- On computing the Mod, 27, of Many Interesting sequences by Shalosh B. Ekhad Theorem Number, 1, : Let C(n) be the constant term, in x, of n (1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 28, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [A[5], A[22], A[23]], [12 A[5] + 16 A[8], 21 A[6] + 7 A[9], A[24]], [A[25], A[26], A[27]], [A[5], 10 A[6] + 18 A[11], A[15]], [15 A[5] + 13 A[8], 6 A[6] + 22 A[9], 24 A[7] + 4 A[10]], [ 9 A[5] + 3 A[8] + 16 A[11], 9 A[6] + 21 A[9] + 25 A[12], 9 A[7] + 12 A[10] + 7 A[13]], [15 A[5] + 13 A[8], 6 A[6] + 22 A[9], 24 A[7] + 4 A[10]], [ 18 A[5] + 3 A[8] + 7 A[11], 18 A[6] + 21 A[9] + 16 A[12], 18 A[7] + 12 A[10] + 25 A[13]], [A[5], 19 A[6] + 9 A[11], 10 A[7] + 18 A[12]], [3 A[5] + 25 A[8], 12 A[6] + 16 A[9], 21 A[7] + 7 A[10]], [6 A[8] + 22 A[11], 15 A[9] + 13 A[12], 24 A[10] + 4 A[13]], [A[5], 10 A[6] + 18 A[11], 19 A[7] + 9 A[12]], [6 A[5] + 22 A[8], 24 A[6] + 4 A[9], 15 A[7] + 13 A[10]], [ 9 A[5] + 12 A[8] + 7 A[11], 9 A[6] + 3 A[9] + 16 A[12], 9 A[7] + 21 A[10] + 25 A[13]], [6 A[5] + 22 A[8], 24 A[6] + 4 A[9], 15 A[7] + 13 A[10]], [ 18 A[5] + 12 A[8] + 25 A[11], 18 A[6] + 3 A[9] + 7 A[12], 18 A[7] + 21 A[10] + 16 A[13]], [15 A[8] + 13 A[11], 24 A[9] + 4 A[12], 6 A[10] + 22 A[13]], [A[5], 10 A[6] + 18 A[11], 19 A[7] + 9 A[12]], [24 A[5] + 4 A[8], 15 A[6] + 13 A[9], 6 A[7] + 22 A[10]], [ 9 A[5] + 21 A[8] + 25 A[11], 9 A[6] + 12 A[9] + 7 A[12], 9 A[7] + 3 A[10] + 16 A[13]], [18 A[5] + 3 A[8] + 7 A[11], 18 A[6] + 21 A[9] + 16 A[12], 18 A[7] + 12 A[10] + 25 A[13]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 27, equals , 1 The congruence classes mod, 27, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 2, : Let C(n) be the constant term, in x, of n (1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 13, states . Here it is: [[[A[2], A[3], A[4]], %1, %1, %1, %1, [ (676 c[5] + 27) A[5] + (26 c[5] + 1) A[6], (676 c[5] + 27) A[5] + (26 c[5] + 1) A[6], (676 c[5] + 27) A[5] + (26 c[5] + 1) A[6]], [ (18928 c[5] + 18252 c[6] + 729) A[5] + (702 c[5] + 676 c[6] + 27) A[6] + (26 c[5] + 26 c[6] + 1) A[7], (18928 c[5] + 18252 c[6] + 729) A[5] + (702 c[5] + 676 c[6] + 27) A[6] + (26 c[5] + 26 c[6] + 1) A[7], (18928 c[5] + 18252 c[6] + 729) A[5] + (702 c[5] + 676 c[6] + 27) A[6] + (26 c[5] + 26 c[6] + 1) A[7]], [ (19683 + 511732 c[5] + 511056 c[6] + 492804 c[7]) A[5] + (18954 c[5] + 18928 c[6] + 18252 c[7] + 729) A[6] + (702 c[5] + 702 c[6] + 676 c[7] + 27) A[7] + (26 c[5] + 26 c[6] + 26 c[7] + 1) A[8], (19683 + 511732 c[5] + 511056 c[6] + 492804 c[7]) A[5] + (18954 c[5] + 18928 c[6] + 18252 c[7] + 729) A[6] + (702 c[5] + 702 c[6] + 676 c[7] + 27) A[7] + (26 c[5] + 26 c[6] + 26 c[7] + 1) A[8], (19683 + 511732 c[5] + 511056 c[6] + 492804 c[7]) A[5] + (18954 c[5] + 18928 c[6] + 18252 c[7] + 729) A[6] + (702 c[5] + 702 c[6] + 676 c[7] + 27) A[7] + (26 c[5] + 26 c[6] + 26 c[7] + 1) A[8]], [ (13817440 c[5] + 13816764 c[6] + 13798512 c[7] + 13305708 c[8] + 531441) A[5] + (511758 c[5] + 19683 + 511732 c[6] + 511056 c[7] + 492804 c[8]) A[6] + (18954 c[5] + 18954 c[6] + 18928 c[7] + 18252 c[8] + 729) A[7] + (702 c[5] + 702 c[6] + 702 c[7] + 676 c[8] + 27) A[8] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 1) A[9], (13817440 c[5] + 13816764 c[6] + 13798512 c[7] + 13305708 c[8] + 531441) A[5] + (511758 c[5] + 19683 + 511732 c[6] + 511056 c[7] + 492804 c[8]) A[6] + (18954 c[5] + 18954 c[6] + 18928 c[7] + 18252 c[8] + 729) A[7] + (702 c[5] + 702 c[6] + 702 c[7] + 676 c[8] + 27) A[8] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 1) A[9], (13817440 c[5] + 13816764 c[6] + 13798512 c[7] + 13305708 c[8] + 531441) A[5] + (511758 c[5] + 19683 + 511732 c[6] + 511056 c[7] + 492804 c[8]) A[6] + (18954 c[5] + 18954 c[6] + 18928 c[7] + 18252 c[8] + 729) A[7] + (702 c[5] + 702 c[6] + 702 c[7] + 676 c[8] + 27) A[8] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 1) A[9]], [(373071556 c[5] + 373070880 c[6] + 373052628 c[7] + 372559824 c[8] + 359254116 c[9] + 14348907) A[5] + (13817466 c[5] + 13817440 c[6] + 13816764 c[7] + 13798512 c[8] + 13305708 c[9] + 531441) A[6] + ( 511758 c[5] + 511758 c[6] + 19683 + 511732 c[7] + 511056 c[8] + 492804 c[9] ) A[7] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18928 c[8] + 18252 c[9] + 729) A[8] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 676 c[9] + 27) A[9] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 1) A[10], ( 373071556 c[5] + 373070880 c[6] + 373052628 c[7] + 372559824 c[8] + 359254116 c[9] + 14348907) A[5] + (13817466 c[5] + 13817440 c[6] + 13816764 c[7] + 13798512 c[8] + 13305708 c[9] + 531441) A[6] + ( 511758 c[5] + 511758 c[6] + 19683 + 511732 c[7] + 511056 c[8] + 492804 c[9] ) A[7] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18928 c[8] + 18252 c[9] + 729) A[8] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 676 c[9] + 27) A[9] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 1) A[10], ( 373071556 c[5] + 373070880 c[6] + 373052628 c[7] + 372559824 c[8] + 359254116 c[9] + 14348907) A[5] + (13817466 c[5] + 13817440 c[6] + 13816764 c[7] + 13798512 c[8] + 13305708 c[9] + 531441) A[6] + ( 511758 c[5] + 511758 c[6] + 19683 + 511732 c[7] + 511056 c[8] + 492804 c[9] ) A[7] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18928 c[8] + 18252 c[9] + 729) A[8] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 676 c[9] + 27) A[9] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 1) A[10]], [( 10072932688 c[5] + 10072932012 c[6] + 10072913760 c[7] + 10072420956 c[8] + 10059115248 c[9] + 9699861132 c[10] + 387420489) A[5] + (373071582 c[5] + 373071556 c[6] + 373070880 c[7] + 373052628 c[8] + 372559824 c[9] + 359254116 c[10] + 14348907) A[6] + (13817466 c[5] + 13817466 c[6] + 13817440 c[7] + 13816764 c[8] + 13798512 c[9] + 13305708 c[10] + 531441) A[7] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 19683 + 511732 c[8] + 511056 c[9] + 492804 c[10]) A[8] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18928 c[9] + 18252 c[10] + 729) A[9] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 676 c[10] + 27) A[10] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 1) A[11], (10072932688 c[5] + 10072932012 c[6] + 10072913760 c[7] + 10072420956 c[8] + 10059115248 c[9] + 9699861132 c[10] + 387420489) A[5] + (373071582 c[5] + 373071556 c[6] + 373070880 c[7] + 373052628 c[8] + 372559824 c[9] + 359254116 c[10] + 14348907) A[6] + (13817466 c[5] + 13817466 c[6] + 13817440 c[7] + 13816764 c[8] + 13798512 c[9] + 13305708 c[10] + 531441) A[7] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 19683 + 511732 c[8] + 511056 c[9] + 492804 c[10]) A[8] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18928 c[9] + 18252 c[10] + 729) A[9] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 676 c[10] + 27) A[10] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 1) A[11], (10072932688 c[5] + 10072932012 c[6] + 10072913760 c[7] + 10072420956 c[8] + 10059115248 c[9] + 9699861132 c[10] + 387420489) A[5] + (373071582 c[5] + 373071556 c[6] + 373070880 c[7] + 373052628 c[8] + 372559824 c[9] + 359254116 c[10] + 14348907) A[6] + (13817466 c[5] + 13817466 c[6] + 13817440 c[7] + 13816764 c[8] + 13798512 c[9] + 13305708 c[10] + 531441) A[7] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 19683 + 511732 c[8] + 511056 c[9] + 492804 c[10]) A[8] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18928 c[9] + 18252 c[10] + 729) A[9] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 676 c[10] + 27) A[10] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 1) A[11]], [(271969183252 c[5] + 271969182576 c[6] + 271969164324 c[7] + 271968671520 c[8] + 10460353203 + 271955365812 c[9] + 271596111696 c[10] + 261896250564 c[11]) A[5] + (10072932714 c[5] + 10072932688 c[6] + 10072932012 c[7] + 10072913760 c[8] + 10072420956 c[9] + 10059115248 c[10] + 9699861132 c[11] + 387420489) A[6] + (373071582 c[5] + 373071582 c[6] + 373071556 c[7] + 373070880 c[8] + 373052628 c[9] + 372559824 c[10] + 359254116 c[11] + 14348907) A[7] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817440 c[8] + 13816764 c[9] + 13798512 c[10] + 13305708 c[11] + 531441) A[8] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 19683 + 511732 c[9] + 511056 c[10] + 492804 c[11]) A[9] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18928 c[10] + 18252 c[11] + 729) A[10] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 676 c[11] + 27) A[11] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 1) A[12], (271969183252 c[5] + 271969182576 c[6] + 271969164324 c[7] + 271968671520 c[8] + 10460353203 + 271955365812 c[9] + 271596111696 c[10] + 261896250564 c[11]) A[5] + (10072932714 c[5] + 10072932688 c[6] + 10072932012 c[7] + 10072913760 c[8] + 10072420956 c[9] + 10059115248 c[10] + 9699861132 c[11] + 387420489) A[6] + (373071582 c[5] + 373071582 c[6] + 373071556 c[7] + 373070880 c[8] + 373052628 c[9] + 372559824 c[10] + 359254116 c[11] + 14348907) A[7] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817440 c[8] + 13816764 c[9] + 13798512 c[10] + 13305708 c[11] + 531441) A[8] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 19683 + 511732 c[9] + 511056 c[10] + 492804 c[11]) A[9] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18928 c[10] + 18252 c[11] + 729) A[10] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 676 c[11] + 27) A[11] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 1) A[12], (271969183252 c[5] + 271969182576 c[6] + 271969164324 c[7] + 271968671520 c[8] + 10460353203 + 271955365812 c[9] + 271596111696 c[10] + 261896250564 c[11]) A[5] + (10072932714 c[5] + 10072932688 c[6] + 10072932012 c[7] + 10072913760 c[8] + 10072420956 c[9] + 10059115248 c[10] + 9699861132 c[11] + 387420489) A[6] + (373071582 c[5] + 373071582 c[6] + 373071556 c[7] + 373070880 c[8] + 373052628 c[9] + 372559824 c[10] + 359254116 c[11] + 14348907) A[7] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817440 c[8] + 13816764 c[9] + 13798512 c[10] + 13305708 c[11] + 531441) A[8] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 19683 + 511732 c[9] + 511056 c[10] + 492804 c[11]) A[9] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18928 c[10] + 18252 c[11] + 729) A[10] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 676 c[11] + 27) A[11] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 1) A[12]], [(7343167948480 c[5] + 7343167947804 c[6] + 7343167929552 c[7] + 7343167436748 c[8] + 7343154131040 c[9] + 7342794876924 c[10] + 7333095015792 c[11] + 7071198765228 c[12] + 282429536481) A[5] + ( 271969183278 c[5] + 271969183252 c[6] + 271969182576 c[7] + 271969164324 c[8] + 271968671520 c[9] + 10460353203 + 271955365812 c[10] + 271596111696 c[11] + 261896250564 c[12]) A[6] + (10072932714 c[5] + 10072932714 c[6] + 10072932688 c[7] + 10072932012 c[8] + 10072913760 c[9] + 10072420956 c[10] + 10059115248 c[11] + 9699861132 c[12] + 387420489) A[7] + (373071582 c[5] + 373071582 c[6] + 373071582 c[7] + 373071556 c[8] + 373070880 c[9] + 373052628 c[10] + 372559824 c[11] + 359254116 c[12] + 14348907) A[8] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817466 c[8] + 13817440 c[9] + 13816764 c[10] + 13798512 c[11] + 13305708 c[12] + 531441) A[9] + ( 511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 511758 c[9] + 19683 + 511732 c[10] + 511056 c[11] + 492804 c[12]) A[10] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18954 c[10] + 18928 c[11] + 18252 c[12] + 729) A[11] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 702 c[11] + 676 c[12] + 27) A[12] + ( 26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 26 c[12] + 1) A[13], (7343167948480 c[5] + 7343167947804 c[6] + 7343167929552 c[7] + 7343167436748 c[8] + 7343154131040 c[9] + 7342794876924 c[10] + 7333095015792 c[11] + 7071198765228 c[12] + 282429536481) A[5] + (271969183278 c[5] + 271969183252 c[6] + 271969182576 c[7] + 271969164324 c[8] + 271968671520 c[9] + 10460353203 + 271955365812 c[10] + 271596111696 c[11] + 261896250564 c[12]) A[6] + ( 10072932714 c[5] + 10072932714 c[6] + 10072932688 c[7] + 10072932012 c[8] + 10072913760 c[9] + 10072420956 c[10] + 10059115248 c[11] + 9699861132 c[12] + 387420489) A[7] + (373071582 c[5] + 373071582 c[6] + 373071582 c[7] + 373071556 c[8] + 373070880 c[9] + 373052628 c[10] + 372559824 c[11] + 359254116 c[12] + 14348907) A[8] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817466 c[8] + 13817440 c[9] + 13816764 c[10] + 13798512 c[11] + 13305708 c[12] + 531441) A[9] + ( 511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 511758 c[9] + 19683 + 511732 c[10] + 511056 c[11] + 492804 c[12]) A[10] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18954 c[10] + 18928 c[11] + 18252 c[12] + 729) A[11] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 702 c[11] + 676 c[12] + 27) A[12] + ( 26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 26 c[12] + 1) A[13], (7343167948480 c[5] + 7343167947804 c[6] + 7343167929552 c[7] + 7343167436748 c[8] + 7343154131040 c[9] + 7342794876924 c[10] + 7333095015792 c[11] + 7071198765228 c[12] + 282429536481) A[5] + (271969183278 c[5] + 271969183252 c[6] + 271969182576 c[7] + 271969164324 c[8] + 271968671520 c[9] + 10460353203 + 271955365812 c[10] + 271596111696 c[11] + 261896250564 c[12]) A[6] + ( 10072932714 c[5] + 10072932714 c[6] + 10072932688 c[7] + 10072932012 c[8] + 10072913760 c[9] + 10072420956 c[10] + 10059115248 c[11] + 9699861132 c[12] + 387420489) A[7] + (373071582 c[5] + 373071582 c[6] + 373071582 c[7] + 373071556 c[8] + 373070880 c[9] + 373052628 c[10] + 372559824 c[11] + 359254116 c[12] + 14348907) A[8] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817466 c[8] + 13817440 c[9] + 13816764 c[10] + 13798512 c[11] + 13305708 c[12] + 531441) A[9] + ( 511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 511758 c[9] + 19683 + 511732 c[10] + 511056 c[11] + 492804 c[12]) A[10] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18954 c[10] + 18928 c[11] + 18252 c[12] + 729) A[11] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 702 c[11] + 676 c[12] + 27) A[12] + ( 26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 26 c[12] + 1) A[13]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] %1 := [A[5], A[5], A[5]] For example, C(100000), mudolo , 27, equals , 1 The congruence classes mod, 27, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 3, : Let C(n) be the constant term, in x, of n (2 + x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 26, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [2 A[5], 20 A[6], 11 A[7]], [8 A[8], 26 A[9], A[22]], [A[23], A[24], A[25]], [4 A[5], 4 A[6] + 18 A[11], A[26]], [6 A[5] + 13 A[8], 24 A[6] + 13 A[9], 15 A[7] + 13 A[10]], [ 9 A[5] + 21 A[8] + 4 A[11], 9 A[6] + 12 A[9] + 4 A[12], 9 A[7] + 3 A[10] + 4 A[13]], [24 A[5] + A[8], 15 A[6] + A[9], 6 A[7] + A[10]], [ 18 A[5] + 3 A[8] + A[11], 18 A[6] + 21 A[9] + A[12], 18 A[7] + 12 A[10] + A[13]], [8 A[5], 26 A[6], 17 A[7]], [9 A[5] + 5 A[8], 9 A[6] + 23 A[9], 9 A[7] + 14 A[10]], [ 9 A[5] + 9 A[8] + 2 A[11], 9 A[6] + 9 A[9] + 20 A[12], 9 A[7] + 9 A[10] + 11 A[13]], [10 A[5], 10 A[6] + 18 A[11], 10 A[7] + 9 A[12]], [6 A[5] + 19 A[8], 24 A[6] + 19 A[9], 15 A[7] + 19 A[10]], [ 9 A[5] + 21 A[8] + 10 A[11], 9 A[6] + 12 A[9] + 10 A[12], 9 A[7] + 3 A[10] + 10 A[13]], [18 A[5] + 9 A[8] + 22 A[11], 18 A[6] + 9 A[9] + 4 A[12], 18 A[7] + 9 A[10] + 13 A[13]], [20 A[5], 20 A[6] + 9 A[11], 20 A[7] + 18 A[12]], [3 A[5] + 2 A[8], 12 A[6] + 2 A[9], 21 A[7] + 2 A[10]], [ 18 A[5] + 24 A[8] + 2 A[11], 18 A[6] + 6 A[9] + 2 A[12], 18 A[7] + 15 A[10] + 2 A[13]], [18 A[5] + 12 A[8] + 4 A[11], 18 A[6] + 3 A[9] + 4 A[12], 18 A[7] + 21 A[10] + 4 A[13]]], [1, 1, 2, 4, 1, 8, 10, 2, 16, 20, 4, 5, 13, 26, 1, 8, 19, 8, 10, 17, 10, 16, 20, 7, 20, 4]] For example, C(100000), mudolo , 27, equals , 25 The congruence classes mod, 27, in the following set , {0, 3, 6, 9, 11, 12, 14, 15, 18, 21, 22, 23, 24, 25}, never show up! Theorem Number, 4, : Let C(n) be the constant term, in x, of n (2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 13, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], 26 A[5], A[5]], [ (190 c[5] + 27) A[5] + (10 c[5] + 1) A[6], (190 c[5] + 513) A[5] + (10 c[5] + 26) A[6], (190 c[5] + 27) A[5] + (10 c[5] + 1) A[6]], [ (3214 c[5] + 3780 c[6] + 540) A[5] + (162 c[5] + 190 c[6] + 27) A[6] + (8 c[5] + 10 c[6] + 1) A[7], (3214 c[5] + 3780 c[6] + 10206) A[5] + (162 c[5] + 190 c[6] + 513) A[6] + (8 c[5] + 10 c[6] + 26) A[7], (3214 c[5] + 3780 c[6] + 540) A[5] + (162 c[5] + 190 c[6] + 27) A[6] + (8 c[5] + 10 c[6] + 1) A[7]], [ (10746 + 87103 c[5] + 152199 c[6] + 141966 c[7]) A[5] + (4374 c[5] + 7642 c[6] + 7128 c[7] + 540) A[6] + (216 c[5] + 378 c[6] + 352 c[7] + 27) A[7] + (13 c[5] + 23 c[6] + 22 c[7] + 1) A[8], (174231 + 87103 c[5] + 152199 c[6] + 141966 c[7]) A[5] + (4374 c[5] + 7642 c[6] + 7128 c[7] + 8748) A[6] + (216 c[5] + 378 c[6] + 352 c[7] + 432) A[7] + (13 c[5] + 23 c[6] + 22 c[7] + 26) A[8], (10746 + 87103 c[5] + 152199 c[6] + 141966 c[7]) A[5] + (4374 c[5] + 7642 c[6] + 7128 c[7] + 540) A[6] + (216 c[5] + 378 c[6] + 352 c[7] + 27) A[7] + (13 c[5] + 23 c[6] + 22 c[7] + 1) A[8]], [ (740368 c[5] + 1850688 c[6] + 3146985 c[7] + 1306530 c[8] + 184977) A[5] + (37179 c[5] + 9288 + 92935 c[6] + 158031 c[7] + 65610 c[8]) A[6] + (1836 c[5] + 4590 c[6] + 7804 c[7] + 3240 c[8] + 459) A[7] + (108 c[5] + 270 c[6] + 459 c[7] + 190 c[8] + 27) A[8] + (5 c[5] + 13 c[6] + 23 c[7] + 10 c[8] + 1) A[9], (740368 c[5] + 1850688 c[6] + 3146985 c[7] + 1306530 c[8] + 3527631) A[5] + (37179 c[5] + 177147 + 92935 c[6] + 158031 c[7] + 65610 c[8]) A[6] + (1836 c[5] + 4590 c[6] + 7804 c[7] + 3240 c[8] + 8748) A[7] + (108 c[5] + 270 c[6] + 459 c[7] + 190 c[8] + 513) A[8] + (5 c[5] + 13 c[6] + 23 c[7] + 10 c[8] + 26) A[9], (740368 c[5] + 1850688 c[6] + 3146985 c[7] + 1306530 c[8] + 184977) A[5] + (37179 c[5] + 9288 + 92935 c[6] + 158031 c[7] + 65610 c[8]) A[6] + (1836 c[5] + 4590 c[6] + 7804 c[7] + 3240 c[8] + 459) A[7] + (108 c[5] + 270 c[6] + 459 c[7] + 190 c[8] + 27) A[8] + (5 c[5] + 13 c[6] + 23 c[7] + 10 c[8] + 1) A[9]], [(11147464 c[5] + 14859558 c[6] + 36970047 c[7] + 22120722 c[8] + 26017740 c[9] + 3712608) A[5] + (559791 c[5] + 746200 c[6] + 1856520 c[7] + 1110834 c[8] + 1306530 c[9] + 186435) A[6] + (27648 c[5] + 36855 c[6] + 9207 + 91693 c[7] + 54864 c[8] + 64530 c[9]) A[7] + (1620 c[5] + 2160 c[6] + 5373 c[7] + 3214 c[8] + 3780 c[9] + 540) A[8] + (81 c[5] + 108 c[6] + 270 c[7] + 162 c[8] + 190 c[9] + 27) A[9] + (4 c[5] + 5 c[6] + 13 c[7] + 8 c[8] + 10 c[9] + 1) A[10], (11147464 c[5] + 14859558 c[6] + 36970047 c[7] + 22120722 c[8] + 26017740 c[9] + 70247898 ) A[5] + (559791 c[5] + 746200 c[6] + 1856520 c[7] + 1110834 c[8] + 1306530 c[9] + 3527631) A[6] + (27648 c[5] + 36855 c[6] + 174231 + 91693 c[7] + 54864 c[8] + 64530 c[9]) A[7] + (1620 c[5] + 2160 c[6] + 5373 c[7] + 3214 c[8] + 3780 c[9] + 10206) A[8] + (81 c[5] + 108 c[6] + 270 c[7] + 162 c[8] + 190 c[9] + 513) A[9] + (4 c[5] + 5 c[6] + 13 c[7] + 8 c[8] + 10 c[9] + 26) A[10], ( 11147464 c[5] + 14859558 c[6] + 36970047 c[7] + 22120722 c[8] + 26017740 c[9] + 3712608) A[5] + (559791 c[5] + 746200 c[6] + 1856520 c[7] + 1110834 c[8] + 1306530 c[9] + 186435) A[6] + (27648 c[5] + 36855 c[6] + 9207 + 91693 c[7] + 54864 c[8] + 64530 c[9]) A[7] + (1620 c[5] + 2160 c[6] + 5373 c[7] + 3214 c[8] + 3780 c[9] + 540) A[8] + (81 c[5] + 108 c[6] + 270 c[7] + 162 c[8] + 190 c[9] + 27) A[9] + (4 c[5] + 5 c[6] + 13 c[7] + 8 c[8] + 10 c[9] + 1) A[10]], [( 899246044 c[5] + 1124982783 c[6] + 525424266 c[7] + 599384799 c[8] + 1047331107 c[9] + 976908438 c[10] + 73960506) A[5] + (45157311 c[5] + 56493100 c[6] + 26385156 c[7] + 30099222 c[8] + 52593678 c[9] + 49057272 c[10] + 3714066) A[6] + (2230308 c[5] + 2790180 c[6] + 1303156 c[7] + 1486593 c[8] + 2597589 c[9] + 2422926 c[10] + 183438) A[7] + (130680 c[5] + 163485 c[6] + 76356 c[7] + 10746 + 87103 c[8] + 152199 c[9] + 141966 c[10]) A[8] + (6561 c[5] + 8208 c[6] + 3834 c[7] + 4374 c[8] + 7642 c[9] + 7128 c[10] + 540) A[9] + (324 c[5] + 405 c[6] + 189 c[7] + 216 c[8] + 378 c[9] + 352 c[10] + 27) A[10] + (20 c[5] + 25 c[6] + 11 c[7] + 13 c[8] + 23 c[9] + 22 c[10] + 1) A[11], (899246044 c[5] + 1124982783 c[6] + 525424266 c[7] + 599384799 c[8] + 1047331107 c[9] + 976908438 c[10] + 1198933083) A[5] + (45157311 c[5] + 56493100 c[6] + 26385156 c[7] + 30099222 c[8] + 52593678 c[9] + 49057272 c[10] + 60206652) A[6] + (2230308 c[5] + 2790180 c[6] + 1303156 c[7] + 1486593 c[8] + 2597589 c[9] + 2422926 c[10] + 2973591) A[7] + (130680 c[5] + 163485 c[6] + 76356 c[7] + 174231 + 87103 c[8] + 152199 c[9] + 141966 c[10]) A[8] + (6561 c[5] + 8208 c[6] + 3834 c[7] + 4374 c[8] + 7642 c[9] + 7128 c[10] + 8748) A[9] + (324 c[5] + 405 c[6] + 189 c[7] + 216 c[8] + 378 c[9] + 352 c[10] + 432) A[10] + (20 c[5] + 25 c[6] + 11 c[7] + 13 c[8] + 23 c[9] + 22 c[10] + 26) A[11], (899246044 c[5] + 1124982783 c[6] + 525424266 c[7] + 599384799 c[8] + 1047331107 c[9] + 976908438 c[10] + 73960506) A[5] + (45157311 c[5] + 56493100 c[6] + 26385156 c[7] + 30099222 c[8] + 52593678 c[9] + 49057272 c[10] + 3714066) A[6] + (2230308 c[5] + 2790180 c[6] + 1303156 c[7] + 1486593 c[8] + 2597589 c[9] + 2422926 c[10] + 183438) A[7] + (130680 c[5] + 163485 c[6] + 76356 c[7] + 10746 + 87103 c[8] + 152199 c[9] + 141966 c[10]) A[8] + (6561 c[5] + 8208 c[6] + 3834 c[7] + 4374 c[8] + 7642 c[9] + 7128 c[10] + 540) A[9] + (324 c[5] + 405 c[6] + 189 c[7] + 216 c[8] + 378 c[9] + 352 c[10] + 27) A[10] + (20 c[5] + 25 c[6] + 11 c[7] + 13 c[8] + 23 c[9] + 22 c[10] + 1) A[11]], [(15284395036 c[5] + 19106419023 c[6] + 23002219566 c[7] + 5094743859 c[8] + 1272893589 + 12735264663 c[9] + 21655560114 c[10] + 8990718120 c[11]) A[5] + (767534319 c[5] + 959464360 c[6] + 1155099231 c[7] + 255842037 c[8] + 639524997 c[9] + 1087474221 c[10] + 451485630 c[11] + 63920718) A[6] + ( 37908324 c[5] + 47387700 c[6] + 57050056 c[7] + 12635973 c[8] + 31585977 c[9] + 53710074 c[10] + 22298760 c[11] + 3157029) A[7] + ( 2221128 c[5] + 2776545 c[6] + 3342681 c[7] + 740368 c[8] + 1850688 c[9] + 3146985 c[10] + 1306530 c[11] + 184977) A[8] + (111537 c[5] + 139428 c[6] + 167859 c[7] + 37179 c[8] + 9288 + 92935 c[9] + 158031 c[10] + 65610 c[11]) A[9] + (5508 c[5] + 6885 c[6] + 8289 c[7] + 1836 c[8] + 4590 c[9] + 7804 c[10] + 3240 c[11] + 459) A[10] + (324 c[5] + 405 c[6] + 486 c[7] + 108 c[8] + 270 c[9] + 459 c[10] + 190 c[11] + 27) A[11] + (16 c[5] + 20 c[6] + 25 c[7] + 5 c[8] + 13 c[9] + 23 c[10] + 10 c[11] + 1) A[12], (15284395036 c[5] + 19106419023 c[6] + 23002219566 c[7] + 5094743859 c[8] + 24274938924 + 12735264663 c[9] + 21655560114 c[10] + 8990718120 c[11]) A[5] + (767534319 c[5] + 959464360 c[6] + 1155099231 c[7] + 255842037 c[8] + 639524997 c[9] + 1087474221 c[10] + 451485630 c[11] + 1219011201) A[6] + (37908324 c[5] + 47387700 c[6] + 57050056 c[7] + 12635973 c[8] + 31585977 c[9] + 53710074 c[10] + 22298760 c[11] + 60206652) A[7] + (2221128 c[5] + 2776545 c[6] + 3342681 c[7] + 740368 c[8] + 1850688 c[9] + 3146985 c[10] + 1306530 c[11] + 3527631) A[8] + (111537 c[5] + 139428 c[6] + 167859 c[7] + 37179 c[8] + 177147 + 92935 c[9] + 158031 c[10] + 65610 c[11]) A[9] + ( 5508 c[5] + 6885 c[6] + 8289 c[7] + 1836 c[8] + 4590 c[9] + 7804 c[10] + 3240 c[11] + 8748) A[10] + (324 c[5] + 405 c[6] + 486 c[7] + 108 c[8] + 270 c[9] + 459 c[10] + 190 c[11] + 513) A[11] + (16 c[5] + 20 c[6] + 25 c[7] + 5 c[8] + 13 c[9] + 23 c[10] + 10 c[11] + 26) A[12], (15284395036 c[5] + 19106419023 c[6] + 23002219566 c[7] + 5094743859 c[8] + 1272893589 + 12735264663 c[9] + 21655560114 c[10] + 8990718120 c[11]) A[5] + (767534319 c[5] + 959464360 c[6] + 1155099231 c[7] + 255842037 c[8] + 639524997 c[9] + 1087474221 c[10] + 451485630 c[11] + 63920718) A[6] + (37908324 c[5] + 47387700 c[6] + 57050056 c[7] + 12635973 c[8] + 31585977 c[9] + 53710074 c[10] + 22298760 c[11] + 3157029) A[7] + (2221128 c[5] + 2776545 c[6] + 3342681 c[7] + 740368 c[8] + 1850688 c[9] + 3146985 c[10] + 1306530 c[11] + 184977) A[8] + (111537 c[5] + 139428 c[6] + 167859 c[7] + 37179 c[8] + 9288 + 92935 c[9] + 158031 c[10] + 65610 c[11]) A[9] + ( 5508 c[5] + 6885 c[6] + 8289 c[7] + 1836 c[8] + 4590 c[9] + 7804 c[10] + 3240 c[11] + 459) A[10] + (324 c[5] + 405 c[6] + 486 c[7] + 108 c[8] + 270 c[9] + 459 c[10] + 190 c[11] + 27) A[11] + (16 c[5] + 20 c[6] + 25 c[7] + 5 c[8] + 13 c[9] + 23 c[10] + 10 c[11] + 1) A[12]], [(51091963138 c[5] + 305566963839 c[6] + 381077900586 c[7] + 76709880063 c[8] + 102254184918 c[9] + 254404752777 c[10] + 152220990528 c[11] + 179037840510 c[12] + 25547832513) A[5] + ( 2565677943 c[5] + 15344613352 c[6] + 19136535471 c[7] + 3852129285 c[8] + 5134884057 c[9] + 1282931919 + 12775407777 c[10] + 7644060126 c[11] + 8990718120 c[12]) A[6] + (126718182 c[5] + 757866564 c[6] + 945148636 c[7] + 190255689 c[8] + 253610622 c[9] + 630974151 c[10] + 377538192 c[11] + 444049290 c[12] + 63363681) A[7] + (7424676 c[5] + 44404929 c[6] + 55378161 c[7] + 11147464 c[8] + 14859558 c[9] + 36970047 c[10] + 22120722 c[11] + 26017740 c[12] + 3712608) A[8] + ( 372843 c[5] + 2229876 c[6] + 2780919 c[7] + 559791 c[8] + 746200 c[9] + 1856520 c[10] + 1110834 c[11] + 1306530 c[12] + 186435) A[9] + ( 18414 c[5] + 110133 c[6] + 137349 c[7] + 27648 c[8] + 36855 c[9] + 9207 + 91693 c[10] + 54864 c[11] + 64530 c[12]) A[10] + (1080 c[5] + 6453 c[6] + 8046 c[7] + 1620 c[8] + 2160 c[9] + 5373 c[10] + 3214 c[11] + 3780 c[12] + 540) A[11] + (54 c[5] + 324 c[6] + 405 c[7] + 81 c[8] + 108 c[9] + 270 c[10] + 162 c[11] + 190 c[12] + 27) A[12] + (2 c[5] + 16 c[6] + 20 c[7] + 4 c[8] + 5 c[9] + 13 c[10] + 8 c[11] + 10 c[12] + 1) A[13], ( 51091963138 c[5] + 305566963839 c[6] + 381077900586 c[7] + 76709880063 c[8] + 102254184918 c[9] + 254404752777 c[10] + 152220990528 c[11] + 179037840510 c[12] + 483402169377) A[5] + (2565677943 c[5] + 15344613352 c[6] + 19136535471 c[7] + 3852129285 c[8] + 5134884057 c[9] + 24274938924 + 12775407777 c[10] + 7644060126 c[11] + 8990718120 c[12]) A[6] + (126718182 c[5] + 757866564 c[6] + 945148636 c[7] + 190255689 c[8] + 253610622 c[9] + 630974151 c[10] + 377538192 c[11] + 444049290 c[12] + 1198933083) A[7] + (7424676 c[5] + 44404929 c[6] + 55378161 c[7] + 11147464 c[8] + 14859558 c[9] + 36970047 c[10] + 22120722 c[11] + 26017740 c[12] + 70247898) A[8] + (372843 c[5] + 2229876 c[6] + 2780919 c[7] + 559791 c[8] + 746200 c[9] + 1856520 c[10] + 1110834 c[11] + 1306530 c[12] + 3527631) A[9] + (18414 c[5] + 110133 c[6] + 137349 c[7] + 27648 c[8] + 36855 c[9] + 174231 + 91693 c[10] + 54864 c[11] + 64530 c[12]) A[10] + (1080 c[5] + 6453 c[6] + 8046 c[7] + 1620 c[8] + 2160 c[9] + 5373 c[10] + 3214 c[11] + 3780 c[12] + 10206) A[11] + ( 54 c[5] + 324 c[6] + 405 c[7] + 81 c[8] + 108 c[9] + 270 c[10] + 162 c[11] + 190 c[12] + 513) A[12] + (2 c[5] + 16 c[6] + 20 c[7] + 4 c[8] + 5 c[9] + 13 c[10] + 8 c[11] + 10 c[12] + 26) A[13], (51091963138 c[5] + 305566963839 c[6] + 381077900586 c[7] + 76709880063 c[8] + 102254184918 c[9] + 254404752777 c[10] + 152220990528 c[11] + 179037840510 c[12] + 25547832513) A[5] + (2565677943 c[5] + 15344613352 c[6] + 19136535471 c[7] + 3852129285 c[8] + 5134884057 c[9] + 1282931919 + 12775407777 c[10] + 7644060126 c[11] + 8990718120 c[12]) A[6] + (126718182 c[5] + 757866564 c[6] + 945148636 c[7] + 190255689 c[8] + 253610622 c[9] + 630974151 c[10] + 377538192 c[11] + 444049290 c[12] + 63363681) A[7] + (7424676 c[5] + 44404929 c[6] + 55378161 c[7] + 11147464 c[8] + 14859558 c[9] + 36970047 c[10] + 22120722 c[11] + 26017740 c[12] + 3712608) A[8] + (372843 c[5] + 2229876 c[6] + 2780919 c[7] + 559791 c[8] + 746200 c[9] + 1856520 c[10] + 1110834 c[11] + 1306530 c[12] + 186435) A[9] + (18414 c[5] + 110133 c[6] + 137349 c[7] + 27648 c[8] + 36855 c[9] + 9207 + 91693 c[10] + 54864 c[11] + 64530 c[12] ) A[10] + (1080 c[5] + 6453 c[6] + 8046 c[7] + 1620 c[8] + 2160 c[9] + 5373 c[10] + 3214 c[11] + 3780 c[12] + 540) A[11] + (54 c[5] + 324 c[6] + 405 c[7] + 81 c[8] + 108 c[9] + 270 c[10] + 162 c[11] + 190 c[12] + 27) A[12] + (2 c[5] + 16 c[6] + 20 c[7] + 4 c[8] + 5 c[9] + 13 c[10] + 8 c[11] + 10 c[12] + 1) A[13]]], [1, 1, 2, 4, 1, 8, 10, 2, 16, 20, 4, 5, 13]] For example, C(100000), mudolo , 27, equals , 25 The congruence classes mod, 27, in the following set , {0, 3, 6, 7, 9, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 5, : Let C(n) be the constant term, in x, of n (3 + x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[6], A[7]], 0, 0, [3 A[5], 3 A[6], 3 A[7]], 0, 0, [9 A[5], 9 A[6], 9 A[7]], 0, 0, 0, 0], [1, 1, 3, 9, 1, 0, 0, 3, 0, 0, 9, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 6, : Let C(n) be the constant term, in x, of n (3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], 0, 0, [3 A[5], 3 A[6], 3 A[7]], 0, 0, [9 A[5], 9 A[6], 9 A[7]], 0, 0, 0, 0], [1, 1, 3, 9, 1, 0, 0, 3, 0, 0, 9, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 7, : Let C(n) be the constant term, in x, of n (1/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 40, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [0, A[22], A[23]], [24 A[5], 15 A[6], 6 A[7]], [A[24], A[25], A[26]], [4 A[5], A[27], A[28]], [A[29], A[17], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [0, 18 A[11], 9 A[12]], [6 A[5], 18 A[6] + 6 A[14], 9 A[7] + 6 A[15]], [18 A[8] + 3 A[16], 9 A[9] + 3 A[17], 3 A[18]], [25 A[5], 12 A[6] + 4 A[14], 7 A[15]], [15 A[8] + 7 A[16], 6 A[9] + 7 A[17], 24 A[10] + 7 A[18]], [ 18 A[5] + 15 A[11] + 4 A[19], 18 A[6] + 6 A[12] + 4 A[20], 18 A[7] + 24 A[13] + 4 A[21]], [18 A[5], 18 A[6], 18 A[7]], [9 A[8], 9 A[9], 9 A[10]], 0, [18 A[5], 18 A[6], 18 A[7]], [3 A[16] + 2 A[24], 6 A[17] + 4 A[25], 3 A[26]], [ 6 A[8] + 4 A[16] + 4 A[24], 6 A[9] + A[17] + 2 A[25], 6 A[10] + A[18] + 8 A[26]], [ 15 A[11] + 7 A[19] + A[22], 15 A[12] + 7 A[20] + 2 A[23], 6 A[13] + 7 A[21] ], [0, A[22], A[23]], [6 A[5], 18 A[6] + 6 A[14], 2 A[15] + 4 A[28] + 15 A[7]], [ 5 A[24] + 2 A[16] + 4 A[29] + 18 A[8], 3 A[25] + 3 A[17], 3 A[26] + 2 A[31] + 9 A[10] + A[18]], [ 2 A[32] + 18 A[11] + 4 A[19] + 4 A[5], A[14] + 3 A[6], A[15] + 3 A[28] + 9 A[7]], [6 A[24] + A[16] + 3 A[29] + 15 A[8], 15 A[9] + 4 A[25] + 5 A[30] + 5 A[17], 4 A[26] + A[31] + 6 A[10]], [ 4 A[32] + A[22] + 24 A[11] + 9 A[19], 5 A[14] + 18 A[6] + 15 A[12] + A[27] + A[23] + 3 A[33] + 7 A[20], 2 A[15] + 4 A[28] + 7 A[21] + 18 A[7] + 3 A[34] + 15 A[13]], [0, A[22], A[23]], [6 A[5], 18 A[6] + 6 A[14], 2 A[15] + 4 A[28] + 15 A[7]], [A[35] + 5 A[24] + 2 A[29] + 12 A[8], A[36] + 12 A[9] + A[25] + 4 A[30] + A[17], 2 A[37] + A[26] + 8 A[31] + 24 A[10] + 2 A[18]], [7 A[5], 8 A[14] + 2 A[27], A[28] + 3 A[7]], [ 2 A[35] + 4 A[24] + 5 A[29] + 21 A[8], 15 A[9] + 5 A[30] + 2 A[17], A[37] + 3 A[31] + 9 A[10]], [8 A[38] + 4 A[32] + 3 A[11] + 7 A[19], 2 A[39] + 4 A[14] + 15 A[6] + 12 A[12] + 2 A[27] + 2 A[23] + 2 A[33], A[21] + 7 A[34] + 21 A[13] + 2 A[40]]], [1, 1, 0, 4, 1, 0, 25, 0, 24, 0, 4, 0, 13, 0, 7, 0, 6, 0, 25, 0, 16, 18, 0, 0, 18, 0, 0, 10, 0, 6, 0, 13, 0, 13, 0, 6, 0, 7, 0, 25]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 3, 5, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, 22, 23, 26}, never show up! Theorem Number, 8, : Let C(n) be the constant term, in x, of n (1/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 12, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], 0, [A[8], A[9], A[10]], [A[5], A[6], A[7]], 0, 0, [6 A[5], 6 A[6], 6 A[7]], 0, 0, 0, 0], [1, 1, 0, 6, 1, 0, 0, 6, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 9, : Let C(n) be the constant term, in x, of n (1/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 47, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [5 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [13 A[5], 21 A[6] + 18 A[11] + A[14], 3 A[7] + 9 A[12] + A[15]], [15 A[5] + 24 A[8] + A[16], 6 A[9] + 6 A[14] + A[17], A[44]], [A[45], A[46], A[47]], [14 A[5], 5 A[14], 12 A[7] + 11 A[15]], [ 18 A[5] + 12 A[8] + 8 A[16], 18 A[6] + 3 A[9] + 8 A[17], 18 A[7] + 21 A[10] + 8 A[18]], [18 A[5] + 3 A[11] + 5 A[19], 18 A[6] + 21 A[12] + 5 A[20], 18 A[7] + 12 A[13] + 5 A[21]], [6 A[8] + 4 A[16], 6 A[9] + 4 A[17], 6 A[10] + 4 A[18]], [15 A[11] + 4 A[19], 15 A[12] + 4 A[20], 15 A[13] + 4 A[21]], [A[5] + 10 A[16] + 11 A[24], 18 A[6] + 3 A[14] + A[22], 21 A[7] + A[23]], [ 24 A[5] + 6 A[8] + 4 A[16] + 15 A[24], 15 A[6] + 15 A[9] + 6 A[14] + 2 A[17] + 11 A[25], 9 A[7] + 6 A[10] + A[18] + 6 A[23]], [21 A[5] + 9 A[6] + 15 A[11] + 14 A[14] + 5 A[16] + 7 A[17] + 4 A[19] + A[22] + 13 A[24] + 14 A[25], 18 A[6] + 9 A[9] + 15 A[12] + 9 A[14] + 6 A[17] + 4 A[20] + 15 A[25], 21 A[7] + 24 A[13] + 8 A[18] + 4 A[21] + 9 A[23] + 10 A[26]], [ 26 A[5] + 10 A[16] + 11 A[24], 9 A[6] + 2 A[14], 21 A[7] + 2 A[15] + 6 A[23]], [12 A[5] + 24 A[6] + 21 A[8] + 18 A[9] + 9 A[11] + 18 A[12] + 10 A[14] + 4 A[16] + 6 A[17] + 2 A[19] + 4 A[20] + 2 A[22] + 4 A[24] + 6 A[25] + A[27] + 2 A[28], 18 A[6] + 12 A[9] + 9 A[12] + 6 A[14] + 3 A[17] + 2 A[20] + 2 A[25] + A[28] , 18 A[7] + 12 A[10] + 6 A[18] + 6 A[23] + 2 A[26]], [18 A[9] + 10 A[14] + 2 A[28] + 24 A[6] + 2 A[22] + 18 A[12] + 12 A[11] + 6 A[25] + 8 A[24] + 7 A[16] + 2 A[27] + 6 A[17] + 6 A[19] + 24 A[5] + 4 A[20], 2 A[15] + 12 A[14] + 2 A[28] + 3 A[6] + 4 A[26] + 12 A[12] + 8 A[25] + 7 A[17] + 18 A[10] + 10 A[23] + 2 A[18] + 6 A[20], A[21] + 8 A[26] + 3 A[7] + 9 A[10] + 12 A[23] + 3 A[13] + 4 A[29] + 10 A[18]], [18 A[9] + 10 A[14] + 2 A[28] + 24 A[6] + 2 A[22] + 18 A[12] + 9 A[11] + 6 A[25] + 8 A[24] + 3 A[16] + A[27] + 6 A[17] + 2 A[19] + 3 A[8] + 24 A[5] + 4 A[20], 21 A[9] + 6 A[14] + A[28] + 12 A[6] + 9 A[12] + 6 A[25] + 2 A[17] + 2 A[20] , 8 A[26] + 3 A[7] + 3 A[10] + 12 A[23] + 6 A[18]], [18 A[9] + 8 A[14] + A[28] + 18 A[6] + A[22] + 9 A[12] + 21 A[11] + 6 A[25] + 8 A[24] + 4 A[16] + 3 A[27] + 9 A[17] + 8 A[19] + 24 A[5] + 2 A[20], 9 A[9] + 12 A[14] + 3 A[28] + 2 A[26] + 24 A[7] + 2 A[31] + 21 A[12] + 10 A[25] + 5 A[17] + 9 A[10] + 8 A[23] + A[18] + 8 A[20], 6 A[21] + 5 A[26] + 24 A[7] + 18 A[10] + 9 A[23] + 12 A[13] + 2 A[29] + 10 A[18]], [ 3 A[32] + 3 A[24] + 9 A[8] + 8 A[5], 6 A[14] + 18 A[6] + 2 A[22], 4 A[26] + 21 A[7] + 18 A[10] + 8 A[23] + 2 A[18]], [ A[32] + 2 A[24] + A[16] + 12 A[8], 3 A[9] + 6 A[14] + A[28] + 12 A[6] + 9 A[12] + 8 A[25] + 3 A[17] + 2 A[20], 4 A[26] + 18 A[7] + 12 A[10] + 6 A[23] + 4 A[18]], [ 4 A[32] + 21 A[11] + 4 A[16] + 4 A[27] + 7 A[19] + 18 A[8], 9 A[9] + 12 A[14] + 5 A[28] + 4 A[26] + 18 A[7] + A[31] + 3 A[12] + 10 A[25] + 5 A[17] + 18 A[10] + 7 A[23] + 2 A[18] + 9 A[20], 7 A[21] + 6 A[26] + 12 A[7] + A[34] + 6 A[23] + 12 A[13] + 4 A[29] + 4 A[18]], [ 3 A[32] + 3 A[24] + 9 A[8] + 4 A[5], A[14] + 3 A[6], 9 A[7] + 2 A[31] + 3 A[23]], [ 9 A[35] + 4 A[32] + A[24] + 7 A[16] + 9 A[27] + 15 A[8] + 3 A[5], 3 A[36] + 15 A[9] + 6 A[14] + 3 A[28] + 15 A[6] + 9 A[12] + 4 A[25] + A[17] + A[33], 4 A[21] + 6 A[26] + 12 A[7] + 2 A[34] + 24 A[10] + 6 A[23] + 2 A[29] + 3 A[18]], [16 A[35] + 4 A[32] + 24 A[11] + 2 A[24] + 8 A[16] + 10 A[27] + 4 A[19] + 18 A[8] + 6 A[5], 6 A[36] + 5 A[37] + 18 A[9] + 10 A[35] + 11 A[14] + 4 A[28] + 3 A[6] + 2 A[21] + 4 A[32] + 2 A[26] + 21 A[7] + A[31] + A[22] + 15 A[12] + 9 A[11] + 5 A[25] + 2 A[24] + 2 A[16] + 6 A[27] + 2 A[17] + 9 A[10] + 7 A[23] + 4 A[19] + 9 A[13] + 3 A[29] + 18 A[8] + 6 A[5] + A[33] + A[18], 4 A[37] + A[21] + 3 A[26] + 6 A[7] + A[34] + 9 A[10] + 3 A[23] + 15 A[13] + A[29] + A[18]], [ 6 A[35] + 2 A[32] + 18 A[11] + 2 A[24] + 6 A[16] + 6 A[27] + 9 A[8] + A[5], 3 A[14] + 9 A[6] + A[22], 7 A[37] + A[21] + 4 A[26] + 18 A[7] + 18 A[10] + 7 A[23] + 18 A[13] + 6 A[29] + 2 A[18]], [ 6 A[35] + 2 A[32] + 18 A[11] + A[24] + 2 A[16] + 6 A[27] + 15 A[8], 3 A[36] + 2 A[38] + 15 A[9] + 8 A[35] + 7 A[14] + 3 A[28] + 18 A[6] + 4 A[32] + 2 A[22] + 9 A[12] + 4 A[25] + 2 A[16] + 6 A[27] + A[17] + 2 A[19] + 15 A[8] + A[33], 6 A[37] + 6 A[26] + 24 A[7] + A[34] + 24 A[10] + 9 A[23] + 18 A[13] + 6 A[29] + 2 A[18]], [ 4 A[35] + 15 A[11] + 6 A[16] + 5 A[27], 4 A[36] + A[39] + A[38] + 21 A[9] + 10 A[35] + 8 A[14] + 5 A[28] + 18 A[6] + 2 A[32] + A[22] + 15 A[12] + 9 A[11] + 6 A[25] + A[16] + 6 A[27] + A[17] + 4 A[19] + 12 A[8] + 2 A[33], 2 A[37] + A[21] + 3 A[26] + 6 A[7] + A[34] + 9 A[10] + 3 A[23] + 6 A[13] + 2 A[29] + A[18]], [2 A[38] + 3 A[35] + 5 A[32] + 9 A[11] + 2 A[24] + A[16] + 3 A[27] + 15 A[8] + 2 A[5], 3 A[36] + A[39] + 2 A[38] + 21 A[9] + 11 A[35] + 10 A[14] + 3 A[28] + 24 A[6] + 3 A[32] + A[22] + 9 A[12] + 12 A[11] + 7 A[25] + A[16] + 7 A[27] + 2 A[17] + 4 A[41] + 15 A[8] + A[33], 3 A[37] + 7 A[26] + 21 A[7] + A[31] + A[34] + 21 A[10] + 9 A[23] + 9 A[13] + 3 A[29] + A[40] + 2 A[18]], [A[38] + 11 A[35] + 4 A[32] + 24 A[11] + 2 A[16] + 7 A[27] + 8 A[41] + 2 A[19] + 15 A[8], 6 A[36] + 2 A[39] + 2 A[38] + 9 A[9] + 11 A[35] + 14 A[14] + 6 A[28] + 3 A[6] + 3 A[32] + A[22] + 18 A[12] + 12 A[11] + 12 A[25] + A[16] + 7 A[27] + 2 A[17] + 4 A[41] + 15 A[8] + 2 A[33], 7 A[37] + A[21] + 12 A[26] + 24 A[7] + 2 A[34] + 9 A[10] + 12 A[23] + 18 A[13] + 6 A[29] + 2 A[40] + 2 A[18]], [A[38] + 8 A[35] + 2 A[32] + 18 A[11] + A[16] + 6 A[27] + 8 A[41] + 2 A[19] + 12 A[8], A[36] + 6 A[37] + A[38] + 9 A[9] + 7 A[35] + 4 A[42] + 7 A[14] + A[28] + 18 A[6] + 2 A[21] + 3 A[32] + 6 A[26] + 18 A[7] + 2 A[31] + 2 A[22] + 15 A[12] + 24 A[11] + 3 A[25] + 2 A[34] + 2 A[16] + 5 A[27] + A[17] + 18 A[10] + 8 A[23] + 2 A[41] + 6 A[13] + 2 A[29] + 12 A[8] + 2 A[43] + A[33] + 2 A[18] + A[20], 4 A[37] + A[21] + 2 A[43]], [ 2 A[38] + 8 A[35] + 3 A[32] + A[16] + 8 A[27] + A[41] + 15 A[8], 2 A[36] + 9 A[9] + 3 A[42] + 3 A[14] + 2 A[28] + 6 A[6] + 15 A[12] + 3 A[25] + A[17] + A[33] + A[20], 24 A[13] + A[29]], [2 A[45] + 2 A[38] + 8 A[35] + 5 A[32] + 21 A[11] + A[24] + 2 A[16] + 8 A[27] + 8 A[41] + 2 A[19] + 15 A[8] + 2 A[5], 2 A[14] + 6 A[6], 9 A[7] + A[31] + 3 A[23]], [2 A[45] + 2 A[38] + 11 A[35] + 4 A[32] + 3 A[11] + A[16] + 11 A[27] + 8 A[41] + 2 A[19] + 18 A[8], 3 A[36] + 2 A[39] + A[45] + A[38] + 7 A[35] + 11 A[14] + 3 A[28] + 24 A[6] + 3 A[32] + A[22] + 9 A[12] + 12 A[11] + 9 A[25] + 2 A[16] + 7 A[27] + A[17] + 6 A[41] + 2 A[19] + 12 A[8] + A[33], 8 A[37] + 2 A[44] + 13 A[26] + 18 A[7] + A[34] + 12 A[10] + 9 A[23] + 21 A[13] + 7 A[29] + A[43] + A[40] + 2 A[18]], [A[35] + 3 A[11], A[36] + 3 A[12], A[37] + 3 A[13]]], [1, 1, 1, 5, 1, 13, 14, 1, 22, 20, 5, 26, 22, 13, 23, 13, 25, 17, 14, 26, 22, 4, 23, 22, 7, 26, 20, 14, 7, 2, 25, 26, 14, 25, 22, 19, 5, 13, 25, 17, 23, 8, 13, 5, 17, 20, 10]] For example, C(100000), mudolo , 27, equals , 20 The congruence classes mod, 27, in the following set , {0, 3, 6, 9, 11, 12, 15, 16, 18, 21, 24}, never show up! Theorem Number, 10, : Let C(n) be the constant term, in x, of n (1/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 46, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [7 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [19 A[5], 10 A[6] + 25 A[7] + 9 A[9] + 2 A[15], 6 A[7] + 9 A[12] + 4 A[15]] , [21 A[5] + 14 A[6] + 2 A[8] + 22 A[9] + 6 A[11] + 4 A[14] + 8 A[16] + 5 A[17], 3 A[5] + 18 A[6] + 26 A[7] + 22 A[8] + 17 A[9] + 3 A[11] + 6 A[14] + A[15] + 8 A[16] + 5 A[17], 18 A[6] + 17 A[9] + 13 A[10] + 24 A[12] + 6 A[14] + 6 A[15] + 7 A[17]], [A[44], A[45], A[46]], [10 A[5], 5 A[6] + 15 A[11] + 5 A[14] + 9 A[16] + 12 A[19], 8 A[6] + A[7] + 18 A[8] + 8 A[9] + 19 A[10] + 6 A[12] + 7 A[14] + 9 A[16] + 7 A[17] + 8 A[18] + 18 A[19]], [6 A[5] + 8 A[6] + 17 A[8] + 16 A[9] + 18 A[10] + 15 A[11] + 18 A[12] + 10 A[14] + 8 A[16] + 11 A[17] + 9 A[18] + 9 A[19] + 9 A[20], 6 A[5] + 4 A[6] + 4 A[7] + 8 A[8] + 11 A[9] + 17 A[10] + 10 A[11] + 14 A[12] + 5 A[14] + 5 A[15] + 4 A[16] + 11 A[17] + 10 A[18] + 2 A[19] + 7 A[20], 18 A[8] + 12 A[9] + 12 A[10] + 12 A[11] + 12 A[12] + 6 A[17] + 7 A[18] + 6 A[19] + 6 A[20]], [15 A[5] + 14 A[6] + 23 A[8] + 25 A[9] + 15 A[10] + 21 A[11] + 24 A[12] + 4 A[14] + 10 A[16] + 11 A[17] + 12 A[18] + 13 A[19] + 12 A[20], 6 A[5] + 9 A[6] + 7 A[7] + 8 A[8] + 20 A[9] + 17 A[10] + 12 A[11] + 21 A[12] + 3 A[14] + 2 A[15] + 7 A[16] + 10 A[17] + 10 A[18] + 3 A[19] + 10 A[20], 6 A[6] + 4 A[7] + 18 A[8] + 24 A[9] + 18 A[10] + 3 A[11] + 7 A[12] + 8 A[13] + 3 A[14] + 2 A[15] + 18 A[17] + 9 A[18] + 15 A[19] + 17 A[20] + 2 A[21]], [9 A[5] + 7 A[6] + 22 A[7] + 21 A[8] + 21 A[9] + 15 A[10] + 15 A[11] + 20 A[12] + 25 A[13] + 2 A[14] + 8 A[15] + 7 A[16] + 9 A[17] + 12 A[18] + 12 A[19] + 10 A[20] + 2 A[21], 3 A[5] + 13 A[6] + 15 A[7] + 4 A[8] + 7 A[9] + 16 A[10] + 5 A[11] + 6 A[12] + 25 A[13] + 4 A[14] + 6 A[15] + 2 A[16] + 3 A[17] + 11 A[18] + A[19] + 3 A[20] + 2 A[21] + A[22], 9 A[10] + A[18]], [3 A[5] + 7 A[6] + 4 A[8] + 14 A[9] + 21 A[10] + 9 A[11] + 12 A[12] + 2 A[14] + 8 A[16] + 4 A[17] + 6 A[18] + 4 A[19] + 6 A[20], 22 A[6] + 23 A[7] + 12 A[9] + 21 A[10] + 2 A[11] + 11 A[12] + 26 A[13] + 7 A[14] + 2 A[15] + 3 A[16] + 3 A[17] + 6 A[18] + A[19] + 5 A[20] + A[21] + A[22] + 8 A[23], 13 A[6] + 14 A[7] + 8 A[9] + 22 A[10] + 2 A[11] + 6 A[12] + 19 A[13] + 4 A[14] + 2 A[15] + 3 A[16] + A[17] + 5 A[18] + A[19] + 3 A[20] + A[22] + 5 A[23]], [7 A[5] + 14 A[6] + 10 A[7] + 22 A[8] + 16 A[9] + 19 A[10] + 11 A[11] + 14 A[12] + 26 A[13] + 4 A[14] + A[15] + 4 A[16] + 5 A[17] + 8 A[18] + 4 A[19] + 7 A[20] + A[21] + 4 A[23] + 4 A[24], 3 A[5] + 21 A[6] + 10 A[7] + 10 A[8] + 10 A[9] + 22 A[10] + 5 A[11] + 8 A[12] + 26 A[13] + 6 A[14] + A[15] + 2 A[16] + 2 A[17] + 5 A[18] + A[19] + 4 A[20] + A[21] + A[22] + 4 A[23] + 3 A[24], 3 A[5] + 8 A[7] + 10 A[8] + 4 A[9] + 25 A[10] + 5 A[11] + 4 A[12] + 2 A[16] + 2 A[17] + 2 A[18] + A[19] + 2 A[20] + 2 A[23] + 3 A[24]], [6 A[5] + 3 A[6] + 10 A[7] + 18 A[8] + 11 A[9] + 25 A[10] + 8 A[11] + 2 A[12] + 26 A[13] + 13 A[14] + A[15] + 4 A[16] + 10 A[17] + 2 A[18] + 4 A[19] + A[20] + A[21] + 2 A[22] + 4 A[23] + 3 A[24] + 9 A[25], 4 A[6] + 20 A[7] + 21 A[9] + 12 A[14] + A[15] + 3 A[17] + 2 A[22] + 6 A[23] + 4 A[25], 26 A[6] + 11 A[7] + 15 A[9] + 9 A[10] + 2 A[12] + 25 A[13] + 9 A[14] + 2 A[15] + 3 A[17] + 4 A[18] + A[20] + 2 A[21] + A[22] + 14 A[23] + 3 A[25]], [24 A[6] + 23 A[7] + 5 A[8] + A[9] + 24 A[10] + 3 A[11] + 4 A[12] + 26 A[13] + 10 A[14] + A[15] + 8 A[17] + 6 A[18] + A[19] + 2 A[20] + A[21] + 2 A[22] + 9 A[23] + A[24] + 6 A[25] + 6 A[26], 17 A[6] + 24 A[7] + 14 A[9] + 4 A[10] + 5 A[12] + 25 A[13] + 6 A[14] + 4 A[17] + 12 A[18] + 2 A[20] + 2 A[21] + A[22] + 12 A[23] + 3 A[25] + 14 A[26], 21 A[7] + 6 A[10] + 24 A[13] + 5 A[18] + A[21] + 9 A[23] + 16 A[26]], [ 4 A[5] + 11 A[6] + 16 A[7] + 8 A[8] + 8 A[9] + 8 A[10] + 4 A[11] + 26 A[13] + 4 A[14] + 2 A[16] + 2 A[17] + 2 A[18] + A[19] + A[21] + 5 A[23] + 2 A[24] + 2 A[25] + 2 A[26] + A[27], 15 A[6] + 4 A[14], 18 A[7] + 9 A[10] + A[15] + A[18] + 6 A[23] + 2 A[26]], [3 A[5] + 18 A[6] + 11 A[7] + 9 A[8] + A[9] + 5 A[10] + 4 A[11] + 4 A[12] + 25 A[13] + 8 A[14] + 2 A[15] + 3 A[16] + 8 A[17] + 8 A[18] + 2 A[19] + 2 A[20] + 2 A[21] + A[22] + 14 A[23] + A[24] + 6 A[25] + 8 A[26], 25 A[6] + 15 A[7] + 15 A[9] + 24 A[10] + 2 A[12] + 25 A[13] + 9 A[14] + 4 A[17] + 6 A[18] + A[20] + 2 A[21] + 2 A[22] + 6 A[23] + 3 A[25] + 6 A[26], A[6] + 13 A[7] + 8 A[9] + 4 A[10] + 4 A[12] + 26 A[13] + 9 A[14] + 2 A[15] + 2 A[17] + 7 A[18] + A[20] + A[21] + 2 A[22] + 15 A[23] + 2 A[25] + 11 A[26] + A[28]], [13 A[9] + 11 A[14] + 25 A[6] + 2 A[26] + 4 A[7] + 2 A[12] + 7 A[11] + 9 A[25] + 4 A[24] + 2 A[16] + A[27] + 8 A[17] + 8 A[10] + 2 A[23] + 2 A[19] + 12 A[8] + 3 A[5] + 2 A[18] + A[20], 11 A[9] + A[15] + 12 A[14] + 2 A[28] + 3 A[6] + 4 A[26] + A[7] + 9 A[12] + 9 A[25] + 7 A[17] + 16 A[10] + 10 A[23] + 4 A[18] + 2 A[20], 23 A[9] + A[15] + 6 A[14] + A[28] + 15 A[6] + 2 A[21] + 8 A[26] + 9 A[7] + 6 A[12] + 5 A[25] + 5 A[17] + A[10] + 14 A[23] + 12 A[13] + 2 A[29] + 6 A[18] + 2 A[20]], [26 A[9] + 10 A[14] + 25 A[6] + 2 A[26] + 4 A[7] + A[22] + 2 A[12] + 10 A[11] + 6 A[25] + 3 A[24] + 5 A[16] + A[27] + 7 A[17] + 8 A[10] + 2 A[23] + 4 A[19] + 17 A[8] + 6 A[5] + 2 A[18] + A[20], 3 A[9] + 2 A[15] + 3 A[14] + 9 A[6] + 19 A[7] + A[25] + 6 A[23], 2 A[9] + 2 A[15] + 4 A[14] + A[28] + 18 A[6] + 3 A[26] + 21 A[7] + 2 A[12] + A[25] + 2 A[30] + 7 A[10] + 7 A[23] + 4 A[13] + 2 A[29]], [26 A[9] + 10 A[14] + A[6] + 2 A[26] + 4 A[7] + 2 A[22] + 2 A[12] + 7 A[11] + 6 A[25] + 3 A[24] + 2 A[30] + 4 A[16] + 7 A[17] + 8 A[10] + 2 A[23] + 3 A[19] + 14 A[8] + 6 A[5] + 2 A[18] + A[20], 18 A[9] + 8 A[14] + 2 A[28] + 22 A[6] + 6 A[26] + 12 A[7] + 2 A[22] + 9 A[12] + 4 A[25] + A[30] + 5 A[17] + 24 A[10] + 6 A[23] + 6 A[18] + 2 A[20], 2 A[9] + 4 A[14] + A[28] + 18 A[6] + 2 A[21] + 3 A[26] + 6 A[7] + 2 A[31] + 2 A[12] + A[25] + 2 A[30] + 10 A[10] + A[23] + 7 A[13] + A[29] + 2 A[18] ], [8 A[9] + 2 A[15] + 5 A[14] + A[28] + 16 A[6] + A[32] + 2 A[26] + 20 A[7] + 2 A[31] + 2 A[22] + 4 A[12] + 2 A[11] + 2 A[25] + 2 A[24] + A[30] + A[16] + A[27] + 2 A[17] + 8 A[10] + 6 A[23] + 8 A[8] + A[5] + 2 A[18] + A[20], 4 A[14] + 15 A[6] + A[22] + 2 A[30], A[15] + 15 A[7] + 2 A[31] + 4 A[23]], [24 A[9] + 2 A[15] + 5 A[14] + A[28] + 12 A[6] + 7 A[32] + 4 A[26] + 24 A[7] + 2 A[31] + 6 A[12] + 6 A[11] + 6 A[25] + 3 A[24] + A[30] + 4 A[16] + A[27] + A[17] + 16 A[10] + 8 A[23] + 2 A[19] + 2 A[8] + 3 A[5] + 5 A[33] + 4 A[18] + 2 A[20], 19 A[9] + A[15] + 5 A[14] + 2 A[28] + 12 A[6] + 2 A[26] + 12 A[7] + A[31] + 6 A[12] + 5 A[25] + A[30] + A[17] + 8 A[10] + 4 A[23] + 3 A[33] + 2 A[18] + A[20], 16 A[9] + A[15] + 3 A[14] + A[28] + 10 A[6] + 2 A[32] + 5 A[26] + 14 A[7] + A[31] + 4 A[12] + 6 A[11] + 4 A[25] + 2 A[24] + 2 A[30] + A[16] + 2 A[27] + A[17] + 19 A[10] + 5 A[23] + A[19] + 2 A[13] + A[29] + 10 A[8] + 3 A[33] + 4 A[18] + A[20]], [22 A[9] + A[15] + 3 A[14] + A[28] + 8 A[6] + 6 A[32] + 6 A[7] + 4 A[12] + 5 A[11] + 4 A[25] + 2 A[24] + A[34] + A[30] + 4 A[16] + A[17] + 2 A[10] + 2 A[23] + 2 A[19] + 24 A[8] + 6 A[5] + 6 A[33] + A[20], 14 A[9] + 2 A[15] + 2 A[14] + A[28] + 6 A[6] + 12 A[7] + 3 A[12] + 3 A[25] + 2 A[34] + A[30] + 2 A[17] + 4 A[10] + 4 A[23] + 2 A[33], 20 A[9] + A[15] + 4 A[14] + 2 A[28] + 8 A[6] + 3 A[21] + 4 A[32] + 2 A[26] + 14 A[7] + A[31] + 8 A[12] + 6 A[11] + 6 A[25] + A[24] + 5 A[34] + 2 A[16] + A[27] + 14 A[10] + 5 A[23] + 2 A[19] + 10 A[13] + 14 A[8] + 4 A[33] + 2 A[20]], [14 A[9] + 4 A[35] + A[15] + 4 A[14] + 10 A[6] + 2 A[32] + 10 A[7] + 2 A[31] + 4 A[12] + 14 A[11] + 4 A[25] + 4 A[24] + 2 A[34] + A[30] + 2 A[27] + 4 A[10] + 2 A[23] + A[19] + 12 A[8] + A[5] + 3 A[33] + 2 A[20], 12 A[9] + 2 A[35] + A[15] + 6 A[14] + 17 A[6] + 4 A[32] + 10 A[7] + 2 A[31] + 4 A[12] + 8 A[11] + 4 A[25] + 4 A[24] + 2 A[34] + 2 A[30] + 2 A[16] + 2 A[27] + 4 A[10] + 2 A[23] + 20 A[8] + 2 A[33] + 2 A[20], 12 A[9] + 2 A[15] + 2 A[14] + 6 A[6] + 19 A[7] + A[31] + 2 A[25] + A[30] + 2 A[17] + 6 A[23] + 2 A[33] ], [A[36] + 10 A[9] + 3 A[35] + 3 A[14] + 6 A[6] + A[32] + 2 A[12] + 10 A[11] + A[25] + 4 A[24] + A[27] + A[17] + A[19] + 11 A[8] + 3 A[33], 11 A[9] + A[35] + 2 A[15] + 2 A[14] + A[28] + 4 A[6] + 2 A[32] + 16 A[7] + 2 A[31] + 2 A[12] + 4 A[11] + 3 A[25] + 2 A[24] + A[16] + A[27] + A[17] + 4 A[23] + 10 A[8] + A[33], 2 A[36] + 8 A[9] + 2 A[35] + A[15] + 4 A[14] + 2 A[28] + 8 A[6] + 4 A[32] + 2 A[26] + 10 A[7] + A[31] + 8 A[12] + 8 A[11] + 2 A[25] + 4 A[24] + 2 A[16] + 2 A[27] + 5 A[10] + 3 A[23] + 2 A[13] + A[29] + 20 A[8] + 2 A[33]], [2 A[36] + A[37] + 12 A[9] + 6 A[35] + 4 A[14] + A[28] + 12 A[6] + 2 A[32] + A[26] + 4 A[7] + 6 A[12] + 17 A[11] + A[25] + 7 A[24] + A[34] + 2 A[30] + A[16] + 4 A[10] + 2 A[23] + 2 A[19] + 4 A[13] + A[29] + 20 A[8] + 5 A[33], A[36] + A[37] + 10 A[9] + 2 A[35] + 2 A[15] + 3 A[14] + 8 A[6] + 4 A[32] + A[26] + 18 A[7] + A[31] + 3 A[12] + 8 A[11] + 2 A[25] + 4 A[24] + 2 A[34] + A[30] + 2 A[16] + 2 A[27] + 6 A[10] + 6 A[23] + 4 A[13] + A[29] + 20 A[8] + 3 A[33], A[37] + 2 A[21] + A[26] + 8 A[7] + 2 A[34] + 6 A[10] + 4 A[23] + 9 A[13] + A[29] ], [A[5], 2 A[37] + A[35] + 2 A[15] + 5 A[6] + 2 A[32] + 2 A[26] + 24 A[7] + 2 A[31] + 4 A[11] + 2 A[24] + 2 A[34] + 2 A[30] + A[16] + A[27] + 8 A[10] + 8 A[23] + 8 A[13] + 2 A[29] + 10 A[8], 2 A[15] + 15 A[7] + A[31] + 4 A[23]], [2 A[37] + A[38] + 3 A[35] + 2 A[15] + 2 A[6] + 4 A[32] + 2 A[26] + 24 A[7] + 2 A[31] + 10 A[11] + 6 A[24] + 2 A[34] + A[30] + A[16] + 2 A[27] + 8 A[10] + 8 A[23] + 8 A[13] + 2 A[29] + 25 A[8], 11 A[9] + A[35] + 2 A[15] + A[14] + A[28] + 4 A[6] + 2 A[32] + 16 A[7] + 2 A[31] + 2 A[12] + 4 A[11] + 3 A[25] + 2 A[24] + A[30] + A[16] + A[27] + 4 A[23] + 10 A[8] + 2 A[33], A[36] + A[37] + 4 A[9] + A[35] + A[15] + 2 A[14] + A[28] + 4 A[6] + 2 A[32] + 3 A[26] + 12 A[7] + 4 A[12] + 4 A[11] + A[25] + 2 A[24] + 3 A[34] + A[16] + A[27] + 13 A[10] + 5 A[23] + 4 A[13] + A[29] + 10 A[8] + A[33]], [A[37] + 4 A[9] + A[28] + 3 A[26] + 12 A[7] + 2 A[31] + 2 A[12] + 3 A[11] + A[25] + 2 A[24] + 3 A[34] + A[27] + 16 A[10] + 4 A[23] + 4 A[13] + A[29] + 4 A[8] + A[33] + 2 A[40], A[36] + 4 A[9] + 2 A[14] + 6 A[6] + A[26] + 2 A[7] + 3 A[12] + A[25] + 2 A[34] + A[30] + 8 A[10] + A[23] + A[33] + A[40], A[37] + A[15] + 6 A[21] + 3 A[26] + 22 A[7] + 2 A[31] + 8 A[34] + 24 A[10] + 8 A[23] + 21 A[13] + A[40]], [A[5], A[37] + 2 A[9] + A[28] + 3 A[6] + 3 A[26] + 12 A[7] + 2 A[31] + 2 A[12] + A[25] + 3 A[34] + A[30] + 16 A[10] + 4 A[23] + 4 A[13] + A[29] + 2 A[40], A[26] + 5 A[7] + A[31] + A[34] + 6 A[10] + A[23] + A[40]], [A[36] + 2 A[37] + A[38] + 2 A[9] + A[35] + 2 A[15] + 2 A[14] + 4 A[6] + 3 A[32] + 4 A[26] + 26 A[7] + A[31] + 2 A[12] + 2 A[11] + A[24] + 4 A[34] + 20 A[10] + 10 A[23] + 8 A[13] + 2 A[29] + 11 A[8] + A[33] + 2 A[40], A[36] + A[37] + 5 A[9] + A[15] + 2 A[14] + 6 A[6] + A[26] + 12 A[7] + A[31] + 2 A[12] + A[34] + A[30] + 4 A[10] + 4 A[23] + 4 A[13] + A[29] + 2 A[33], 2 A[37] + 2 A[15] + A[26] + 22 A[7] + A[31] + 3 A[34] + 9 A[10] + 8 A[23] + 6 A[13] + A[29]], [A[36] + A[38] + 6 A[9] + 2 A[35] + A[15] + 2 A[14] + 2 A[28] + 4 A[6] + 2 A[32] + 8 A[7] + A[31] + 6 A[12] + 9 A[11] + 2 A[25] + 3 A[24] + 2 A[27] + 2 A[23] + 12 A[8] + A[33], A[36] + 2 A[37] + 10 A[9] + 3 A[14] + 2 A[28] + 6 A[6] + 2 A[26] + 8 A[7] + 7 A[12] + 2 A[25] + 2 A[34] + 8 A[10] + 4 A[23] + 8 A[13] + 2 A[29] + 3 A[33], 2 A[37] + 3 A[26] + 12 A[7] + 5 A[34] + 18 A[10] + 6 A[23] + 9 A[13] + 2 A[29] + A[40]], [2 A[36] + 2 A[37] + 2 A[38] + 8 A[9] + 2 A[35] + 4 A[14] + 2 A[28] + 8 A[6] + 4 A[32] + 4 A[26] + 16 A[7] + 2 A[31] + 8 A[12] + 8 A[11] + 2 A[25] + 4 A[24] + 4 A[34] + 2 A[27] + 20 A[10] + 6 A[23] + 8 A[13] + 2 A[29] + 20 A[8] + A[5] + 2 A[33] + 2 A[40], 2 A[37] + 4 A[9] + 2 A[28] + 3 A[6] + 3 A[26] + 12 A[7] + A[31] + 4 A[12] + 2 A[25] + 3 A[34] + A[30] + 14 A[10] + 5 A[23] + 8 A[13] + 2 A[29] + A[40], 2 A[26] + 7 A[7] + A[31] + 2 A[34] + 12 A[10] + 2 A[23] + 2 A[40] ], [A[37] + 4 A[9] + 2 A[15] + 2 A[28] + A[32] + 2 A[26] + 18 A[7] + 4 A[12] + 4 A[11] + 2 A[25] + 2 A[24] + 2 A[34] + 2 A[27] + 10 A[10] + 7 A[23] + 4 A[13] + A[29] + 7 A[8] + A[40], A[36] + 2 A[37] + 7 A[9] + A[15] + 2 A[14] + A[28] + 6 A[6] + 4 A[26] + 18 A[7] + 4 A[12] + A[25] + 4 A[34] + A[30] + 20 A[10] + 8 A[23] + 8 A[13] + 2 A[29] + 2 A[33] + 2 A[40], 2 A[37] + 2 A[15] + 2 A[26] + 24 A[7] + A[31] + 4 A[34] + 15 A[10] + 9 A[23] + 6 A[13] + A[29] + A[40]], [ 2 A[38] + 3 A[35] + 4 A[32] + 9 A[11] + 3 A[24] + A[27] + 18 A[8], 2 A[37] + 4 A[9] + A[15] + A[14] + 6 A[6] + 4 A[26] + 18 A[7] + A[12] + 4 A[34] + 2 A[30] + 20 A[10] + 8 A[23] + 8 A[13] + 2 A[29] + 2 A[33] + 2 A[40], 2 A[15] + 5 A[26] + 18 A[7] + A[31] + 3 A[34] + 18 A[10] + 6 A[23] + 9 A[13] + 2 A[29] + 2 A[46] + A[40]]], [1, 1, 1, 7, 1, 19, 10, 1, 10, 19, 7, 7, 7, 1, 1, 19, 19, 19, 10, 10, 10, 1, 1, 10, 10, 10, 19, 19, 19, 7, 7, 7, 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 19, 19, 19]] For example, C(100000), mudolo , 27, equals , 1 The congruence classes mod, 27, in the following set , {0, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 11, : Let C(n) be the constant term, in x, of n (1/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 26, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [2 A[5], 20 A[6], 11 A[7]], [8 A[8], 26 A[9], A[22]], [A[23], A[24], A[25]], [4 A[5], 4 A[6] + 18 A[11], A[26]], [6 A[5] + 13 A[8], 24 A[6] + 13 A[9], 15 A[7] + 13 A[10]], [ 9 A[5] + 21 A[8] + 4 A[11], 9 A[6] + 12 A[9] + 4 A[12], 9 A[7] + 3 A[10] + 4 A[13]], [24 A[5] + A[8], 15 A[6] + A[9], 6 A[7] + A[10]], [ 18 A[5] + 3 A[8] + A[11], 18 A[6] + 21 A[9] + A[12], 18 A[7] + 12 A[10] + A[13]], [8 A[5], 26 A[6], 17 A[7]], [9 A[5] + 5 A[8], 9 A[6] + 23 A[9], 9 A[7] + 14 A[10]], [ 9 A[5] + 9 A[8] + 2 A[11], 9 A[6] + 9 A[9] + 20 A[12], 9 A[7] + 9 A[10] + 11 A[13]], [10 A[5], 10 A[6] + 18 A[11], 10 A[7] + 9 A[12]], [6 A[5] + 19 A[8], 24 A[6] + 19 A[9], 15 A[7] + 19 A[10]], [ 9 A[5] + 21 A[8] + 10 A[11], 9 A[6] + 12 A[9] + 10 A[12], 9 A[7] + 3 A[10] + 10 A[13]], [18 A[5] + 9 A[8] + 22 A[11], 18 A[6] + 9 A[9] + 4 A[12], 18 A[7] + 9 A[10] + 13 A[13]], [20 A[5], 20 A[6] + 9 A[11], 20 A[7] + 18 A[12]], [3 A[5] + 2 A[8], 12 A[6] + 2 A[9], 21 A[7] + 2 A[10]], [ 18 A[5] + 24 A[8] + 2 A[11], 18 A[6] + 6 A[9] + 2 A[12], 18 A[7] + 15 A[10] + 2 A[13]], [18 A[5] + 12 A[8] + 4 A[11], 18 A[6] + 3 A[9] + 4 A[12], 18 A[7] + 21 A[10] + 4 A[13]]], [1, 1, 2, 4, 1, 8, 10, 2, 16, 20, 4, 5, 13, 26, 1, 8, 19, 8, 10, 17, 10, 16, 20, 7, 20, 4]] For example, C(100000), mudolo , 27, equals , 25 The congruence classes mod, 27, in the following set , {0, 3, 6, 9, 11, 12, 14, 15, 18, 21, 22, 23, 24, 25}, never show up! Theorem Number, 12, : Let C(n) be the constant term, in x, of n (1/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 25, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[6], A[7]], [A[16], A[17], A[18]], [A[11], A[12], A[13]], [2 A[5], 20 A[6], 11 A[7]], [8 A[8], 26 A[9], A[22]], [A[23], A[24], A[25]], [6 A[5], 15 A[6], 24 A[7]], [18 A[8], 0, 9 A[10]], [12 A[11], 21 A[12], 3 A[13]], [18 A[5] + A[8], 18 A[6] + A[9], 18 A[7] + A[10]], [A[11], A[12], A[13]], [20 A[5], 11 A[6], 2 A[7]], [26 A[8], 17 A[9], 8 A[10]], [5 A[11], 23 A[12], 14 A[13]], [6 A[5], 15 A[6], 24 A[7]], [18 A[8], 0, 9 A[10]], [12 A[11], 21 A[12], 3 A[13]], [22 A[11], 4 A[12], 13 A[13]], [3 A[5], 21 A[6], 12 A[7]], [9 A[8], 0, 18 A[10]], [6 A[11], 24 A[12], 15 A[13]]], [1, 1, 2, 6, 1, 20, 6, 2, 16, 3, 6, 9, 18, 20, 6, 20, 25, 3, 6, 9, 18, 24, 3, 18, 9]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 26}, never show up! Theorem Number, 13, : Let C(n) be the constant term, in x, of n (1/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 47, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [2 A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [8 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [5 A[5], 18 A[6] + 18 A[11] + 5 A[14], 12 A[7] + 9 A[12] + 2 A[15]], [24 A[5] + 12 A[8] + 8 A[16], 9 A[6] + 3 A[9] + 6 A[14] + 8 A[17], A[44]], [A[45], A[46], A[47]], [5 A[5], 18 A[6] + 5 A[14], 3 A[7] + 11 A[15]], [ 18 A[5] + 3 A[8] + 8 A[16], 18 A[6] + 21 A[9] + 8 A[17], 18 A[7] + 12 A[10] + 8 A[18]], [18 A[5] + 18 A[8] + 21 A[11] + 5 A[19], 18 A[6] + 18 A[9] + 12 A[12] + 5 A[20], 18 A[7] + 18 A[10] + 3 A[13] + 5 A[21]], [9 A[5] + 12 A[8] + 8 A[16], 18 A[6] + 12 A[9] + 6 A[14] + 8 A[17] + 6 A[22], 9 A[7] + 12 A[10] + 8 A[18]], [ 21 A[6] + 18 A[8] + 21 A[11] + 8 A[14] + 8 A[19] + 8 A[22], 9 A[7] + 18 A[9] + 21 A[12] + A[15] + 8 A[20] + 4 A[23], 18 A[10] + 3 A[13] + 8 A[21]], [A[5], 6 A[14] + 18 A[6] + 2 A[22], 12 A[7] + 8 A[23]], [ 18 A[5] + 24 A[8] + 6 A[16] + 8 A[24], 21 A[6] + 15 A[9] + 9 A[14] + 11 A[17] + 6 A[22] + 4 A[25], 3 A[7] + 24 A[10] + 7 A[18] + 9 A[23]], [18 A[5] + 21 A[6] + 18 A[8] + 6 A[11] + 7 A[14] + 9 A[16] + A[19] + 4 A[22] + 12 A[24], 3 A[6] + 18 A[9] + 6 A[12] + 12 A[14] + 12 A[17] + A[20] + 6 A[22] + 6 A[25], 15 A[7] + 9 A[10] + 15 A[13] + 16 A[18] + A[21] + 12 A[23] + 10 A[26]], [ 7 A[5] + 18 A[8] + 18 A[11] + 10 A[16] + 13 A[19] + 4 A[24] + 19 A[27], 21 A[6] + 7 A[14] + 6 A[22], 18 A[7] + A[15] + 15 A[23]], [ 6 A[8] + 9 A[11] + A[16] + 3 A[19] + 3 A[27], 21 A[6] + 18 A[7] + 24 A[9] + 18 A[12] + 9 A[14] + 2 A[15] + 11 A[17] + 8 A[20] + 8 A[23] + 7 A[25] + 2 A[28], 12 A[7] + 24 A[10] + 6 A[18] + 15 A[23] + 5 A[26]], [ 15 A[11] + 3 A[16] + 2 A[27] + 3 A[19], 18 A[9] + 2 A[15] + 15 A[14] + 4 A[28] + 6 A[6] + 18 A[7] + 15 A[12] + 13 A[25] + 19 A[17] + 8 A[23] + 2 A[20], 3 A[21] + 3 A[26] + 3 A[7] + 9 A[10] + 3 A[23] + 15 A[13] + 2 A[29] + 6 A[18]], [ 18 A[11] + 2 A[24] + 4 A[16] + 10 A[27] + 4 A[19] + 12 A[8] + 6 A[5], 21 A[9] + 8 A[14] + 24 A[6] + 4 A[25] + 2 A[30] + 9 A[17], 8 A[21] + 8 A[26] + 18 A[7] + 3 A[10] + 12 A[23] + 2 A[29] + 16 A[18]], [ 3 A[11] + A[24] + 4 A[16] + 8 A[27] + 10 A[19] + 9 A[8] + 3 A[5], 9 A[9] + A[15] + 10 A[14] + A[28] + 21 A[6] + 9 A[7] + 3 A[12] + 10 A[25] + A[30] + 16 A[17] + 4 A[23], 2 A[15] + 6 A[26] + 21 A[7] + 2 A[31] + 18 A[10] + 12 A[23] + 21 A[13] + A[29] + 12 A[18]], [ 6 A[32] + 18 A[11] + 3 A[24] + 6 A[27] + 6 A[19] + 9 A[8] + 7 A[5], 6 A[14] + 15 A[6] + 2 A[22], A[15] + 18 A[7] + A[31] + 8 A[23]], [ A[32] + 18 A[11] + 2 A[24] + 7 A[16] + 4 A[27] + 7 A[19] + 6 A[8], 15 A[9] + 8 A[14] + 24 A[6] + 6 A[25] + 2 A[30] + 2 A[33], 6 A[26] + 9 A[7] + 15 A[10] + 6 A[23] + 10 A[18]], [4 A[14] + 12 A[6] + 5 A[32] + 2 A[22] + 6 A[11] + 2 A[24] + 2 A[30] + 7 A[16] + 10 A[27] + 11 A[19] + 9 A[8] + 6 A[5], 9 A[9] + 2 A[15] + 8 A[14] + 3 A[28] + 24 A[6] + 18 A[7] + 6 A[12] + 4 A[25] + 2 A[30] + 8 A[23] + 2 A[33] + A[20], A[21] + 4 A[26] + 9 A[7] + 3 A[34] + 9 A[10] + 6 A[23] + 6 A[13] + 3 A[29] + 4 A[18]], [5 A[32] + 18 A[11] + 2 A[24] + 4 A[16] + 6 A[27] + 6 A[19] + 9 A[8] + 7 A[5], 3 A[6] + 2 A[30], 3 A[7] + 2 A[31]], [2 A[35] + 4 A[14] + 12 A[6] + 4 A[32] + 2 A[22] + 18 A[11] + A[24] + 2 A[30] + 3 A[16] + 4 A[27] + 12 A[19] + 6 A[8] + 3 A[5], 6 A[9] + 2 A[14] + 9 A[6] + 2 A[30] + A[17] + 3 A[33], A[15] + 2 A[26] + 12 A[7] + A[31] + 3 A[34] + 6 A[10] + 6 A[23] + 3 A[18]], [4 A[35] + 4 A[32] + 15 A[11] + A[24] + 2 A[16] + 2 A[27] + 10 A[19] + 9 A[8] + 3 A[5], 2 A[36] + 18 A[9] + 10 A[14] + A[28] + 6 A[12] + 6 A[25] + A[30] + A[17] + 4 A[33] + A[20], A[37] + A[21] + 6 A[26] + 6 A[7] + 8 A[34] + 18 A[10] + 6 A[23] + 6 A[13] + 5 A[29] + 5 A[18]], [ 4 A[32] + 9 A[11] + 3 A[24] + A[16] + 3 A[27] + 3 A[19] + 9 A[8] + 5 A[5], 3 A[14] + 9 A[6] + A[22], A[37] + A[21] + 3 A[26] + 6 A[7] + 4 A[34] + 9 A[10] + 4 A[23] + 9 A[13] + 3 A[29] + A[18]], [A[38] + 2 A[35] + 4 A[14] + 12 A[6] + 6 A[32] + 2 A[22] + 18 A[11] + 3 A[24] + 2 A[30] + A[16] + 4 A[27] + 12 A[19] + 15 A[8] + 6 A[5], A[39] + 6 A[9] + 4 A[14] + 15 A[6] + A[25] + A[30] + A[33], A[37] + A[15] + A[21] + 8 A[26] + 12 A[7] + A[31] + 7 A[34] + 21 A[10] + 9 A[23] + 9 A[13] + 3 A[29] + 8 A[18]], [2 A[38] + 2 A[35] + 4 A[14] + 12 A[6] + 7 A[32] + 2 A[22] + 21 A[11] + 2 A[24] + 2 A[30] + A[16] + 5 A[27] + 12 A[19] + 15 A[8] + 6 A[5], A[39] + 3 A[9] + 2 A[14] + A[28] + 9 A[6] + 3 A[12] + 2 A[30] + A[33], 2 A[37] + 2 A[21] + 6 A[26] + 6 A[7] + 10 A[34] + 24 A[10] + 6 A[23] + 21 A[13] + 7 A[29] + 2 A[40] + 2 A[18]], [2 A[38] + 6 A[32] + 9 A[11] + 3 A[24] + A[16] + 3 A[27] + 3 A[41] + 15 A[8] + 5 A[5], 3 A[14] + 12 A[6] + A[30], A[37] + A[15] + A[21] + 3 A[26] + 12 A[7] + 2 A[31] + 4 A[34] + 9 A[10] + 6 A[23] + 9 A[13] + 3 A[29] + A[18]], [ 6 A[32] + 2 A[24] + A[16] + 5 A[27] + 9 A[41] + 2 A[19] + 12 A[8] + 6 A[5], A[39] + 6 A[9] + 5 A[14] + 18 A[6] + A[25] + 2 A[30] + A[33], A[39] + 12 A[9] + A[42] + 2 A[15] + 2 A[14] + A[28] + 6 A[6] + A[26] + 9 A[7] + 3 A[12] + 2 A[25] + 2 A[34] + 2 A[30] + 2 A[17] + 9 A[10] + 5 A[23] + 4 A[33] + 2 A[40]], [A[35] + 18 A[11] + 5 A[41] + A[19], 2 A[39] + 15 A[9] + 2 A[14] + A[28] + 6 A[6] + 3 A[12] + 2 A[25] + 2 A[30] + 2 A[17] + 5 A[33], A[39] + 12 A[9] + A[42] + A[15] + 2 A[14] + A[28] + 6 A[6] + 6 A[26] + 12 A[7] + 2 A[31] + 3 A[12] + 2 A[25] + 5 A[34] + 2 A[30] + 2 A[17] + 15 A[10] + 8 A[23] + 9 A[13] + 3 A[29] + 2 A[43] + 4 A[33] + 2 A[40]], [A[38] + A[35] + 5 A[32] + 15 A[11] + A[24] + 2 A[16] + 4 A[27] + 3 A[41] + 12 A[8] + 3 A[5], 9 A[9] + 6 A[14] + 3 A[28] + 18 A[6] + 6 A[12] + 2 A[25] + 2 A[17] + 3 A[33] + A[20], 2 A[37] + 2 A[39] + 15 A[9] + 2 A[42] + 4 A[14] + 2 A[28] + 9 A[6] + 4 A[26] + 6 A[7] + 2 A[31] + 6 A[12] + 4 A[25] + 7 A[34] + A[30] + A[17] + 15 A[10] + 4 A[23] + 7 A[29] + A[43] + 5 A[33] + 2 A[40]], [A[38] + A[32] + 3 A[8] + A[5], 3 A[6] + 2 A[30], 3 A[7] + 2 A[31]], [ 3 A[32] + A[16] + 6 A[8], 2 A[36] + 2 A[39] + 21 A[9] + 6 A[14] + 4 A[28] + 12 A[6] + 12 A[12] + 6 A[25] + 10 A[33] + 2 A[46] + 2 A[20], 2 A[39] + 15 A[9] + 2 A[42] + A[15] + 4 A[14] + 2 A[28] + 9 A[6] + 2 A[26] + 6 A[7] + 6 A[12] + 4 A[25] + 4 A[34] + A[30] + A[17] + 9 A[10] + 4 A[23] + 3 A[13] + A[29] + A[43] + 5 A[33] + A[40]], [2 A[45] + A[38] + 4 A[35] + A[32] + 21 A[11] + 4 A[27] + 4 A[41] + A[19] + 3 A[8], 4 A[36] + A[47] + A[37] + A[39] + 12 A[9] + A[42] + A[15] + 4 A[14] + 3 A[28] + 9 A[6] + 5 A[26] + 9 A[7] + 12 A[12] + 4 A[25] + 6 A[34] + A[30] + 15 A[10] + 7 A[23] + 3 A[13] + 6 A[29] + 2 A[43] + 6 A[33] + 2 A[46] + 2 A[40], 2 A[15] + 5 A[26] + 15 A[7] + 2 A[31] + 4 A[34] + 9 A[10] + 9 A[23] + 21 A[13] + 4 A[29] + 2 A[43]]], [1, 1, 2, 8, 1, 5, 5, 2, 1, 4, 8, 25, 19, 14, 23, 5, 7, 19, 5, 10, 4, 19, 19, 1, 23, 11, 4, 8, 23, 13, 13, 25, 26, 23, 19, 2, 8, 14, 25, 10, 23, 19, 13, 23, 19, 11, 8]] For example, C(100000), mudolo , 27, equals , 14 The congruence classes mod, 27, in the following set , {0, 3, 6, 9, 12, 15, 16, 17, 18, 20, 21, 22, 24}, never show up! Theorem Number, 14, : Let C(n) be the constant term, in x, of n (1/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 46, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [2 A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [10 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[32], A[33], A[34]], [A[41], A[42], A[43]], [ 17 A[5], 9 A[5] + 2 A[6] + 3 A[8] + 6 A[14] + 6 A[16], 18 A[5] + 5 A[6] + 20 A[7] + 15 A[8] + 18 A[9] + 4 A[14] + 6 A[15] + 3 A[16]], [ 15 A[5] + 9 A[8] + 5 A[16], 9 A[5] + 13 A[6] + 3 A[8] + 24 A[9] + 2 A[14] + 6 A[16] + 8 A[17], 9 A[5] + 13 A[6] + 9 A[7] + 3 A[8] + 5 A[9] + 23 A[10] + 3 A[12] + 2 A[14] + 6 A[16] + A[17]], [A[44], A[45], A[46]], [7 A[5] + 4 A[6] + 6 A[7] + 17 A[8] + 4 A[9] + 23 A[10] + 9 A[11] + 3 A[12] + 5 A[14] + 6 A[15] + 7 A[16] + 2 A[17] + 4 A[18] + 3 A[19], 9 A[6] + 10 A[14], 9 A[7] + 10 A[15]], [18 A[5] + 20 A[6] + 18 A[8] + 15 A[9] + 16 A[14] + 10 A[16] + 12 A[17] + 18 A[19], 6 A[6] + 12 A[7] + 8 A[9] + 20 A[10] + 2 A[11] + 8 A[12] + 6 A[14] + 9 A[15] + 3 A[16] + 5 A[17] + 7 A[18] + A[19] + 4 A[20], 24 A[5] + A[6] + 8 A[7] + 11 A[8] + 3 A[9] + A[10] + 15 A[11] + 12 A[12] + 14 A[14] + 7 A[15] + 4 A[16] + 18 A[17] + 24 A[19] + 12 A[20]], [12 A[5] + 14 A[6] + 9 A[7] + 4 A[8] + 3 A[9] + 23 A[10] + 8 A[11] + 2 A[12] + 4 A[14] + 3 A[15] + 2 A[16] + 4 A[18] + 2 A[19] + A[20], 24 A[6] + 13 A[7] + 9 A[9] + 21 A[10] + 2 A[11] + 14 A[12] + 9 A[14] + 5 A[15] + 3 A[16] + 6 A[17] + 6 A[18] + A[19] + 5 A[20], 24 A[6] + 22 A[7] + 6 A[9] + 24 A[10] + 2 A[11] + 18 A[12] + 19 A[13] + 9 A[14] + 8 A[15] + 3 A[16] + 6 A[17] + 6 A[18] + A[19] + 6 A[20]], [ 12 A[5] + 22 A[6] + 12 A[7] + 16 A[8] + 13 A[9] + 21 A[10] + 24 A[11] + 12 A[12] + 9 A[14] + 6 A[15] + 7 A[16] + 5 A[17] + 6 A[18] + 9 A[19] + 6 A[20] + A[22], 12 A[5] + 17 A[6] + 19 A[8] + 16 A[9] + 21 A[11] + 7 A[14] + 2 A[16] + 4 A[17] + 12 A[19], 6 A[5] + A[6] + A[7] + 23 A[8] + 14 A[9] + 24 A[10] + 17 A[11] + 24 A[12] + 25 A[13] + 13 A[14] + 11 A[15] + 4 A[16] + 7 A[17] + 8 A[18] + 7 A[19] + 9 A[20] + 2 A[21] + 2 A[22]], [15 A[5] + 4 A[6] + 13 A[7] + 8 A[8] + 2 A[9] + 16 A[11] + 7 A[12] + 2 A[14] + 2 A[15] + 4 A[16] + A[17] + 4 A[19] + 2 A[20] + 6 A[23], 24 A[6] + 2 A[7] + 8 A[9] + 8 A[10] + 16 A[12] + 25 A[13] + 9 A[14] + 4 A[17] + 4 A[18] + 4 A[20] + 2 A[21] + 2 A[23], 18 A[6] + 14 A[7] + 5 A[9] + 11 A[10] + 9 A[12] + 9 A[13] + 6 A[14] + A[17] + 7 A[18] + 3 A[20] + 2 A[21] + 8 A[23]], [19 A[5] + 12 A[6] + 11 A[7] + 4 A[8] + 8 A[10] + 14 A[11] + 25 A[13] + 4 A[14] + A[15] + 2 A[16] + 4 A[18] + 4 A[19] + 2 A[21] + A[22] + 6 A[23], 6 A[14] + 15 A[6] + 2 A[22], 15 A[6] + 18 A[7] + 10 A[9] + 25 A[10] + 18 A[12] + 6 A[14] + 2 A[15] + 2 A[17] + 2 A[18] + 6 A[20] + 10 A[23]], [6 A[5] + 5 A[7] + 12 A[8] + 24 A[9] + 6 A[11] + 10 A[14] + 2 A[16] + 4 A[17] + 3 A[19] + A[22] + 2 A[23] + 4 A[24] + 10 A[25], 15 A[6] + 10 A[7] + 21 A[9] + 6 A[14] + 4 A[17] + 4 A[23] + 6 A[25], 12 A[6] + 9 A[7] + 18 A[9] + 12 A[10] + 12 A[12] + 6 A[14] + 6 A[17] + 7 A[18] + 6 A[20] + 6 A[23] + 6 A[25]], [6 A[5] + 18 A[6] + 9 A[7] + 12 A[8] + A[9] + 4 A[10] + 13 A[11] + 26 A[13] + 8 A[14] + 2 A[15] + 2 A[16] + 4 A[17] + 2 A[18] + 6 A[19] + A[21] + 2 A[22] + 5 A[23] + 2 A[24] + 14 A[25], 20 A[6] + 15 A[9] + 3 A[10] + 3 A[12] + 8 A[14] + 4 A[17] + A[20] + A[22] + 4 A[25] + 3 A[26], 11 A[6] + 19 A[7] + 12 A[9] + 3 A[10] + 2 A[12] + 17 A[13] + 5 A[14] + A[15] + 3 A[17] + 8 A[18] + A[20] + 5 A[21] + A[22] + 14 A[23] + 3 A[25] + 8 A[26]], [2 A[5], 7 A[6] + 5 A[7] + 18 A[9] + 12 A[14] + 4 A[17] + 2 A[22] + 2 A[23] + 4 A[25], 15 A[7] + 2 A[15] + 6 A[23]], [ 3 A[5] + 9 A[8] + 3 A[9] + 3 A[11] + 3 A[16] + A[24] + 3 A[25] + 3 A[27], 18 A[6] + 4 A[7] + 6 A[9] + 3 A[10] + 9 A[12] + 6 A[14] + A[15] + A[17] + 2 A[20] + 2 A[23] + 2 A[25] + 3 A[26] + 5 A[28], 3 A[7] + 9 A[10] + 5 A[18] + 3 A[23] + 3 A[26]], [9 A[9] + 2 A[15] + 7 A[14] + 3 A[28] + 19 A[6] + 4 A[26] + 10 A[7] + 2 A[22] + 5 A[12] + 11 A[11] + 7 A[25] + 4 A[24] + 3 A[16] + 7 A[27] + A[17] + 11 A[10] + 6 A[23] + A[19] + 13 A[8] + 6 A[5] + 2 A[18] + A[20], 11 A[9] + 2 A[15] + 10 A[14] + 7 A[28] + 7 A[26] + 10 A[7] + A[22] + 13 A[12] + 4 A[11] + 7 A[25] + 2 A[24] + 2 A[16] + 4 A[27] + 2 A[17] + 14 A[10] + 6 A[23] + 12 A[8] + 6 A[5] + 2 A[18] + 2 A[20], 8 A[9] + 2 A[15] + 4 A[14] + 4 A[28] + 11 A[6] + 4 A[26] + 11 A[7] + 6 A[12] + 2 A[11] + 4 A[25] + A[24] + A[16] + 2 A[27] + 2 A[17] + 8 A[10] + 7 A[23] + 4 A[13] + 5 A[29] + 6 A[8] + 3 A[5] + 2 A[18] + A[20]], [3 A[9] + 6 A[14] + 18 A[6] + A[22] + 9 A[11] + 3 A[25] + A[30] + A[16] + 5 A[27] + 2 A[19] + 3 A[8], 6 A[9] + A[15] + 7 A[14] + 5 A[28] + 18 A[6] + 3 A[26] + 4 A[7] + 9 A[12] + 3 A[25] + 2 A[30] + A[17] + 3 A[10] + 2 A[23] + 2 A[20], 2 A[9] + 10 A[14] + 4 A[28] + 3 A[6] + A[21] + 6 A[26] + 4 A[7] + 2 A[22] + 8 A[12] + 2 A[11] + 2 A[25] + A[24] + A[30] + A[16] + 2 A[27] + 11 A[10] + 4 A[23] + 3 A[13] + A[29] + 6 A[8] + 3 A[5] + 2 A[18] + 2 A[20]], [ 2 A[9] + 6 A[14] + 17 A[6] + A[22] + 3 A[11] + 2 A[25] + 2 A[24] + 2 A[30] + 3 A[16] + 2 A[27] + 8 A[8] + 3 A[5], 2 A[9] + 5 A[14] + 2 A[28] + 14 A[6] + A[21] + 4 A[26] + 6 A[7] + A[31] + A[22] + 3 A[12] + 2 A[25] + 8 A[10] + 4 A[23] + 3 A[13] + A[29] + 2 A[18], 3 A[9] + A[15] + 7 A[14] + 5 A[28] + 19 A[6] + 4 A[26] + 12 A[7] + A[31] + 2 A[22] + 9 A[12] + A[25] + A[17] + 6 A[10] + 8 A[23] + 3 A[13] + 5 A[29] + A[18] + 2 A[20]] , [5 A[9] + A[15] + 7 A[14] + 3 A[28] + 18 A[6] + A[21] + 5 A[32] + 4 A[26] + 10 A[7] + A[31] + 2 A[22] + 5 A[12] + 5 A[11] + 3 A[25] + A[24] + A[30] + 2 A[16] + 3 A[27] + A[17] + 8 A[10] + 6 A[23] + A[19] + 3 A[13] + A[29] + 15 A[8] + 2 A[5] + 2 A[18] + A[20], 6 A[14] + 18 A[6] + A[22], 3 A[9] + 7 A[14] + 5 A[28] + 19 A[6] + 3 A[26] + 8 A[7] + A[31] + 2 A[22] + 9 A[12] + A[25] + A[17] + 3 A[10] + 4 A[23] + 2 A[20]], [10 A[9] + 2 A[15] + 10 A[14] + 6 A[28] + A[6] + 2 A[21] + 2 A[32] + 8 A[26] + 20 A[7] + 2 A[31] + 2 A[22] + 10 A[12] + 10 A[11] + 6 A[25] + A[24] + A[16] + 6 A[27] + 16 A[10] + 12 A[23] + 2 A[19] + 6 A[13] + 2 A[29] + 9 A[8] + 2 A[33] + 4 A[18] + 2 A[20], 3 A[9] + 2 A[15] + 2 A[14] + 4 A[28] + 6 A[6] + 3 A[26] + 8 A[7] + 6 A[12] + A[25] + A[30] + 3 A[10] + 4 A[23] + A[20], A[21] + 4 A[26] + 7 A[7] + A[31] + 12 A[10] + 5 A[23] + 3 A[13] + A[29] + 3 A[18]], [8 A[9] + 10 A[14] + 6 A[28] + 26 A[6] + 2 A[21] + 3 A[32] + 8 A[26] + 8 A[7] + A[31] + 2 A[22] + 10 A[12] + 7 A[11] + 4 A[25] + 2 A[24] + 4 A[34] + 2 A[30] + A[16] + 3 A[27] + 16 A[10] + 6 A[23] + A[19] + 6 A[13] + 2 A[29] + 10 A[8] + 2 A[33] + 2 A[20], 4 A[9] + 5 A[14] + A[28] + 11 A[6] + 2 A[21] + 5 A[26] + 8 A[7] + A[31] + A[22] + 3 A[12] + 2 A[25] + 4 A[34] + 13 A[10] + 6 A[23] + 6 A[13] + 2 A[29] + A[33], 2 A[15] + 2 A[14] + 4 A[28] + 6 A[6] + 2 A[21] + 7 A[26] + 18 A[7] + A[31] + 8 A[12] + 4 A[34] + A[30] + 15 A[10] + 12 A[23] + 8 A[13] + 2 A[29] + 2 A[20]], [5 A[9] + 2 A[35] + 5 A[14] + 3 A[28] + 14 A[6] + A[21] + 2 A[32] + 4 A[26] + 10 A[7] + 2 A[31] + 5 A[12] + 8 A[11] + 3 A[25] + 2 A[34] + 2 A[30] + A[16] + 4 A[27] + 8 A[10] + 6 A[23] + 3 A[13] + A[29] + 6 A[8] + A[5] + A[33] + A[20], 3 A[9] + 7 A[14] + 5 A[28] + 19 A[6] + 2 A[21] + 7 A[26] + 4 A[7] + 2 A[22] + 9 A[12] + A[25] + 8 A[34] + A[30] + 4 A[23] + 6 A[13] + 2 A[29] + A[33] + 2 A[18] + 2 A[20], A[15] + 2 A[21] + 4 A[26] + 17 A[7] + 2 A[31] + 8 A[34] + 24 A[10] + 10 A[23] + 6 A[13] + 2 A[29] + 2 A[18]], [2 A[36] + 8 A[9] + A[35] + 2 A[15] + 4 A[14] + 8 A[28] + 10 A[6] + 2 A[21] + A[32] + 9 A[26] + 20 A[7] + 2 A[31] + 14 A[12] + 5 A[11] + 4 A[25] + 6 A[34] + A[30] + 3 A[27] + 23 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + 3 A[8] + 2 A[33] + A[18] + A[20], A[36] + 6 A[9] + A[15] + 5 A[14] + 5 A[28] + 12 A[6] + 3 A[26] + 4 A[7] + 9 A[12] + 3 A[25] + A[30] + 3 A[10] + 2 A[23] + A[33] + A[20], 2 A[15] + 2 A[21] + 6 A[26] + 18 A[7] + A[31] + 6 A[34] + 21 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + A[18]], [ A[37] + 4 A[9] + A[15] + 3 A[14] + 3 A[28] + 8 A[6] + 6 A[32] + 4 A[26] + 14 A[7] + 2 A[31] + 5 A[12] + 2 A[11] + 2 A[25] + A[24] + 4 A[34] + A[30] + 2 A[16] + A[27] + 14 A[10] + 8 A[23] + 3 A[13] + A[29] + 17 A[8] + A[33] + A[18] + A[20], 2 A[37] + 8 A[9] + 2 A[15] + 5 A[14] + 6 A[28] + 12 A[6] + 9 A[26] + 20 A[7] + 2 A[31] + 9 A[12] + 4 A[25] + 4 A[34] + A[30] + 17 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + 2 A[33] + A[20], 2 A[14] + 4 A[28] + 6 A[6] + 4 A[26] + 10 A[7] + A[31] + 8 A[12] + A[34] + A[30] + 6 A[10] + 8 A[23] + 5 A[13] + 10 A[29] + 2 A[20]], [2 A[36] + A[37] + A[38] + 2 A[9] + A[35] + 2 A[14] + 4 A[28] + 6 A[6] + 3 A[32] + 4 A[26] + 10 A[7] + 2 A[31] + 8 A[12] + 5 A[11] + 2 A[25] + A[24] + 4 A[34] + A[30] + 3 A[27] + 14 A[10] + 6 A[23] + 3 A[13] + A[29] + 9 A[8] + 2 A[5] + A[18], 2 A[36] + A[37] + A[38] + 2 A[9] + 2 A[35] + 2 A[14] + 4 A[28] + 6 A[6] + 2 A[32] + 4 A[26] + 10 A[7] + 2 A[31] + 8 A[12] + 8 A[11] + 2 A[25] + 4 A[34] + 4 A[27] + 14 A[10] + 6 A[23] + 3 A[13] + A[29] + 6 A[8] + A[18], A[37] + A[15] + 2 A[26] + 14 A[7] + A[31] + 6 A[34] + 18 A[10] + 8 A[23] + 3 A[13] + A[29] + 2 A[18]], [ A[36] + 2 A[37] + 9 A[9] + A[35] + 4 A[14] + 4 A[28] + 10 A[6] + 7 A[26] + 4 A[7] + 6 A[12] + 6 A[11] + 5 A[25] + A[24] + 6 A[34] + A[30] + 4 A[27] + 21 A[10] + 4 A[23] + 6 A[13] + 2 A[29] + 3 A[8] + 2 A[33] + A[18], 2 A[37] + 2 A[38] + 10 A[9] + A[35] + A[15] + 6 A[14] + 4 A[28] + 16 A[6] + 4 A[32] + 8 A[26] + 12 A[7] + A[31] + 4 A[12] + 4 A[11] + 4 A[25] + 8 A[34] + 2 A[30] + 2 A[27] + A[10] + 8 A[23] + 6 A[13] + 2 A[29] + 12 A[8] + 2 A[33] + 2 A[18], A[37] + A[15] + 4 A[26] + 11 A[7] + A[31] + 5 A[34] + 18 A[10] + 7 A[23] + 3 A[13] + A[29] + A[18]], [2 A[36] + A[37] + 3 A[9] + 2 A[35] + A[15] + 4 A[14] + 4 A[28] + 12 A[6] + 5 A[32] + 4 A[26] + 10 A[7] + A[31] + 8 A[12] + 11 A[11] + 3 A[25] + 2 A[24] + 4 A[34] + 2 A[30] + 2 A[16] + 5 A[27] + 14 A[10] + 6 A[23] + 3 A[13] + A[29] + 16 A[8] + A[40], A[36] + A[37] + A[38] + 9 A[9] + 2 A[35] + A[15] + 4 A[14] + 4 A[28] + 8 A[6] + 2 A[32] + 5 A[26] + 10 A[7] + A[31] + 8 A[12] + 8 A[11] + 5 A[25] + 6 A[34] + 4 A[27] + 21 A[10] + 6 A[23] + 3 A[13] + A[29] + 6 A[8] + 2 A[33] + 2 A[40], A[38] + 5 A[9] + 2 A[35] + 2 A[15] + A[14] + A[28] + 2 A[6] + 2 A[32] + 2 A[26] + 10 A[7] + A[12] + 8 A[11] + 3 A[25] + 4 A[27] + 2 A[10] + 6 A[23] + 2 A[13] + 6 A[8] + A[33]], [A[5], 2 A[36] + A[37] + 3 A[9] + 2 A[15] + 5 A[14] + 5 A[28] + 15 A[6] + 5 A[26] + 14 A[7] + A[31] + 9 A[12] + A[25] + 6 A[34] + 2 A[30] + 21 A[10] + 8 A[23] + 3 A[13] + A[29] + A[33] + 2 A[40], 2 A[37] + 4 A[26] + 13 A[7] + 2 A[31] + 6 A[34] + 18 A[10] + 8 A[23] + 6 A[13] + 2 A[29] + A[40]], [2 A[36] + A[37] + 10 A[9] + 2 A[35] + 2 A[14] + 6 A[28] + 4 A[6] + A[32] + 6 A[26] + 6 A[7] + A[31] + 10 A[12] + 13 A[11] + 6 A[25] + 2 A[34] + 7 A[27] + 10 A[10] + 4 A[23] + A[41] + 3 A[13] + A[29] + 3 A[8] + 2 A[33], A[37] + A[38] + 11 A[9] + 2 A[35] + 6 A[14] + 2 A[28] + 14 A[6] + 2 A[32] + 4 A[26] + 6 A[7] + A[31] + 2 A[12] + 8 A[11] + 6 A[25] + 4 A[34] + A[30] + 4 A[27] + 14 A[10] + 4 A[23] + 3 A[13] + A[29] + 6 A[8] + 2 A[33] + A[40], 2 A[37] + A[15] + 6 A[26] + 18 A[7] + 2 A[31] + 6 A[34] + 21 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + A[40]], [A[36] + 2 A[37] + 3 A[9] + A[35] + 2 A[15] + 2 A[14] + 2 A[28] + 6 A[6] + 2 A[32] + 5 A[26] + 20 A[7] + 2 A[31] + 4 A[12] + 10 A[11] + 3 A[25] + A[24] + 8 A[34] + A[30] + 5 A[27] + 25 A[10] + 12 A[23] + A[41] + 6 A[13] + 2 A[29] + 5 A[8] + 2 A[40], 2 A[37] + 2 A[38] + 6 A[9] + A[35] + 2 A[15] + 6 A[14] + 3 A[28] + 16 A[6] + 4 A[32] + 7 A[26] + 20 A[7] + 2 A[31] + 4 A[12] + 4 A[11] + 4 A[25] + 6 A[34] + 2 A[30] + 2 A[27] + 21 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + 12 A[8] + A[33] + A[40], A[37] + 2 A[38] + 7 A[9] + A[35] + A[15] + 2 A[14] + 2 A[28] + 4 A[6] + 4 A[32] + 4 A[26] + 16 A[7] + 2 A[31] + 2 A[12] + 4 A[11] + 3 A[25] + 5 A[34] + 2 A[27] + 16 A[10] + 10 A[23] + 4 A[13] + A[29] + 12 A[8] + 2 A[33] + A[40]], [2 A[36] + A[37] + A[38] + 5 A[9] + 2 A[35] + A[15] + 2 A[14] + 4 A[28] + 6 A[6] + 3 A[32] + 4 A[26] + 10 A[7] + A[31] + 8 A[12] + 17 A[11] + 5 A[25] + A[24] + 4 A[34] + A[30] + 9 A[27] + 14 A[10] + 6 A[23] + 2 A[41] + 3 A[13] + A[29] + 9 A[8] + 2 A[5] + A[40], 6 A[14] + 18 A[6] + 2 A[30], 12 A[7] + 2 A[31] + 6 A[23]], [2 A[38] + 3 A[9] + 2 A[35] + 6 A[32] + 12 A[11] + 3 A[25] + 6 A[27] + A[41] + 18 A[8], 9 A[9] + 7 A[14] + 3 A[28] + 18 A[6] + 3 A[26] + 3 A[12] + 3 A[25] + 2 A[30] + 3 A[10] + 2 A[33], 2 A[15] + 12 A[7] + A[31] + 6 A[34] + 18 A[10] + 6 A[23] + 2 A[40]], [ 2 A[38] + 11 A[9] + 2 A[35] + 2 A[15] + 6 A[14] + 2 A[28] + 16 A[6] + 4 A[32] + 2 A[26] + 8 A[7] + 2 A[12] + 20 A[11] + 7 A[25] + 4 A[34] + 2 A[30] + 10 A[27] + 14 A[10] + 4 A[23] + 2 A[41] + 12 A[8] + 2 A[33] + 2 A[40], A[37] + A[38] + 5 A[9] + 2 A[35] + 2 A[15] + 3 A[14] + 3 A[28] + 8 A[6] + 2 A[32] + 4 A[26] + 14 A[7] + A[31] + 5 A[12] + 8 A[11] + 3 A[25] + 4 A[34] + A[30] + 4 A[27] + 14 A[10] + 8 A[23] + 3 A[13] + A[29] + 6 A[8] + A[33] + A[40], 2 A[37] + 6 A[26] + 14 A[7] + 2 A[31] + 10 A[34] + 3 A[10] + 10 A[23] + 9 A[13] + 3 A[29] + 2 A[40]]], [1, 1, 2, 10, 1, 17, 19, 2, 25, 2, 10, 26, 19, 26, 1, 17, 10, 17, 19, 8, 19, 25, 2, 25, 2, 25, 2, 25, 2, 17, 10, 26, 1, 26, 19, 8, 19, 26, 1, 26, 1, 26, 1, 17, 10, 17]] For example, C(100000), mudolo , 27, equals , 25 The congruence classes mod, 27, in the following set , {0, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 23, 24}, never show up! Theorem Number, 15, : Let C(n) be the constant term, in x, of n (1/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[6], A[7]], 0, 0, [3 A[5], 3 A[6], 3 A[7]], 0, 0, [9 A[5], 9 A[6], 9 A[7]], 0, 0, 0, 0], [1, 1, 3, 9, 1, 0, 0, 3, 0, 0, 9, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 16, : Let C(n) be the constant term, in x, of n (1/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 28, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [3 A[5], A[22], A[23]], [24 A[5] + 12 A[8], 15 A[6] + 12 A[9], A[24]], [A[25], A[26], A[27]], [11 A[5], 20 A[6] + 9 A[11], A[28]], [21 A[5] + 14 A[8], 3 A[6] + 23 A[9], 12 A[7] + 5 A[10]], [ 18 A[5] + 24 A[8] + 17 A[11], 18 A[6] + 6 A[9] + 26 A[12], 18 A[7] + 15 A[10] + 8 A[13]], [24 A[5] + A[8], 15 A[6] + A[9], 6 A[7] + A[10]], [21 A[8] + A[11], 12 A[9] + A[12], 3 A[10] + A[13]], [18 A[5], 18 A[6] + 18 A[11], 18 A[7] + 9 A[12]], [6 A[5] + 18 A[8], 24 A[6] + 18 A[9], 15 A[7] + 18 A[10]], [ 18 A[5] + 21 A[8] + 18 A[11], 18 A[6] + 12 A[9] + 18 A[12], 18 A[7] + 3 A[10] + 18 A[13]], [20 A[5], 2 A[6] + 9 A[11], 11 A[7] + 18 A[12]], [12 A[5] + 23 A[8], 21 A[6] + 5 A[9], 3 A[7] + 14 A[10]], [ 18 A[5] + 6 A[8] + 26 A[11], 18 A[6] + 15 A[9] + 8 A[12], 18 A[7] + 24 A[10] + 17 A[13]], [3 A[8], 3 A[9], 3 A[10]], [3 A[11], 3 A[12], 3 A[13]], [15 A[11], 15 A[12], 15 A[13]], [15 A[5], 15 A[6], 15 A[7]], [24 A[8], 24 A[9], 24 A[10]], [6 A[11], 6 A[12], 6 A[13]], [15 A[8] + 2 A[11], 24 A[9] + 2 A[12], 6 A[10] + 2 A[13]]], [1, 1, 3, 11, 1, 18, 20, 3, 6, 15, 11, 9, 7, 0, 20, 18, 6, 9, 20, 0, 25, 9, 6, 3, 15, 18, 12, 13]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 8, 10, 14, 16, 17, 19, 21, 22, 23, 24, 26}, never show up! Theorem Number, 17, : Let C(n) be the constant term, in x, of n (1/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 43, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [3 A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [13 A[5], A[30], A[31]], [A[32], A[33], A[18]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [9 A[5], 9 A[6] + 18 A[11], 9 A[7] + 9 A[12]], [6 A[5] + 9 A[8], 18 A[6] + 9 A[9] + 6 A[14], 15 A[7] + 9 A[10]], [ 18 A[8] + 9 A[11] + 3 A[16], 9 A[9] + 9 A[12] + 3 A[17], 9 A[13] + 3 A[18]] , [25 A[5], 12 A[6] + 4 A[14], 7 A[15]], [15 A[8] + 7 A[16], 6 A[9] + 7 A[17], 24 A[10] + 7 A[18]], [ 18 A[5] + 15 A[11] + 4 A[19], 18 A[6] + 6 A[12] + 4 A[20], 18 A[7] + 24 A[13] + 4 A[21]], [18 A[5] + 3 A[16], 3 A[17] + 6 A[22], 18 A[7] + 3 A[18]], [ 18 A[6] + 9 A[8] + 9 A[11] + 6 A[14] + 3 A[19] + A[22], 3 A[7] + 9 A[9] + 9 A[12] + 3 A[20] + 8 A[23], 9 A[10] + 3 A[21]], [24 A[5], 18 A[6] + 6 A[14], 6 A[7] + 6 A[23]], [15 A[5] + 8 A[24], 18 A[6] + 6 A[14] + 3 A[22] + 8 A[25], 9 A[10] + 6 A[18] + 6 A[23]], [ 9 A[5] + 15 A[6] + 18 A[11] + 6 A[14] + 6 A[19] + 2 A[22] + 3 A[24], 3 A[7] + 18 A[12] + 3 A[18] + 6 A[20] + 3 A[22] + 4 A[23] + 3 A[25] + 2 A[26], 6 A[7] + 3 A[18] + 6 A[21] + 6 A[23] + 5 A[26]], [ 9 A[5] + 18 A[6] + 6 A[14] + 6 A[19] + A[22] + 8 A[24] + 4 A[27], 9 A[6] + 3 A[14], 3 A[7] + 3 A[23]], [ 9 A[5] + 18 A[6] + 6 A[14] + 6 A[19] + A[22] + 6 A[24] + 4 A[27], 9 A[6] + 3 A[14] + 3 A[22] + 7 A[25], 6 A[7] + 6 A[18] + 9 A[23] + 11 A[26] ], [6 A[14] + 18 A[6] + A[22] + 4 A[24] + 5 A[27] + 6 A[19] + 12 A[5], 6 A[14] + A[28] + 18 A[6] + 6 A[22] + 8 A[25], 6 A[26] + 6 A[23] + A[29] + 6 A[18]], [2 A[24] + A[16] + 6 A[8] + 6 A[5], 6 A[9] + 4 A[14] + 12 A[6] + 2 A[25] + 2 A[30] + A[17], 6 A[26] + 6 A[10] + 6 A[23] + 7 A[18]], [4 A[14] + 12 A[6] + A[22] + 6 A[11] + 3 A[24] + 2 A[30] + 6 A[27] + 7 A[19] + 9 A[5], 2 A[14] + 2 A[28] + 6 A[6] + 6 A[12] + A[25] + A[30] + A[20], 7 A[21] + 15 A[13]], [4 A[32] + 10 A[24] + 2 A[16] + 18 A[8] + 12 A[5], 4 A[14] + 12 A[6] + 4 A[22] + 2 A[30], A[15] + 6 A[7] + 2 A[31] + 4 A[23]], [4 A[32] + 4 A[24] + 2 A[16] + 18 A[8] + 18 A[5], 4 A[14] + 12 A[6] + 3 A[22] + 3 A[25] + 2 A[30], 2 A[15] + 6 A[21] + 2 A[26] + 6 A[7] + A[31] + 9 A[23] + A[29]], [5 A[14] + 12 A[6] + 4 A[32] + 2 A[22] + 5 A[24] + A[30] + 2 A[16] + 18 A[8] + 15 A[5], 2 A[15] + 2 A[14] + 6 A[6] + 9 A[7] + A[31] + A[25] + A[30] + 7 A[23], A[15] + 5 A[26] + 6 A[7] + 2 A[31] + 4 A[34] + 18 A[10] + 9 A[23] + 9 A[13] + 2 A[18]], [6 A[35] + 5 A[14] + 12 A[6] + 4 A[32] + 2 A[22] + 2 A[24] + A[30] + 2 A[16] + 5 A[27] + 6 A[19] + 18 A[8] + A[5], A[14] + 3 A[6] + 3 A[22], 2 A[15] + 12 A[7] + 2 A[31] + 9 A[23]], [6 A[35] + 5 A[32] + 18 A[11] + 2 A[24] + 2 A[16] + 6 A[27] + 3 A[19] + 15 A[8] + 6 A[5], 15 A[9] + 4 A[14] + 12 A[6] + 2 A[25] + 2 A[30] + 2 A[17] + 5 A[33], 2 A[15] + 4 A[26] + 6 A[7] + A[31] + A[34] + 6 A[10] + 9 A[23]], [4 A[35] + 4 A[14] + 12 A[6] + 2 A[32] + A[22] + 15 A[11] + 2 A[30] + A[16] + 2 A[27] + 3 A[19] + 9 A[8], A[36] + 2 A[15] + 4 A[14] + 4 A[28] + 12 A[6] + 9 A[7] + A[31] + 6 A[12] + 2 A[25] + 2 A[30] + 7 A[23], 5 A[37] + 2 A[15] + 2 A[21] + 12 A[7] + A[31] + 9 A[23] + 15 A[13]], [ 0, 4 A[14] + 12 A[6] + A[22] + 2 A[30], 2 A[15] + 9 A[7] + A[31] + 7 A[23]] , [8 A[35] + 5 A[14] + 12 A[6] + 2 A[22] + 9 A[11] + 3 A[24] + A[30] + 6 A[27] + 7 A[19] + 6 A[5], 4 A[14] + 12 A[6] + 2 A[30], A[15] + 6 A[7] + 2 A[31] + 6 A[23]], [A[38] + 6 A[35] + 5 A[14] + 12 A[6] + 4 A[32] + 2 A[22] + 2 A[24] + A[30] + A[16] + 5 A[27] + 6 A[19] + 12 A[8] + 6 A[5], 2 A[14] + 6 A[6] + A[25] + A[30], 2 A[15] + A[26] + 6 A[7] + A[31] + 4 A[34] + 12 A[10] + 6 A[23] + A[40] + A[18]], [2 A[38] + 5 A[35] + 8 A[32] + 2 A[24] + 2 A[16] + 4 A[27] + 6 A[41] + A[19] + 24 A[8] + 4 A[5], 2 A[36] + 4 A[37] + 4 A[14] + 4 A[28] + 9 A[6] + 2 A[21] + 4 A[26] + 9 A[12] + 2 A[25] + 8 A[34] + 24 A[10] + 4 A[23] + 9 A[13] + 2 A[29] + 2 A[40] + 2 A[18] + A[20], 2 A[37] + A[15] + A[21] + 2 A[26] + 3 A[7] + 4 A[34] + 12 A[10] + 6 A[23] + 18 A[13] + A[29] + A[40] + A[18]], [5 A[32] + 2 A[16] + 15 A[8], 8 A[36] + A[39] + 9 A[9] + 2 A[42] + 5 A[14] + 8 A[28] + 12 A[6] + 24 A[12] + 7 A[25] + A[30] + 3 A[33] + 2 A[20], 3 A[34] + 9 A[10] + A[40]], [2 A[38] + 7 A[35] + 8 A[32] + 6 A[11] + 2 A[24] + 2 A[16] + 4 A[27] + 6 A[41] + 24 A[8] + 6 A[5], 7 A[36] + 2 A[42] + 3 A[14] + 6 A[28] + 6 A[6] + 21 A[12] + 3 A[25] + A[20] , A[37] + 2 A[15] + 4 A[26] + 6 A[7] + A[31] + 8 A[34] + 24 A[10] + 9 A[23] + 15 A[13] + 2 A[29] + 2 A[40] + 2 A[18]]], [1, 1, 3, 13, 1, 9, 25, 3, 24, 12, 13, 9, 22, 0, 7, 9, 6, 9, 25, 0, 16, 18, 21, 24, 18, 15, 12, 18, 12, 0, 19, 9, 6, 9, 22, 0, 22, 0, 6, 0, 7, 0, 25]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 8, 10, 11, 14, 17, 20, 23, 26}, never show up! Theorem Number, 18, : Let C(n) be the constant term, in x, of n (1/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[6], A[7]], 0, 0, [3 A[5], 3 A[6], 3 A[7]], 0, 0, [15 A[5], 15 A[6], 15 A[7]], 0, 0, 0, 0], [1, 1, 3, 15, 1, 0, 0, 3, 0, 0, 15, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 19, : Let C(n) be the constant term, in x, of n (2/x + x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 40, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [0, A[22], A[23]], [24 A[5], 15 A[6], 6 A[7]], [A[24], A[25], A[26]], [4 A[5], A[27], A[28]], [A[29], A[17], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [0, 18 A[11], 9 A[12]], [6 A[5], 18 A[6] + 6 A[14], 9 A[7] + 6 A[15]], [18 A[8] + 3 A[16], 9 A[9] + 3 A[17], 3 A[18]], [25 A[5], 12 A[6] + 4 A[14], 7 A[15]], [15 A[8] + 7 A[16], 6 A[9] + 7 A[17], 24 A[10] + 7 A[18]], [ 18 A[5] + 15 A[11] + 4 A[19], 18 A[6] + 6 A[12] + 4 A[20], 18 A[7] + 24 A[13] + 4 A[21]], [18 A[5], 18 A[6], 18 A[7]], [9 A[8], 9 A[9], 9 A[10]], 0, [18 A[5], 18 A[6], 18 A[7]], [3 A[16] + 2 A[24], 6 A[17] + 4 A[25], 3 A[26]], [ 6 A[8] + 4 A[16] + 4 A[24], 6 A[9] + A[17] + 2 A[25], 6 A[10] + A[18] + 8 A[26]], [ 15 A[11] + 7 A[19] + A[22], 15 A[12] + 7 A[20] + 2 A[23], 6 A[13] + 7 A[21] ], [0, A[22], A[23]], [6 A[5], 18 A[6] + 6 A[14], 2 A[15] + 4 A[28] + 15 A[7]], [ 5 A[24] + 2 A[16] + 4 A[29] + 18 A[8], 3 A[25] + 3 A[17], 3 A[26] + 2 A[31] + 9 A[10] + A[18]], [ 2 A[32] + 18 A[11] + 4 A[19] + 4 A[5], A[14] + 3 A[6], A[15] + 3 A[28] + 9 A[7]], [6 A[24] + A[16] + 3 A[29] + 15 A[8], 15 A[9] + 4 A[25] + 5 A[30] + 5 A[17], 4 A[26] + A[31] + 6 A[10]], [ 4 A[32] + A[22] + 24 A[11] + 9 A[19], 5 A[14] + 18 A[6] + 15 A[12] + A[27] + A[23] + 3 A[33] + 7 A[20], A[15] + 2 A[28] + 2 A[21] + 9 A[7] + 5 A[34] + 24 A[13]], [0, A[22], A[23]], [6 A[5], 18 A[6] + 6 A[14], 2 A[15] + 4 A[28] + 15 A[7]], [ A[35] + 5 A[24] + 2 A[29] + 12 A[8], A[36] + 12 A[9] + A[25] + 4 A[30] + A[17], 2 A[37] + A[26] + 8 A[31] + 24 A[10] + 2 A[18]], [7 A[5], 8 A[14] + 2 A[27], A[28] + 3 A[7]], [ 2 A[35] + 4 A[24] + 5 A[29] + 21 A[8], 15 A[9] + 5 A[30] + 2 A[17], A[37] + 3 A[31] + 9 A[10]], [8 A[38] + 4 A[32] + 3 A[11] + 7 A[19], 2 A[39] + 4 A[14] + 15 A[6] + 12 A[12] + 2 A[27] + 2 A[23] + 2 A[33], A[15] + 2 A[28] + 9 A[7] + 2 A[34] + 12 A[13] + 2 A[40]]], [1, 1, 0, 4, 1, 0, 25, 0, 24, 0, 4, 0, 13, 0, 7, 0, 6, 0, 25, 0, 16, 18, 0, 0, 18, 0, 0, 10, 0, 6, 0, 13, 0, 13, 0, 6, 0, 7, 0, 25]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 3, 5, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, 22, 23, 26}, never show up! Theorem Number, 20, : Let C(n) be the constant term, in x, of n (2/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 12, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], 0, [A[8], A[9], A[10]], [A[5], A[11], A[12]], 0, 0, [12 A[5], 12 A[6], 12 A[7]], 0, 0, 0, 0], [1, 1, 0, 12, 1, 0, 0, 12, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 21, : Let C(n) be the constant term, in x, of n (2/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 28, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [A[5], A[22], A[23]], [12 A[5] + 16 A[8], 21 A[6] + 7 A[9], A[24]], [A[25], A[26], A[27]], [A[5], 10 A[6] + 18 A[11], A[15]], [15 A[5] + 13 A[8], 6 A[6] + 22 A[9], 24 A[7] + 4 A[10]], [ 9 A[5] + 3 A[8] + 16 A[11], 9 A[6] + 21 A[9] + 25 A[12], 9 A[7] + 12 A[10] + 7 A[13]], [15 A[5] + 13 A[8], 6 A[6] + 22 A[9], 24 A[7] + 4 A[10]], [ 18 A[5] + 3 A[8] + 7 A[11], 18 A[6] + 21 A[9] + 16 A[12], 18 A[7] + 12 A[10] + 25 A[13]], [A[5], 19 A[6] + 9 A[11], 10 A[7] + 18 A[12]], [3 A[5] + 25 A[8], 12 A[6] + 16 A[9], 21 A[7] + 7 A[10]], [6 A[8] + 22 A[11], 15 A[9] + 13 A[12], 24 A[10] + 4 A[13]], [A[5], 10 A[6] + 18 A[11], 19 A[7] + 9 A[12]], [6 A[5] + 22 A[8], 24 A[6] + 4 A[9], 15 A[7] + 13 A[10]], [ 9 A[5] + 12 A[8] + 7 A[11], 9 A[6] + 3 A[9] + 16 A[12], 9 A[7] + 21 A[10] + 25 A[13]], [6 A[5] + 22 A[8], 24 A[6] + 4 A[9], 15 A[7] + 13 A[10]], [ 18 A[5] + 12 A[8] + 25 A[11], 18 A[6] + 3 A[9] + 7 A[12], 18 A[7] + 21 A[10] + 16 A[13]], [15 A[8] + 13 A[11], 24 A[9] + 4 A[12], 6 A[10] + 22 A[13]], [A[5], 10 A[6] + 18 A[11], 19 A[7] + 9 A[12]], [24 A[5] + 4 A[8], 15 A[6] + 13 A[9], 6 A[7] + 22 A[10]], [ 9 A[5] + 21 A[8] + 25 A[11], 9 A[6] + 12 A[9] + 7 A[12], 9 A[7] + 3 A[10] + 16 A[13]], [18 A[5] + 3 A[8] + 7 A[11], 18 A[6] + 21 A[9] + 16 A[12], 18 A[7] + 12 A[10] + 25 A[13]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] For example, C(100000), mudolo , 27, equals , 1 The congruence classes mod, 27, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 22, : Let C(n) be the constant term, in x, of n (2/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 47, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [5 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [13 A[5], 21 A[6] + 18 A[11] + A[14], 3 A[7] + 9 A[12] + A[15]], [15 A[5] + 24 A[8] + A[16], 6 A[9] + 6 A[14] + A[17], A[44]], [A[45], A[46], A[47]], [14 A[5], 5 A[14], 12 A[7] + 11 A[15]], [ 18 A[5] + 12 A[8] + 8 A[16], 18 A[6] + 3 A[9] + 8 A[17], 18 A[7] + 21 A[10] + 8 A[18]], [18 A[5] + 3 A[11] + 5 A[19], 18 A[6] + 21 A[12] + 5 A[20], 18 A[7] + 12 A[13] + 5 A[21]], [6 A[8] + 4 A[16], 6 A[9] + 4 A[17], 6 A[10] + 4 A[18]], [15 A[11] + 4 A[19], 15 A[12] + 4 A[20], 15 A[13] + 4 A[21]], [A[5] + 10 A[16] + 11 A[24], 18 A[6] + 3 A[14] + A[22], 21 A[7] + A[23]], [ 24 A[5] + 6 A[8] + 4 A[16] + 15 A[24], 15 A[6] + 15 A[9] + 6 A[14] + 2 A[17] + 11 A[25], 9 A[7] + 6 A[10] + A[18] + 6 A[23]], [21 A[5] + 9 A[6] + 15 A[11] + 14 A[14] + 5 A[16] + 7 A[17] + 4 A[19] + A[22] + 13 A[24] + 14 A[25], 18 A[6] + 9 A[9] + 15 A[12] + 9 A[14] + 6 A[17] + 4 A[20] + 15 A[25], 21 A[7] + 24 A[13] + 8 A[18] + 4 A[21] + 9 A[23] + 10 A[26]], [ 26 A[5] + 10 A[16] + 11 A[24], 9 A[6] + 2 A[14], 21 A[7] + 2 A[15] + 6 A[23]], [12 A[5] + 24 A[6] + 21 A[8] + 18 A[9] + 9 A[11] + 18 A[12] + 10 A[14] + 4 A[16] + 6 A[17] + 2 A[19] + 4 A[20] + 2 A[22] + 4 A[24] + 6 A[25] + A[27] + 2 A[28], 18 A[6] + 12 A[9] + 9 A[12] + 6 A[14] + 3 A[17] + 2 A[20] + 2 A[25] + A[28] , 18 A[7] + 12 A[10] + 6 A[18] + 6 A[23] + 2 A[26]], [18 A[9] + 10 A[14] + 2 A[28] + 24 A[6] + 2 A[22] + 18 A[12] + 12 A[11] + 6 A[25] + 8 A[24] + 7 A[16] + 2 A[27] + 6 A[17] + 6 A[19] + 24 A[5] + 4 A[20], 2 A[15] + 12 A[14] + 2 A[28] + 3 A[6] + 4 A[26] + 12 A[12] + 8 A[25] + 7 A[17] + 18 A[10] + 10 A[23] + 2 A[18] + 6 A[20], A[21] + 8 A[26] + 3 A[7] + 9 A[10] + 12 A[23] + 3 A[13] + 4 A[29] + 10 A[18]], [18 A[9] + 10 A[14] + 2 A[28] + 24 A[6] + 2 A[22] + 18 A[12] + 9 A[11] + 6 A[25] + 8 A[24] + 3 A[16] + A[27] + 6 A[17] + 2 A[19] + 3 A[8] + 24 A[5] + 4 A[20], 21 A[9] + 6 A[14] + A[28] + 12 A[6] + 9 A[12] + 6 A[25] + 2 A[17] + 2 A[20] , 8 A[26] + 3 A[7] + 3 A[10] + 12 A[23] + 6 A[18]], [18 A[9] + 8 A[14] + A[28] + 18 A[6] + A[22] + 9 A[12] + 21 A[11] + 6 A[25] + 8 A[24] + 4 A[16] + 3 A[27] + 9 A[17] + 8 A[19] + 24 A[5] + 2 A[20], 9 A[9] + 12 A[14] + 3 A[28] + 2 A[26] + 24 A[7] + 2 A[31] + 21 A[12] + 10 A[25] + 5 A[17] + 9 A[10] + 8 A[23] + A[18] + 8 A[20], 6 A[21] + 5 A[26] + 24 A[7] + 18 A[10] + 9 A[23] + 12 A[13] + 2 A[29] + 10 A[18]], [ 3 A[32] + 3 A[24] + 9 A[8] + 8 A[5], 6 A[14] + 18 A[6] + 2 A[22], 4 A[26] + 21 A[7] + 18 A[10] + 8 A[23] + 2 A[18]], [ A[32] + 2 A[24] + A[16] + 12 A[8], 3 A[9] + 6 A[14] + A[28] + 12 A[6] + 9 A[12] + 8 A[25] + 3 A[17] + 2 A[20], 4 A[26] + 18 A[7] + 12 A[10] + 6 A[23] + 4 A[18]], [ 4 A[32] + 21 A[11] + 4 A[16] + 4 A[27] + 7 A[19] + 18 A[8], 9 A[9] + 12 A[14] + 5 A[28] + 4 A[26] + 18 A[7] + A[31] + 3 A[12] + 10 A[25] + 5 A[17] + 18 A[10] + 7 A[23] + 2 A[18] + 9 A[20], 7 A[21] + 6 A[26] + 12 A[7] + A[34] + 6 A[23] + 12 A[13] + 4 A[29] + 4 A[18]], [ 3 A[32] + 3 A[24] + 9 A[8] + 4 A[5], A[14] + 3 A[6], 9 A[7] + 2 A[31] + 3 A[23]], [ 9 A[35] + 4 A[32] + A[24] + 7 A[16] + 9 A[27] + 15 A[8] + 3 A[5], 3 A[36] + 15 A[9] + 6 A[14] + 3 A[28] + 15 A[6] + 9 A[12] + 4 A[25] + A[17] + A[33], 4 A[21] + 6 A[26] + 12 A[7] + 2 A[34] + 24 A[10] + 6 A[23] + 2 A[29] + 3 A[18]], [16 A[35] + 4 A[32] + 24 A[11] + 2 A[24] + 8 A[16] + 10 A[27] + 4 A[19] + 18 A[8] + 6 A[5], 6 A[36] + 5 A[37] + 18 A[9] + 10 A[35] + 11 A[14] + 4 A[28] + 3 A[6] + 2 A[21] + 4 A[32] + 2 A[26] + 21 A[7] + A[31] + A[22] + 15 A[12] + 9 A[11] + 5 A[25] + 2 A[24] + 2 A[16] + 6 A[27] + 2 A[17] + 9 A[10] + 7 A[23] + 4 A[19] + 9 A[13] + 3 A[29] + 18 A[8] + 6 A[5] + A[33] + A[18], 4 A[37] + A[21] + 3 A[26] + 6 A[7] + A[34] + 9 A[10] + 3 A[23] + 15 A[13] + A[29] + A[18]], [ 6 A[35] + 2 A[32] + 18 A[11] + 2 A[24] + 6 A[16] + 6 A[27] + 9 A[8] + A[5], 3 A[14] + 9 A[6] + A[22], 7 A[37] + A[21] + 4 A[26] + 18 A[7] + 18 A[10] + 7 A[23] + 18 A[13] + 6 A[29] + 2 A[18]], [ 6 A[35] + 2 A[32] + 18 A[11] + A[24] + 2 A[16] + 6 A[27] + 15 A[8], 3 A[36] + 2 A[38] + 15 A[9] + 8 A[35] + 7 A[14] + 3 A[28] + 18 A[6] + 4 A[32] + 2 A[22] + 9 A[12] + 4 A[25] + 2 A[16] + 6 A[27] + A[17] + 2 A[19] + 15 A[8] + A[33], 6 A[37] + 6 A[26] + 24 A[7] + A[34] + 24 A[10] + 9 A[23] + 18 A[13] + 6 A[29] + 2 A[18]], [ 4 A[35] + 15 A[11] + 6 A[16] + 5 A[27], 4 A[36] + A[39] + A[38] + 21 A[9] + 10 A[35] + 8 A[14] + 5 A[28] + 18 A[6] + 2 A[32] + A[22] + 15 A[12] + 9 A[11] + 6 A[25] + A[16] + 6 A[27] + A[17] + 4 A[19] + 12 A[8] + 2 A[33], 2 A[37] + A[21] + 3 A[26] + 6 A[7] + A[34] + 9 A[10] + 3 A[23] + 6 A[13] + 2 A[29] + A[18]], [2 A[38] + 3 A[35] + 5 A[32] + 9 A[11] + 2 A[24] + A[16] + 3 A[27] + 15 A[8] + 2 A[5], 3 A[36] + A[39] + 2 A[38] + 21 A[9] + 11 A[35] + 10 A[14] + 3 A[28] + 24 A[6] + 3 A[32] + A[22] + 9 A[12] + 12 A[11] + 7 A[25] + A[16] + 7 A[27] + 2 A[17] + 4 A[41] + 15 A[8] + A[33], 3 A[37] + 7 A[26] + 21 A[7] + A[31] + A[34] + 21 A[10] + 9 A[23] + 9 A[13] + 3 A[29] + A[40] + 2 A[18]], [A[38] + 11 A[35] + 4 A[32] + 24 A[11] + 2 A[16] + 7 A[27] + 8 A[41] + 2 A[19] + 15 A[8], 6 A[36] + 2 A[39] + 2 A[38] + 9 A[9] + 11 A[35] + 14 A[14] + 6 A[28] + 3 A[6] + 3 A[32] + A[22] + 18 A[12] + 12 A[11] + 12 A[25] + A[16] + 7 A[27] + 2 A[17] + 4 A[41] + 15 A[8] + 2 A[33], 7 A[37] + A[21] + 12 A[26] + 24 A[7] + 2 A[34] + 9 A[10] + 12 A[23] + 18 A[13] + 6 A[29] + 2 A[40] + 2 A[18]], [A[38] + 8 A[35] + 2 A[32] + 18 A[11] + A[16] + 6 A[27] + 8 A[41] + 2 A[19] + 12 A[8], A[36] + 6 A[37] + A[38] + 9 A[9] + 7 A[35] + 4 A[42] + 7 A[14] + A[28] + 18 A[6] + 2 A[21] + 3 A[32] + 6 A[26] + 18 A[7] + 2 A[31] + 2 A[22] + 15 A[12] + 24 A[11] + 3 A[25] + 2 A[34] + 2 A[16] + 5 A[27] + A[17] + 18 A[10] + 8 A[23] + 2 A[41] + 6 A[13] + 2 A[29] + 12 A[8] + 2 A[43] + A[33] + 2 A[18] + A[20], 4 A[37] + A[21] + 2 A[43]], [ 2 A[38] + 8 A[35] + 3 A[32] + A[16] + 8 A[27] + A[41] + 15 A[8], 2 A[36] + 9 A[9] + 3 A[42] + 3 A[14] + 2 A[28] + 6 A[6] + 15 A[12] + 3 A[25] + A[17] + A[33] + A[20], 24 A[13] + A[29]], [2 A[45] + 2 A[38] + 8 A[35] + 5 A[32] + 21 A[11] + A[24] + 2 A[16] + 8 A[27] + 8 A[41] + 2 A[19] + 15 A[8] + 2 A[5], 2 A[14] + 6 A[6], 9 A[7] + A[31] + 3 A[23]], [2 A[45] + 2 A[38] + 11 A[35] + 4 A[32] + 3 A[11] + A[16] + 11 A[27] + 8 A[41] + 2 A[19] + 18 A[8], 3 A[36] + 2 A[39] + A[45] + A[38] + 7 A[35] + 11 A[14] + 3 A[28] + 24 A[6] + 3 A[32] + A[22] + 9 A[12] + 12 A[11] + 9 A[25] + 2 A[16] + 7 A[27] + A[17] + 6 A[41] + 2 A[19] + 12 A[8] + A[33], 8 A[37] + 2 A[44] + 13 A[26] + 18 A[7] + A[34] + 12 A[10] + 9 A[23] + 21 A[13] + 7 A[29] + A[43] + A[40] + 2 A[18]], [A[35] + 3 A[11], A[36] + 3 A[12], A[37] + 3 A[13]]], [1, 1, 1, 5, 1, 13, 14, 1, 22, 20, 5, 26, 22, 13, 23, 13, 25, 17, 14, 26, 22, 4, 23, 22, 7, 26, 20, 14, 7, 2, 25, 26, 14, 25, 22, 19, 5, 13, 25, 17, 23, 8, 13, 5, 17, 20, 10]] For example, C(100000), mudolo , 27, equals , 20 The congruence classes mod, 27, in the following set , {0, 3, 6, 9, 11, 12, 15, 16, 18, 21, 24}, never show up! Theorem Number, 23, : Let C(n) be the constant term, in x, of n (2/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 27, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [A[5], A[22], A[23]], [21 A[5] + 16 A[8], 3 A[6] + 7 A[9], A[24]], [A[25], A[26], A[27]], [9 A[5], 18 A[6], 0], [9 A[5] + 12 A[8], 9 A[6] + 21 A[9], 9 A[7] + 3 A[10]], [9 A[8] + 15 A[11], 9 A[9] + 24 A[12], 9 A[10] + 6 A[13]], [21 A[5] + 13 A[8], 3 A[6] + 22 A[9], 12 A[7] + 4 A[10]], [ 9 A[5] + 15 A[8] + 7 A[11], 9 A[6] + 24 A[9] + 16 A[12], 9 A[7] + 6 A[10] + 25 A[13]], [25 A[5], 16 A[6] + 9 A[11], 7 A[7] + 18 A[12]], [12 A[5] + 4 A[8], 21 A[6] + 22 A[9], 3 A[7] + 13 A[10]], [ 9 A[5] + 6 A[8] + 10 A[11], 9 A[6] + 15 A[9] + A[12], 9 A[7] + 24 A[10] + 19 A[13]], [6 A[5], 15 A[6], 24 A[7]], [18 A[8], 0, 9 A[10]], [18 A[8] + 3 A[11], 18 A[9] + 12 A[12], 18 A[10] + 21 A[13]], [21 A[5] + 22 A[8], 3 A[6] + 4 A[9], 12 A[7] + 13 A[10]], [ 9 A[5] + 15 A[8] + 25 A[11], 9 A[6] + 24 A[9] + 7 A[12], 9 A[7] + 6 A[10] + 16 A[13]], [9 A[5] + 24 A[8] + 22 A[11], 9 A[6] + 6 A[9] + 13 A[12], 9 A[7] + 15 A[10] + 4 A[13]], [15 A[5], 24 A[6], 6 A[7]], [18 A[8], 0, 9 A[10]], [18 A[8] + 21 A[11], 18 A[9] + 3 A[12], 18 A[10] + 12 A[13]]], [1, 1, 1, 9, 1, 25, 6, 1, 10, 15, 9, 21, 9, 7, 6, 25, 16, 24, 6, 18, 18, 16, 6, 15, 15, 18, 18]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 3, 4, 5, 8, 11, 12, 13, 14, 17, 19, 20, 22, 23, 26}, never show up! Theorem Number, 24, : Let C(n) be the constant term, in x, of n (2/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 46, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [13 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [ 10 A[5], 9 A[5] + 8 A[6] + 21 A[8] + 2 A[14] + 6 A[16], 9 A[5] + 16 A[6] + 8 A[7] + 21 A[8] + 9 A[9] + 2 A[14] + 2 A[15] + 6 A[16]] , [18 A[5] + 4 A[6] + 19 A[8] + 5 A[9] + 6 A[11] + 8 A[14] + 3 A[16] + 10 A[17], 24 A[5] + 22 A[6] + 20 A[8] + 13 A[9] + 3 A[11] + 8 A[14] + 7 A[16] + 3 A[17], 9 A[6] + 18 A[7] + 20 A[9] + 16 A[10] + 3 A[12] + 6 A[14] + 3 A[15] + 7 A[17]], [A[27], A[28], A[29]], [7 A[5] + 5 A[6] + 8 A[7] + 17 A[8] + 2 A[9] + 8 A[10] + 3 A[11] + 6 A[12] + 22 A[14] + 4 A[15] + 10 A[16] + 10 A[17] + 7 A[18] + 18 A[19], 9 A[6] + 10 A[14], 9 A[7] + 10 A[15]], [9 A[8] + A[16], 9 A[9] + A[17], 9 A[10] + A[18]], [ 20 A[6] + 15 A[7] + 3 A[8] + 19 A[9] + 20 A[10] + 15 A[11] + 3 A[12] + 10 A[14] + 6 A[15] + 6 A[16] + 8 A[17] + 4 A[18] + 4 A[19] + 3 A[20], 6 A[5] + 22 A[6] + 26 A[7] + 14 A[8] + 3 A[9] + 4 A[10] + 15 A[11] + 4 A[12] + 20 A[13] + 8 A[14] + 10 A[15] + 4 A[16] + 6 A[17] + 5 A[18] + 6 A[19] + A[21], 3 A[5] + 26 A[6] + 17 A[7] + 16 A[8] + 13 A[9] + A[10] + 18 A[11] + 3 A[12] + 9 A[13] + 10 A[14] + 7 A[15] + 2 A[16] + 11 A[17] + 11 A[18] + 6 A[19] + A[21]], [6 A[5] + 5 A[6] + 22 A[7] + 14 A[8] + 15 A[9] + 22 A[10] + 3 A[11] + 2 A[12] + 13 A[13] + 14 A[14] + 8 A[15] + 5 A[16] + 9 A[17] + 2 A[18] + 18 A[19] + A[20] + 2 A[21] + 2 A[22], 6 A[5] + 23 A[6] + 14 A[8] + 19 A[9] + 15 A[11] + 8 A[14] + 4 A[16] + 3 A[17] + 6 A[19] + 2 A[22], 6 A[5] + 23 A[6] + 4 A[7] + 14 A[8] + 17 A[9] + 26 A[10] + 15 A[11] + 8 A[12] + 13 A[13] + 20 A[14] + 14 A[15] + 4 A[16] + 7 A[17] + 11 A[18] + 6 A[19] + 7 A[20] + 2 A[21] + 2 A[22]], [ 3 A[5] + 19 A[6] + 7 A[7] + 16 A[8] + 5 A[9] + 21 A[10] + 9 A[11] + 7 A[12] + 20 A[13] + 18 A[14] + 2 A[15] + 2 A[16] + 7 A[17] + 3 A[18] + 7 A[19] + 2 A[20] + A[21] + 2 A[22] + 12 A[23], 17 A[6] + 11 A[7] + 8 A[9] + 10 A[10] + 3 A[12] + 6 A[14] + A[17] + 2 A[18] + A[20] + A[22] + 4 A[23] , 17 A[6] + 23 A[7] + 8 A[9] + A[10] + 3 A[12] + 9 A[13] + 6 A[14] + A[15] + A[17] + 11 A[18] + A[21] + A[22] + 9 A[23]], [7 A[5] + 15 A[6] + 3 A[8] + 24 A[9] + 26 A[11] + 4 A[12] + 16 A[14] + 4 A[16] + 6 A[17] + 10 A[19] + 2 A[20] + 2 A[22] + 8 A[24], 6 A[5] + 5 A[6] + 3 A[7] + 26 A[8] + 14 A[9] + 11 A[10] + 21 A[11] + 4 A[12] + 20 A[13] + 12 A[14] + 2 A[15] + 4 A[16] + 4 A[17] + A[18] + 9 A[19] + 2 A[20] + A[21] + 2 A[22] + 10 A[23] + 6 A[24], 15 A[7] + 4 A[23]], [22 A[6] + 19 A[8] + 8 A[9] + 17 A[11] + 4 A[12] + 9 A[14] + 2 A[17] + 7 A[19] + 2 A[20] + 2 A[22] + 6 A[24] + 2 A[25], A[6] + 3 A[7] + 16 A[9] + 11 A[10] + 4 A[12] + 20 A[13] + 12 A[14] + 2 A[15] + 4 A[17] + A[18] + 2 A[20] + A[21] + 2 A[22] + 10 A[23] + 2 A[25], 6 A[5] + 6 A[6] + 25 A[7] + 26 A[8] + 6 A[9] + 7 A[10] + 21 A[11] + 2 A[12] + 13 A[13] + 3 A[14] + A[15] + 4 A[16] + A[17] + 3 A[18] + 9 A[19] + A[20] + 2 A[21] + 10 A[23] + 6 A[24] + 2 A[25]], [6 A[5] + 5 A[6] + 3 A[8] + 6 A[9] + 24 A[11] + 11 A[14] + 4 A[16] + 10 A[19] + 2 A[22] + 8 A[24] + 3 A[25], 26 A[6] + 16 A[7] + 10 A[9] + 2 A[10] + 3 A[12] + 9 A[14] + 2 A[15] + 3 A[17] + A[20] + A[22] + 6 A[23] + 2 A[25] + A[26], 7 A[6] + 2 A[7] + 18 A[9] + 18 A[10] + 6 A[13] + 12 A[14] + A[15] + 2 A[17] + 10 A[18] + A[21] + 2 A[22] + 12 A[23] + 4 A[25] + 5 A[26]], [A[5] + 22 A[6] + 14 A[8] + 4 A[9] + 10 A[11] + 2 A[12] + 8 A[14] + A[17] + 4 A[19] + A[20] + 4 A[24] + A[25] + A[27], 21 A[6] + 18 A[8] + 6 A[9] + 15 A[11] + 7 A[14] + 5 A[19] + 6 A[24] + 3 A[25] + A[27], 3 A[5] + 18 A[6] + 8 A[7] + 4 A[8] + 2 A[9] + 6 A[11] + 6 A[14] + 2 A[16] + 2 A[19] + 2 A[23] + A[25] + A[27] ], [6 A[5] + 22 A[6] + 19 A[8] + 8 A[9] + 17 A[11] + 4 A[12] + 9 A[14] + 5 A[16] + 2 A[17] + 7 A[19] + 2 A[20] + 2 A[22] + 4 A[24] + 2 A[25], 18 A[6] + 9 A[9] + 6 A[14] + 2 A[17] + 2 A[25], 20 A[6] + 20 A[7] + 18 A[8] + 10 A[9] + 17 A[10] + 15 A[11] + 8 A[12] + 13 A[13] + 9 A[14] + 2 A[17] + 4 A[18] + 5 A[19] + 2 A[20] + 2 A[21] + A[22] + 10 A[23] + 6 A[24] + 3 A[25] + 4 A[26] + A[27] + 2 A[28]], [20 A[9] + 11 A[14] + 22 A[6] + 4 A[12] + 8 A[11] + 8 A[25] + 10 A[24] + 4 A[16] + 2 A[27] + 2 A[17] + 12 A[19] + 7 A[8] + 6 A[5] + 2 A[20], 14 A[9] + 2 A[15] + 12 A[14] + A[28] + 7 A[6] + 3 A[26] + 18 A[7] + 2 A[22] + 3 A[12] + 18 A[11] + 6 A[25] + 6 A[24] + 2 A[16] + A[17] + 6 A[10] + 7 A[23] + 6 A[19] + 19 A[8] + 3 A[5], 8 A[9] + 9 A[14] + 24 A[6] + 6 A[26] + 8 A[7] + 2 A[12] + 15 A[11] + 3 A[25] + 6 A[24] + A[27] + A[17] + 22 A[10] + 4 A[23] + 5 A[19] + 13 A[13] + 3 A[29] + 18 A[8] + 5 A[18] + A[20]], [ 8 A[9] + 10 A[14] + 23 A[6] + 4 A[12] + 8 A[11] + 2 A[25] + 3 A[16] + 2 A[27] + 2 A[17] + 2 A[19] + 7 A[8] + 3 A[5] + 2 A[20], 3 A[9] + 5 A[14] + 17 A[6] + A[22] + A[30] + A[17], 6 A[9] + A[15] + 12 A[14] + 2 A[28] + 5 A[6] + 2 A[26] + 14 A[7] + A[22] + 6 A[12] + 12 A[11] + 2 A[25] + 4 A[16] + 2 A[27] + A[17] + 7 A[10] + 6 A[23] + 4 A[19] + 8 A[8] + 6 A[5] + A[18] + A[20]], [6 A[9] + A[15] + 7 A[14] + A[28] + 16 A[6] + A[26] + 14 A[7] + A[31] + 6 A[12] + 9 A[11] + A[25] + 2 A[24] + A[30] + A[27] + 2 A[17] + 2 A[10] + 5 A[23] + 3 A[19] + 7 A[8] + 2 A[20], 10 A[9] + A[15] + 8 A[14] + 18 A[6] + 14 A[7] + 2 A[31] + 3 A[12] + 12 A[11] + 2 A[25] + A[30] + 4 A[16] + 2 A[27] + 3 A[17] + 4 A[23] + 4 A[19] + 8 A[8] + 6 A[5] + A[20], 4 A[9] + 10 A[14] + A[28] + A[6] + A[26] + 10 A[7] + A[31] + 2 A[22] + 2 A[12] + 12 A[11] + 2 A[25] + 2 A[30] + 4 A[16] + 2 A[27] + 6 A[10] + 4 A[23] + 4 A[19] + A[13] + 8 A[8] + 6 A[5] + 2 A[18]], [ 3 A[14] + 10 A[6] + A[32] + A[22] + 2 A[24] + A[30] + 6 A[8] + A[5], 3 A[14] + 9 A[6] + A[22], 4 A[9] + 10 A[14] + A[6] + 2 A[32] + 11 A[7] + 2 A[31] + 2 A[22] + 4 A[12] + 2 A[11] + A[24] + 2 A[30] + A[16] + 2 A[17] + 3 A[23] + A[19] + 8 A[8] + 2 A[20]], [ 6 A[14] + 20 A[6] + 2 A[22] + A[24] + 2 A[30] + 3 A[8], 9 A[9] + A[15] + 6 A[14] + A[28] + 16 A[6] + 4 A[32] + A[26] + 14 A[7] + A[31] + 2 A[22] + 2 A[12] + 4 A[11] + 3 A[25] + 2 A[24] + 2 A[16] + 2 A[10] + 5 A[23] + 2 A[19] + 16 A[8] + A[33], 6 A[9] + 5 A[14] + A[28] + 14 A[6] + A[21] + 4 A[26] + 10 A[7] + A[22] + 2 A[12] + 2 A[25] + A[30] + 15 A[10] + 5 A[23] + 7 A[13] + A[29] + A[33] + 3 A[18]], [12 A[9] + 9 A[14] + 24 A[6] + A[22] + 3 A[11] + 5 A[25] + 2 A[30] + A[27] + A[33], 12 A[9] + 2 A[15] + 6 A[14] + 14 A[6] + 2 A[32] + 2 A[26] + 18 A[7] + A[31] + A[22] + A[12] + 2 A[11] + 4 A[25] + A[24] + A[16] + 4 A[10] + 6 A[23] + A[19] + 8 A[8] + 2 A[33], 6 A[9] + 2 A[15] + 4 A[14] + A[28] + 12 A[6] + A[21] + 2 A[32] + A[26] + 20 A[7] + 2 A[12] + 2 A[11] + 2 A[25] + A[24] + 2 A[30] + A[16] + 4 A[10] + 8 A[23] + A[19] + 5 A[13] + A[29] + 8 A[8] + A[33] + A[18]], [ 3 A[14] + 10 A[6] + A[32] + A[22] + 2 A[24] + A[30] + 6 A[8] + A[5], 6 A[9] + 7 A[14] + 19 A[6] + 2 A[22] + 2 A[25] + A[33], A[15] + 9 A[7] + 3 A[23]], [A[36] + 14 A[9] + A[35] + 2 A[15] + 5 A[14] + A[28] + 14 A[6] + 3 A[26] + 18 A[7] + 4 A[12] + 2 A[11] + 6 A[25] + 2 A[24] + 2 A[30] + A[16] + 6 A[10] + 7 A[23] + 7 A[8] + A[33], A[36] + 13 A[9] + 2 A[15] + 4 A[14] + A[28] + 12 A[6] + 2 A[32] + 3 A[26] + 18 A[7] + 4 A[12] + 2 A[11] + 4 A[25] + A[24] + 2 A[30] + A[16] + A[17] + 6 A[10] + 7 A[23] + A[19] + 8 A[8] + A[33], 4 A[26] + 8 A[7] + 2 A[34] + 15 A[10] + 4 A[23] + A[18]], [A[36] + 4 A[9] + A[15] + A[32] + 2 A[26] + 16 A[7] + 2 A[31] + 2 A[12] + A[11] + 2 A[25] + 2 A[24] + 2 A[16] + 4 A[10] + 5 A[23] + 10 A[8], 2 A[36] + 12 A[9] + 2 A[15] + 2 A[14] + 2 A[28] + 6 A[6] + 6 A[26] + 5 A[7] + 2 A[31] + 9 A[12] + 4 A[25] + A[30] + 2 A[17] + 12 A[10] + 12 A[23], 2 A[36] + 14 A[9] + 2 A[15] + 4 A[14] + 12 A[6] + 4 A[32] + 8 A[26] + 26 A[7] + 4 A[12] + 4 A[11] + 6 A[25] + 2 A[24] + 2 A[34] + 2 A[30] + 2 A[16] + 24 A[10] + 11 A[23] + 2 A[19] + A[13] + 16 A[8] + A[33] + 2 A[18]], [ A[5], 6 A[9] + 4 A[14] + 13 A[6] + 2 A[25] + 2 A[30] + A[33], 2 A[36] + 20 A[9] + A[15] + 5 A[14] + A[28] + 12 A[6] + 5 A[26] + 15 A[7] + 6 A[12] + 8 A[25] + A[30] + 10 A[10] + 6 A[23] + 2 A[33]], [2 A[36] + 5 A[38] + 3 A[9] + 2 A[35] + 8 A[14] + A[28] + 16 A[6] + 2 A[32] + 5 A[26] + 16 A[7] + 2 A[31] + 6 A[12] + 6 A[11] + 13 A[25] + 5 A[24] + 2 A[16] + A[27] + 10 A[10] + 6 A[23] + 2 A[8] + 2 A[33], A[36] + A[39] + 4 A[38] + 19 A[9] + 7 A[14] + 2 A[28] + 16 A[6] + 4 A[26] + 14 A[7] + A[31] + 6 A[12] + 6 A[25] + A[30] + 2 A[16] + 8 A[10] + 6 A[23] + 12 A[8] + 2 A[33], A[36] + 10 A[9] + A[15] + 4 A[14] + 2 A[28] + 12 A[6] + 8 A[26] + 5 A[7] + A[31] + 6 A[12] + 4 A[25] + 2 A[34] + 2 A[30] + 23 A[10] + 14 A[23] + A[33] + A[18]], [A[36] + 6 A[38] + 20 A[9] + 2 A[35] + 2 A[15] + 6 A[14] + A[28] + 12 A[6] + A[32] + 3 A[26] + 18 A[7] + 4 A[12] + 5 A[11] + 9 A[25] + 5 A[24] + 2 A[16] + 6 A[10] + 7 A[23] + A[8] + A[33], A[36] + 2 A[39] + 18 A[9] + A[15] + 3 A[14] + 6 A[6] + 3 A[26] + 16 A[7] + A[31] + 3 A[12] + 6 A[25] + 6 A[10] + 6 A[23] + A[33], A[36] + 2 A[38] + 4 A[9] + A[15] + 2 A[14] + A[28] + 8 A[6] + 7 A[26] + 22 A[7] + A[31] + 4 A[12] + 2 A[25] + A[34] + 2 A[30] + A[16] + 20 A[10] + 9 A[23] + 5 A[13] + 2 A[29] + 6 A[8] + 2 A[40]], [ A[5], 4 A[38] + 12 A[9] + 5 A[14] + 13 A[6] + 4 A[25] + A[30] + 2 A[16] + 12 A[8] + 2 A[33], 2 A[15] + 21 A[7] + 2 A[31] + 6 A[23]], [A[36] + 2 A[38] + 16 A[9] + A[35] + A[15] + 6 A[14] + 14 A[6] + 2 A[32] + 2 A[26] + 16 A[7] + 2 A[31] + 2 A[12] + 4 A[11] + 7 A[25] + 2 A[24] + A[30] + A[27] + 4 A[10] + 5 A[23] + 13 A[8] + A[33], A[36] + 2 A[39] + 4 A[38] + 19 A[9] + 6 A[14] + 2 A[28] + 16 A[6] + 4 A[26] + 14 A[7] + A[31] + 6 A[12] + 6 A[25] + 2 A[30] + 2 A[16] + 8 A[10] + 6 A[23] + 12 A[8] + A[33], 2 A[36] + 20 A[9] + A[15] + 5 A[14] + A[28] + 12 A[6] + 7 A[26] + 18 A[7] + 6 A[12] + 8 A[25] + A[30] + 19 A[10] + 8 A[23] + 2 A[33] + 2 A[40]], [2 A[36] + 3 A[38] + 26 A[9] + 2 A[35] + 2 A[15] + 5 A[14] + 10 A[6] + 2 A[32] + 4 A[26] + 20 A[7] + A[31] + 4 A[12] + 5 A[11] + 12 A[25] + 4 A[24] + A[16] + 8 A[10] + 7 A[23] + 20 A[8] + A[33], A[36] + A[39] + 4 A[38] + 16 A[9] + 2 A[15] + 6 A[14] + 16 A[6] + 4 A[26] + 20 A[7] + A[31] + 3 A[12] + 6 A[25] + 2 A[30] + 2 A[16] + 8 A[10] + 7 A[23] + 12 A[8] + A[33], 2 A[36] + A[37] + 4 A[38] + 8 A[9] + A[14] + 2 A[28] + 4 A[6] + 14 A[26] + A[7] + A[31] + 8 A[12] + 4 A[25] + 2 A[34] + A[30] + 2 A[16] + 7 A[10] + 13 A[23] + 7 A[13] + 2 A[29] + 12 A[8] + A[40]], [A[36] + 4 A[38] + 14 A[9] + A[35] + 2 A[15] + 5 A[14] + A[28] + 14 A[6] + 3 A[26] + 18 A[7] + 4 A[12] + 2 A[11] + 6 A[25] + 2 A[24] + 2 A[30] + 2 A[16] + 6 A[10] + 7 A[23] + 16 A[8] + A[5] + A[33] , 2 A[14] + 9 A[6] + 2 A[30], 2 A[36] + 20 A[9] + 2 A[15] + 5 A[14] + A[28] + 12 A[6] + 5 A[26] + 21 A[7] + 6 A[12] + 8 A[25] + A[30] + 10 A[10] + 8 A[23] + 2 A[33]], [A[36] + 4 A[38] + 18 A[9] + A[35] + 7 A[14] + 2 A[28] + 14 A[6] + 2 A[32] + 4 A[26] + 14 A[7] + A[31] + 6 A[12] + 6 A[11] + 8 A[25] + 2 A[24] + A[16] + 2 A[27] + 8 A[10] + 6 A[23] + 19 A[8] + A[33], 2 A[36] + 2 A[39] + 23 A[9] + 5 A[14] + A[28] + 12 A[6] + 5 A[26] + 16 A[7] + 2 A[31] + 6 A[12] + 8 A[25] + A[30] + 10 A[10] + 6 A[23] + A[33], A[36] + 10 A[9] + 4 A[14] + 2 A[28] + 12 A[6] + 8 A[26] + 26 A[7] + 2 A[31] + 6 A[12] + 4 A[25] + A[34] + 2 A[30] + 23 A[10] + 11 A[23] + A[33] + 2 A[40]], [A[36] + 3 A[38] + 16 A[9] + A[35] + A[15] + 8 A[14] + 20 A[6] + 2 A[32] + 2 A[26] + 16 A[7] + 2 A[31] + 2 A[12] + 7 A[11] + 8 A[25] + 2 A[24] + 2 A[30] + A[16] + 2 A[27] + 4 A[10] + 5 A[23] + 16 A[8], A[39] + 6 A[9] + 2 A[15] + 4 A[14] + A[28] + 12 A[6] + 3 A[26] + 18 A[7] + 3 A[12] + 2 A[25] + 2 A[30] + 6 A[10] + 7 A[23], A[36] + A[37] + 4 A[38] + 16 A[9] + A[15] + 6 A[14] + 16 A[6] + 8 A[26] + 20 A[7] + A[31] + 2 A[12] + 6 A[25] + A[34] + 2 A[30] + 2 A[16] + 20 A[10] + 8 A[23] + 7 A[13] + 2 A[29] + 12 A[8] + 2 A[33] + A[40]]], [ 1, 1, 1, 13, 1, 10, 19, 1, 19, 10, 13, 13, 13, 1, 1, 10, 10, 10, 19, 19, 19, 1, 1, 19, 19, 19, 10, 10, 10, 13, 13, 13, 13, 13, 13, 13, 13, 1, 1, 1, 1, 1, 1, 10, 10, 10]] For example, C(100000), mudolo , 27, equals , 1 The congruence classes mod, 27, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 25, : Let C(n) be the constant term, in x, of n (2/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 47, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [2 A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [8 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [5 A[5], 18 A[6] + 18 A[11] + 5 A[14], 12 A[7] + 9 A[12] + 2 A[15]], [24 A[5] + 12 A[8] + 8 A[16], 9 A[6] + 3 A[9] + 6 A[14] + 8 A[17], A[44]], [A[45], A[46], A[47]], [5 A[5], 18 A[6] + 5 A[14], 3 A[7] + 11 A[15]], [ 18 A[5] + 3 A[8] + 8 A[16], 18 A[6] + 21 A[9] + 8 A[17], 18 A[7] + 12 A[10] + 8 A[18]], [18 A[5] + 18 A[8] + 21 A[11] + 5 A[19], 18 A[6] + 18 A[9] + 12 A[12] + 5 A[20], 18 A[7] + 18 A[10] + 3 A[13] + 5 A[21]], [9 A[5] + 12 A[8] + 8 A[16], 18 A[6] + 12 A[9] + 6 A[14] + 8 A[17] + 6 A[22], 9 A[7] + 12 A[10] + 8 A[18]], [ 21 A[6] + 18 A[8] + 21 A[11] + 8 A[14] + 8 A[19] + 8 A[22], 9 A[7] + 18 A[9] + 21 A[12] + A[15] + 8 A[20] + 4 A[23], 18 A[10] + 3 A[13] + 8 A[21]], [A[5], 6 A[14] + 18 A[6] + 2 A[22], 12 A[7] + 8 A[23]], [ 18 A[5] + 24 A[8] + 6 A[16] + 8 A[24], 21 A[6] + 15 A[9] + 9 A[14] + 11 A[17] + 6 A[22] + 4 A[25], 3 A[7] + 24 A[10] + 7 A[18] + 9 A[23]], [18 A[5] + 21 A[6] + 18 A[8] + 6 A[11] + 7 A[14] + 9 A[16] + A[19] + 4 A[22] + 12 A[24], 3 A[6] + 18 A[9] + 6 A[12] + 12 A[14] + 12 A[17] + A[20] + 6 A[22] + 6 A[25], 15 A[7] + 9 A[10] + 15 A[13] + 16 A[18] + A[21] + 12 A[23] + 10 A[26]], [ 7 A[5] + 18 A[8] + 18 A[11] + 10 A[16] + 13 A[19] + 4 A[24] + 19 A[27], 21 A[6] + 7 A[14] + 6 A[22], 18 A[7] + A[15] + 15 A[23]], [ 6 A[8] + 9 A[11] + A[16] + 3 A[19] + 3 A[27], 21 A[6] + 18 A[7] + 24 A[9] + 18 A[12] + 9 A[14] + 2 A[15] + 11 A[17] + 8 A[20] + 8 A[23] + 7 A[25] + 2 A[28], 12 A[7] + 24 A[10] + 6 A[18] + 15 A[23] + 5 A[26]], [ 15 A[11] + 3 A[16] + 2 A[27] + 3 A[19], 18 A[9] + 2 A[15] + 15 A[14] + 4 A[28] + 6 A[6] + 18 A[7] + 15 A[12] + 13 A[25] + 19 A[17] + 8 A[23] + 2 A[20], 3 A[21] + 3 A[26] + 3 A[7] + 9 A[10] + 3 A[23] + 15 A[13] + 2 A[29] + 6 A[18]], [ 18 A[11] + 2 A[24] + 4 A[16] + 10 A[27] + 4 A[19] + 12 A[8] + 6 A[5], 21 A[9] + 8 A[14] + 24 A[6] + 4 A[25] + 2 A[30] + 9 A[17], 8 A[21] + 8 A[26] + 18 A[7] + 3 A[10] + 12 A[23] + 2 A[29] + 16 A[18]], [ 3 A[11] + A[24] + 4 A[16] + 8 A[27] + 10 A[19] + 9 A[8] + 3 A[5], 9 A[9] + A[15] + 10 A[14] + A[28] + 21 A[6] + 9 A[7] + 3 A[12] + 10 A[25] + A[30] + 16 A[17] + 4 A[23], 2 A[15] + 6 A[26] + 21 A[7] + 2 A[31] + 18 A[10] + 12 A[23] + 21 A[13] + A[29] + 12 A[18]], [ 6 A[32] + 18 A[11] + 3 A[24] + 6 A[27] + 6 A[19] + 9 A[8] + 7 A[5], 6 A[14] + 15 A[6] + 2 A[22], A[15] + 18 A[7] + A[31] + 8 A[23]], [ A[32] + 18 A[11] + 2 A[24] + 7 A[16] + 4 A[27] + 7 A[19] + 6 A[8], 15 A[9] + 8 A[14] + 24 A[6] + 6 A[25] + 2 A[30] + 2 A[33], 6 A[26] + 9 A[7] + 15 A[10] + 6 A[23] + 10 A[18]], [4 A[14] + 12 A[6] + 5 A[32] + 2 A[22] + 6 A[11] + 2 A[24] + 2 A[30] + 7 A[16] + 10 A[27] + 11 A[19] + 9 A[8] + 6 A[5], 9 A[9] + 2 A[15] + 8 A[14] + 3 A[28] + 24 A[6] + 18 A[7] + 6 A[12] + 4 A[25] + 2 A[30] + 8 A[23] + 2 A[33] + A[20], A[21] + 4 A[26] + 9 A[7] + 3 A[34] + 9 A[10] + 6 A[23] + 6 A[13] + 3 A[29] + 4 A[18]], [5 A[32] + 18 A[11] + 2 A[24] + 4 A[16] + 6 A[27] + 6 A[19] + 9 A[8] + 7 A[5], 3 A[6] + 2 A[30], 3 A[7] + 2 A[31]], [2 A[35] + 4 A[14] + 12 A[6] + 4 A[32] + 2 A[22] + 18 A[11] + A[24] + 2 A[30] + 3 A[16] + 4 A[27] + 12 A[19] + 6 A[8] + 3 A[5], 6 A[9] + 2 A[14] + 9 A[6] + 2 A[30] + A[17] + 3 A[33], A[15] + 2 A[26] + 12 A[7] + A[31] + 3 A[34] + 6 A[10] + 6 A[23] + 3 A[18]], [4 A[35] + 4 A[32] + 15 A[11] + A[24] + 2 A[16] + 2 A[27] + 10 A[19] + 9 A[8] + 3 A[5], 2 A[36] + 18 A[9] + 10 A[14] + A[28] + 6 A[12] + 6 A[25] + A[30] + A[17] + 4 A[33] + A[20], A[37] + A[21] + 6 A[26] + 6 A[7] + 8 A[34] + 18 A[10] + 6 A[23] + 6 A[13] + 5 A[29] + 5 A[18]], [ 4 A[32] + 9 A[11] + 3 A[24] + A[16] + 3 A[27] + 3 A[19] + 9 A[8] + 5 A[5], 3 A[14] + 9 A[6] + A[22], A[37] + A[21] + 3 A[26] + 6 A[7] + 4 A[34] + 9 A[10] + 4 A[23] + 9 A[13] + 3 A[29] + A[18]], [A[38] + 2 A[35] + 4 A[14] + 12 A[6] + 6 A[32] + 2 A[22] + 18 A[11] + 3 A[24] + 2 A[30] + A[16] + 4 A[27] + 12 A[19] + 15 A[8] + 6 A[5], A[39] + 6 A[9] + 4 A[14] + 15 A[6] + A[25] + A[30] + A[33], A[37] + A[15] + A[21] + 8 A[26] + 12 A[7] + A[31] + 7 A[34] + 21 A[10] + 9 A[23] + 9 A[13] + 3 A[29] + 8 A[18]], [2 A[38] + 2 A[35] + 4 A[14] + 12 A[6] + 7 A[32] + 2 A[22] + 21 A[11] + 2 A[24] + 2 A[30] + A[16] + 5 A[27] + 12 A[19] + 15 A[8] + 6 A[5], A[39] + 3 A[9] + 2 A[14] + A[28] + 9 A[6] + 3 A[12] + 2 A[30] + A[33], 2 A[37] + 2 A[21] + 6 A[26] + 6 A[7] + 10 A[34] + 24 A[10] + 6 A[23] + 21 A[13] + 7 A[29] + 2 A[40] + 2 A[18]], [2 A[38] + 6 A[32] + 9 A[11] + 3 A[24] + A[16] + 3 A[27] + 3 A[41] + 15 A[8] + 5 A[5], 3 A[14] + 12 A[6] + A[30], A[37] + A[15] + A[21] + 3 A[26] + 12 A[7] + 2 A[31] + 4 A[34] + 9 A[10] + 6 A[23] + 9 A[13] + 3 A[29] + A[18]], [ 6 A[32] + 2 A[24] + A[16] + 5 A[27] + 9 A[41] + 2 A[19] + 12 A[8] + 6 A[5], A[39] + 6 A[9] + 5 A[14] + 18 A[6] + A[25] + 2 A[30] + A[33], A[39] + 12 A[9] + A[42] + 2 A[15] + 2 A[14] + A[28] + 6 A[6] + A[26] + 9 A[7] + 3 A[12] + 2 A[25] + 2 A[34] + 2 A[30] + 2 A[17] + 9 A[10] + 5 A[23] + 4 A[33] + 2 A[40]], [A[35] + 18 A[11] + 5 A[41] + A[19], 2 A[39] + 15 A[9] + 2 A[14] + A[28] + 6 A[6] + 3 A[12] + 2 A[25] + 2 A[30] + 2 A[17] + 5 A[33], A[39] + 12 A[9] + A[42] + A[15] + 2 A[14] + A[28] + 6 A[6] + 6 A[26] + 12 A[7] + 2 A[31] + 3 A[12] + 2 A[25] + 5 A[34] + 2 A[30] + 2 A[17] + 15 A[10] + 8 A[23] + 9 A[13] + 3 A[29] + 2 A[43] + 4 A[33] + 2 A[40]], [A[38] + A[35] + 5 A[32] + 15 A[11] + A[24] + 2 A[16] + 4 A[27] + 3 A[41] + 12 A[8] + 3 A[5], 9 A[9] + 6 A[14] + 3 A[28] + 18 A[6] + 6 A[12] + 2 A[25] + 2 A[17] + 3 A[33] + A[20], 2 A[37] + 2 A[39] + 15 A[9] + 2 A[42] + 4 A[14] + 2 A[28] + 9 A[6] + 4 A[26] + 6 A[7] + 2 A[31] + 6 A[12] + 4 A[25] + 7 A[34] + A[30] + A[17] + 15 A[10] + 4 A[23] + 7 A[29] + A[43] + 5 A[33] + 2 A[40]], [A[38] + A[32] + 3 A[8] + A[5], 3 A[6] + 2 A[30], 3 A[7] + 2 A[31]], [ 3 A[32] + A[16] + 6 A[8], 2 A[36] + 2 A[39] + 21 A[9] + 6 A[14] + 4 A[28] + 12 A[6] + 12 A[12] + 6 A[25] + 10 A[33] + 2 A[46] + 2 A[20], 2 A[39] + 15 A[9] + 2 A[42] + A[15] + 4 A[14] + 2 A[28] + 9 A[6] + 2 A[26] + 6 A[7] + 6 A[12] + 4 A[25] + 4 A[34] + A[30] + A[17] + 9 A[10] + 4 A[23] + 3 A[13] + A[29] + A[43] + 5 A[33] + A[40]], [2 A[45] + A[38] + 4 A[35] + A[32] + 21 A[11] + 4 A[27] + 4 A[41] + A[19] + 3 A[8], 4 A[36] + A[47] + A[37] + A[39] + 12 A[9] + A[42] + A[15] + 4 A[14] + 3 A[28] + 9 A[6] + 5 A[26] + 9 A[7] + 12 A[12] + 4 A[25] + 6 A[34] + A[30] + 15 A[10] + 7 A[23] + 3 A[13] + 6 A[29] + 2 A[43] + 6 A[33] + 2 A[46] + 2 A[40], 2 A[15] + 5 A[26] + 15 A[7] + 2 A[31] + 4 A[34] + 9 A[10] + 9 A[23] + 21 A[13] + 4 A[29] + 2 A[43]]], [1, 1, 2, 8, 1, 5, 5, 2, 1, 4, 8, 25, 19, 14, 23, 5, 7, 19, 5, 10, 4, 19, 19, 1, 23, 11, 4, 8, 23, 13, 13, 25, 26, 23, 19, 2, 8, 14, 25, 10, 23, 19, 13, 23, 19, 11, 8]] For example, C(100000), mudolo , 27, equals , 14 The congruence classes mod, 27, in the following set , {0, 3, 6, 9, 12, 15, 16, 17, 18, 20, 21, 22, 24}, never show up! Theorem Number, 26, : Let C(n) be the constant term, in x, of n (2/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 46, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [2 A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [16 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [ 26 A[5], 13 A[6] + 8 A[7] + 9 A[9] + 4 A[14] + A[15], 6 A[7] + 9 A[12] + 2 A[15]], [9 A[5] + 16 A[7] + 19 A[8] + 22 A[9] + 3 A[11] + 2 A[15] + 4 A[16] + 8 A[17], 18 A[6] + 8 A[7] + 24 A[9] + 6 A[14] + A[15] + 8 A[17], 12 A[5] + 22 A[6] + 11 A[7] + 7 A[8] + 18 A[9] + 20 A[10] + 3 A[11] + 24 A[12] + 2 A[14] + 4 A[15] + 2 A[16] + 3 A[17]], [A[44], A[45], A[46]], [A[5], 18 A[6] + 10 A[14], 15 A[6] + 9 A[7] + 7 A[9] + 25 A[10] + 3 A[12] + 6 A[14] + 10 A[15] + 2 A[17] + 5 A[18]], [3 A[5] + 16 A[6] + 8 A[7] + 16 A[8] + 9 A[9] + 23 A[10] + 21 A[11] + 18 A[12] + 8 A[14] + 10 A[15] + 12 A[16] + 3 A[17] + 10 A[18] + 9 A[19] + 6 A[20], 24 A[5] + 16 A[6] + 14 A[8] + 5 A[9] + 15 A[11] + 12 A[12] + 11 A[14] + 4 A[16] + 2 A[17] + 9 A[19] + 6 A[20], 15 A[5] + 20 A[6] + 20 A[8] + 4 A[9] + 14 A[10] + A[11] + 22 A[12] + 10 A[14] + 13 A[16] + 2 A[17] + 11 A[18] + 14 A[19] + 8 A[20]], [9 A[5] + 24 A[6] + 23 A[7] + 15 A[8] + 12 A[9] + 9 A[10] + 21 A[11] + 25 A[12] + 25 A[13] + 12 A[14] + 10 A[15] + 12 A[16] + 3 A[17] + 9 A[18] + 16 A[19] + 11 A[20] + 8 A[21], 24 A[5] + 7 A[6] + 7 A[7] + 14 A[8] + 12 A[9] + 10 A[10] + 15 A[11] + 21 A[12] + 14 A[14] + 2 A[15] + 4 A[16] + 3 A[17] + 2 A[18] + 9 A[19] + 10 A[20], 18 A[5] + 26 A[6] + 18 A[7] + 6 A[8] + 5 A[9] + 20 A[10] + 14 A[11] + 12 A[12] + 25 A[13] + 10 A[14] + 6 A[15] + 6 A[16] + A[17] + 4 A[18] + 4 A[19] + 6 A[20]], [15 A[5] + 8 A[6] + 16 A[7] + 8 A[8] + 14 A[9] + 12 A[10] + 10 A[11] + 18 A[12] + 20 A[13] + 14 A[14] + 5 A[15] + 9 A[16] + 4 A[17] + 6 A[18] + 5 A[19] + 6 A[20] + A[21] + A[22], 11 A[6] + 9 A[9] + 6 A[12] + 5 A[14] + 2 A[17] + 3 A[20] + A[22], 18 A[5] + 26 A[6] + 24 A[7] + 6 A[8] + 5 A[9] + 8 A[10] + 14 A[11] + 12 A[12] + 19 A[13] + 10 A[14] + 9 A[15] + 6 A[16] + A[17] + 9 A[18] + 4 A[19] + 6 A[20] + 5 A[21]], [6 A[5] + 8 A[6] + 2 A[7] + 17 A[8] + 10 A[9] + 11 A[10] + 26 A[11] + 18 A[12] + 13 A[13] + 15 A[14] + 13 A[16] + 2 A[17] + 4 A[18] + 9 A[19] + 9 A[20] + 2 A[21] + 2 A[22] + 2 A[23], 18 A[5] + 15 A[6] + 3 A[7] + 6 A[8] + 6 A[9] + 13 A[11] + 13 A[12] + 6 A[13] + 7 A[14] + 9 A[16] + 3 A[17] + 5 A[19] + 4 A[20] + 3 A[21] + A[22] + 3 A[23], 18 A[5] + 26 A[6] + 17 A[7] + 6 A[8] + 9 A[9] + 13 A[11] + 18 A[12] + 17 A[13] + 12 A[14] + 2 A[15] + 9 A[16] + 3 A[17] + 5 A[19] + 6 A[20] + 6 A[21] + 2 A[22] + 10 A[23]], [7 A[5], 6 A[5] + 23 A[6] + 2 A[7] + A[8] + 11 A[10] + 3 A[11] + 4 A[12] + 13 A[13] + 8 A[14] + 4 A[18] + 2 A[20] + 2 A[21] + 2 A[23] + A[24], 3 A[7] + 2 A[23]], [ 12 A[5] + 14 A[6] + 21 A[8] + 4 A[14] + 4 A[16] + 6 A[24], 20 A[6] + A[7] + 25 A[9] + 19 A[10] + 2 A[12] + 20 A[13] + 8 A[14] + 5 A[17] + 2 A[18] + A[20] + A[21] + A[22] + A[23] + 8 A[25], 6 A[5] + 17 A[6] + 14 A[7] + A[8] + 3 A[9] + 5 A[10] + 3 A[11] + 4 A[12] + 13 A[13] + 8 A[14] + A[15] + 7 A[17] + 2 A[18] + 2 A[20] + 2 A[21] + A[22] + 6 A[23] + A[24] + 10 A[25]], [6 A[5] + 8 A[6] + 10 A[8] + 22 A[9] + 5 A[11] + 2 A[12] + 4 A[14] + 3 A[16] + 5 A[17] + 2 A[19] + A[20] + 4 A[24] + 6 A[25], 6 A[5] + 13 A[6] + A[8] + A[9] + 3 A[11] + 3 A[12] + 16 A[14] + 7 A[17] + A[20] + 2 A[22] + A[24] + 8 A[25], 12 A[5] + 22 A[6] + 2 A[7] + 2 A[8] + 21 A[9] + 10 A[10] + 6 A[11] + 2 A[12] + 5 A[13] + 10 A[14] + 5 A[17] + 3 A[18] + A[20] + 2 A[21] + 2 A[22] + 2 A[23] + 2 A[24] + 8 A[25] + 4 A[26]], [ 2 A[5] + 14 A[6] + 15 A[8] + 6 A[11] + 4 A[14] + 3 A[16] + 2 A[19] + 6 A[24] + 2 A[27], 9 A[6] + 2 A[14], 3 A[5] + 17 A[6] + 16 A[7] + 11 A[8] + 8 A[9] + 4 A[11] + 4 A[12] + 6 A[14] + A[15] + 3 A[16] + 2 A[17] + 2 A[20] + 2 A[22] + 6 A[23] + 2 A[24] + 4 A[25] + 4 A[27]], [ 3 A[5] + 7 A[6] + 24 A[8] + 9 A[11] + 2 A[14] + 6 A[16] + 2 A[19] + 7 A[24] + 5 A[27], 6 A[5] + 3 A[6] + 9 A[7] + 22 A[8] + 7 A[9] + 8 A[11] + 8 A[12] + 11 A[14] + A[15] + 6 A[16] + 2 A[20] + 2 A[22] + 4 A[23] + 4 A[24] + 5 A[25] + 8 A[27] + 4 A[28], 14 A[7] + 3 A[10] + 2 A[15] + 7 A[23] + A[26]], [5 A[9] + 2 A[15] + 8 A[14] + 5 A[28] + 21 A[6] + 18 A[7] + 2 A[22] + 9 A[12] + 9 A[11] + 5 A[25] + 7 A[24] + 5 A[16] + 3 A[27] + 8 A[23] + 2 A[19] + 20 A[8] + 2 A[20], 7 A[9] + A[15] + 10 A[14] + 7 A[28] + 24 A[6] + A[26] + 13 A[7] + A[22] + 13 A[12] + 4 A[11] + 5 A[25] + 2 A[24] + 3 A[16] + 4 A[27] + A[17] + 5 A[10] + 8 A[23] + 3 A[13] + 3 A[29] + 11 A[8] + 3 A[5] + 2 A[18] + 2 A[20], A[21] + 4 A[26] + 9 A[7] + 6 A[10] + 6 A[23] + 12 A[13] + 5 A[29] + A[18]], [3 A[11] + 3 A[24] + A[16] + A[27] + A[19] + 6 A[8], 6 A[9] + 4 A[14] + 3 A[28] + 12 A[6] + 3 A[12] + A[25] + 2 A[30] + 2 A[17], 2 A[9] + 2 A[15] + 4 A[14] + A[28] + 11 A[6] + A[26] + 18 A[7] + A[22] + A[12] + A[30] + A[17] + 2 A[10] + 8 A[23] + A[13] + A[29]], [5 A[9] + A[15] + 5 A[14] + 4 A[28] + 13 A[6] + 9 A[7] + A[22] + 6 A[12] + 6 A[11] + 3 A[25] + 2 A[24] + A[30] + 3 A[16] + 3 A[27] + A[17] + 4 A[23] + A[19] + 8 A[8] + 3 A[5] + A[20], 8 A[9] + 10 A[14] + 7 A[28] + 22 A[6] + A[26] + 4 A[7] + 2 A[22] + 12 A[12] + 4 A[25] + 2 A[17] + 5 A[10] + 4 A[23] + 3 A[13] + 3 A[29] + 2 A[18] + 2 A[20], 4 A[9] + 2 A[15] + 8 A[14] + 2 A[28] + 22 A[6] + 2 A[21] + 5 A[26] + 19 A[7] + 2 A[22] + 2 A[12] + 2 A[30] + 2 A[17] + 7 A[10] + 12 A[23] + 14 A[13] + 6 A[29] + A[18]], [ 2 A[6] + A[32] + A[24] + A[30] + 3 A[8] + 5 A[5], 4 A[9] + 2 A[15] + 6 A[14] + 4 A[28] + 20 A[6] + 4 A[32] + 10 A[7] + A[31] + 2 A[22] + 8 A[12] + 5 A[11] + 4 A[25] + 3 A[24] + 2 A[30] + A[16] + 3 A[27] + 4 A[23] + A[19] + 13 A[8] + 3 A[5] + 2 A[20], 4 A[9] + 4 A[14] + 2 A[28] + 12 A[6] + 4 A[32] + 4 A[7] + A[31] + 2 A[12] + 5 A[11] + 3 A[24] + 2 A[30] + A[16] + 3 A[27] + 2 A[17] + A[19] + 13 A[8] + 3 A[5]], [ 6 A[32] + 9 A[11] + 4 A[24] + 5 A[27] + 2 A[19] + 18 A[8], 12 A[9] + A[15] + 7 A[14] + 5 A[28] + 17 A[6] + 8 A[7] + 2 A[31] + A[22] + 9 A[12] + 6 A[25] + A[30] + 2 A[23] + 2 A[33] + 2 A[20], 2 A[15] + 15 A[7] + A[31] + 6 A[10] + 9 A[23] + 3 A[13] + 3 A[29] + 2 A[18] ], [5 A[9] + A[15] + 5 A[14] + 2 A[28] + 14 A[6] + 3 A[32] + 2 A[26] + 10 A[7] + 2 A[31] + 2 A[22] + 2 A[12] + 5 A[11] + 3 A[25] + 4 A[24] + 2 A[34] + A[30] + 2 A[16] + 3 A[27] + 6 A[10] + 4 A[23] + 14 A[8] + A[33], A[9] + 2 A[15] + 2 A[14] + A[28] + 6 A[6] + 4 A[26] + 14 A[7] + A[31] + 3 A[12] + A[25] + 2 A[34] + A[30] + 10 A[10] + 8 A[23] + 3 A[13] + 3 A[29] + A[18], 4 A[9] + 2 A[15] + 8 A[14] + 2 A[28] + 22 A[6] + 2 A[21] + 4 A[26] + 10 A[7] + 2 A[22] + 2 A[12] + 2 A[30] + 2 A[17] + 4 A[10] + 6 A[23] + 17 A[13] + 5 A[29]], [4 A[9] + A[15] + 4 A[14] + 2 A[28] + 12 A[6] + 4 A[32] + A[26] + 9 A[7] + 2 A[31] + 2 A[12] + 5 A[11] + 3 A[24] + A[34] + 2 A[30] + A[16] + 3 A[27] + 2 A[17] + 3 A[10] + 3 A[23] + A[19] + 13 A[8] + A[5], 9 A[9] + 2 A[35] + 2 A[15] + 7 A[14] + 4 A[28] + 17 A[6] + 4 A[32] + 10 A[7] + A[31] + 4 A[12] + 10 A[11] + 3 A[25] + 6 A[24] + A[30] + 2 A[16] + 6 A[27] + 2 A[17] + 4 A[23] + 18 A[8] + A[33], 2 A[15] + A[26] + 14 A[7] + A[31] + A[34] + 3 A[10] + 7 A[23]], [ 3 A[32] + 6 A[11] + 3 A[24] + A[16] + 4 A[27] + A[19] + 12 A[8], 3 A[9] + A[35] + A[15] + 2 A[28] + 4 A[6] + 2 A[32] + 2 A[26] + 10 A[7] + 2 A[31] + 2 A[12] + 5 A[11] + 2 A[25] + 3 A[24] + 2 A[34] + 2 A[30] + A[16] + 3 A[27] + 6 A[10] + 4 A[23] + 9 A[8], 2 A[36] + 7 A[9] + A[35] + 8 A[14] + 7 A[28] + 16 A[6] + 2 A[32] + 4 A[26] + 3 A[7] + A[31] + 11 A[12] + 5 A[11] + 3 A[25] + 3 A[24] + A[34] + A[16] + 3 A[27] + 2 A[17] + 7 A[10] + A[23] + 3 A[13] + 3 A[29] + 9 A[8]], [3 A[9] + 2 A[35] + 3 A[28] + 2 A[6] + 6 A[32] + 3 A[12] + 18 A[11] + 3 A[25] + 6 A[24] + A[30] + 9 A[27] + 2 A[19] + 18 A[8], 2 A[36] + 9 A[9] + 2 A[15] + 9 A[14] + 7 A[28] + 18 A[6] + 5 A[26] + 12 A[7] + A[31] + 12 A[12] + 3 A[25] + 2 A[34] + 2 A[17] + 9 A[10] + 6 A[23] + 3 A[13] + 3 A[29] + A[33], 2 A[36] + 2 A[37] + 7 A[9] + A[35] + 2 A[15] + 8 A[14] + 7 A[28] + 16 A[6] + 2 A[32] + 6 A[26] + 15 A[7] + 2 A[31] + 11 A[12] + 5 A[11] + 3 A[25] + 3 A[24] + 2 A[34] + A[16] + 3 A[27] + 2 A[17] + 10 A[10] + 7 A[23] + 9 A[13] + 4 A[29] + 9 A[8]], [2 A[38] + 4 A[6] + 5 A[32] + 6 A[11] + 4 A[24] + 2 A[30] + 4 A[27] + A[19] + 18 A[8] + 2 A[5], A[38] + 4 A[9] + A[35] + 2 A[15] + 3 A[14] + A[28] + 10 A[6] + 4 A[32] + 10 A[7] + A[31] + A[12] + 7 A[11] + 2 A[25] + 5 A[24] + A[30] + 2 A[16] + 5 A[27] + 4 A[23] + 19 A[8] + A[33], A[15] + 12 A[7] + 2 A[31] + 4 A[23]], [A[39] + A[38] + 5 A[9] + 2 A[15] + 3 A[14] + A[28] + 8 A[6] + 4 A[32] + A[26] + 11 A[7] + A[31] + A[12] + 7 A[11] + A[25] + 2 A[24] + A[34] + A[30] + 5 A[27] + 3 A[10] + 5 A[23] + A[19] + 14 A[8] + A[33], 2 A[39] + A[38] + 16 A[9] + 2 A[35] + 8 A[14] + 6 A[28] + 16 A[6] + 3 A[32] + 6 A[12] + 9 A[11] + 6 A[25] + 5 A[24] + 5 A[27] + 13 A[8] + 2 A[33], A[38] + 8 A[9] + 2 A[35] + A[15] + 2 A[14] + 2 A[28] + 4 A[6] + 3 A[32] + 3 A[26] + 9 A[7] + A[31] + 2 A[12] + 9 A[11] + 4 A[25] + 5 A[24] + A[34] + 5 A[27] + 7 A[10] + 5 A[23] + 2 A[13] + 2 A[29] + 13 A[8] + 2 A[33]], [ 2 A[39] + 2 A[38] + 13 A[9] + 2 A[35] + A[15] + 6 A[14] + 5 A[28] + 14 A[6] + 8 A[32] + 2 A[26] + 10 A[7] + 2 A[31] + 5 A[12] + 17 A[11] + 5 A[25] + 8 A[24] + 2 A[34] + A[30] + 9 A[27] + 6 A[10] + 4 A[23] + A[19] + A[8] + 2 A[33], 2 A[36] + A[39] + 12 A[9] + A[15] + 9 A[14] + 6 A[28] + 18 A[6] + 5 A[26] + 10 A[7] + 2 A[31] + 12 A[12] + 6 A[25] + 2 A[34] + 9 A[10] + 4 A[23] + 3 A[13] + 3 A[29] + 2 A[33], A[36] + 2 A[37] + 2 A[38] + 8 A[9] + A[35] + A[15] + 4 A[14] + 3 A[28] + 8 A[6] + 2 A[21] + 3 A[32] + 6 A[26] + 10 A[7] + A[31] + 5 A[12] + 6 A[11] + 4 A[25] + 4 A[24] + 4 A[27] + 6 A[10] + 6 A[23] + 17 A[13] + 7 A[29] + 14 A[8] + 2 A[33]], [ A[5], A[39] + A[38] + 10 A[9] + 2 A[35] + 4 A[14] + 3 A[28] + 13 A[6] + 3 A[32] + 3 A[12] + 9 A[11] + 4 A[25] + 5 A[24] + 2 A[30] + 5 A[27] + 13 A[8] + 2 A[33], 2 A[26] + 7 A[7] + 2 A[31] + 2 A[34] + 6 A[10] + 2 A[23]], [A[39] + 6 A[9] + A[35] + A[15] + 3 A[14] + 2 A[28] + 10 A[6] + 4 A[32] + 8 A[7] + 2 A[31] + 2 A[12] + 8 A[11] + 2 A[25] + 3 A[24] + 2 A[30] + 4 A[27] + 2 A[23] + A[41] + 12 A[8] + A[33], 6 A[9] + 3 A[14] + 6 A[6] + A[26] + A[7] + 3 A[25] + A[34] + 3 A[10] + A[23] + A[33], 2 A[36] + 8 A[9] + 2 A[15] + 6 A[14] + 4 A[28] + 12 A[6] + 6 A[26] + 16 A[7] + 2 A[31] + 8 A[12] + 4 A[25] + 2 A[34] + 11 A[10] + 8 A[23] + 2 A[13] + 2 A[29] + 2 A[33]], [ 2 A[39] + A[38] + 14 A[9] + 2 A[35] + 6 A[14] + 6 A[28] + 12 A[6] + 7 A[32] + A[26] + A[7] + 6 A[12] + 15 A[11] + 6 A[25] + 7 A[24] + A[34] + 8 A[27] + 3 A[10] + A[23] + A[41] + 23 A[8] + 2 A[33], A[36] + 2 A[39] + 2 A[38] + 14 A[9] + A[35] + 9 A[14] + 6 A[28] + 20 A[6] + 3 A[32] + 4 A[26] + A[7] + 9 A[12] + 6 A[11] + 6 A[25] + 4 A[24] + A[34] + A[30] + 4 A[27] + 6 A[10] + A[23] + 3 A[13] + 3 A[29] + 14 A[8] + 2 A[33], A[36] + A[37] + 2 A[38] + 8 A[9] + A[35] + 2 A[15] + 4 A[14] + 3 A[28] + 8 A[6] + A[21] + 3 A[32] + 3 A[26] + 11 A[7] + A[31] + 5 A[12] + 6 A[11] + 4 A[25] + 4 A[24] + 4 A[27] + 3 A[10] + 5 A[23] + 8 A[13] + 3 A[29] + 14 A[8] + 2 A[33]], [2 A[39] + 12 A[9] + 2 A[35] + 2 A[15] + 6 A[14] + 4 A[28] + 12 A[6] + 7 A[32] + 10 A[7] + A[31] + 4 A[12] + 16 A[11] + 4 A[25] + 7 A[24] + 8 A[27] + 4 A[23] + 2 A[41] + 21 A[8] + 2 A[5] + 2 A[33], 2 A[39] + 2 A[38] + 8 A[9] + A[35] + 7 A[14] + 3 A[28] + 20 A[6] + 3 A[32] + 3 A[12] + 6 A[11] + 2 A[25] + 4 A[24] + 2 A[30] + 4 A[27] + 14 A[8] + A[33], 2 A[15] + 12 A[7] + 6 A[23]], [2 A[39] + A[38] + 10 A[9] + A[15] + 6 A[14] + 2 A[28] + 14 A[6] + 2 A[32] + 2 A[26] + 10 A[7] + 2 A[31] + 2 A[12] + 2 A[11] + 2 A[25] + 2 A[24] + 2 A[34] + A[30] + 2 A[27] + 6 A[10] + 4 A[23] + 10 A[8] + 2 A[33], 2 A[39] + 12 A[9] + 8 A[14] + 3 A[28] + 18 A[6] + A[26] + A[7] + 3 A[12] + 2 A[25] + A[34] + A[30] + 3 A[10] + A[23] + 2 A[33], 2 A[36] + 2 A[38] + 3 A[9] + A[35] + A[15] + 4 A[14] + 5 A[28] + 8 A[6] + 3 A[32] + 3 A[26] + 11 A[7] + 2 A[31] + 9 A[12] + 6 A[11] + 3 A[25] + 4 A[24] + A[34] + 4 A[27] + 7 A[10] + 5 A[23] + 3 A[13] + 3 A[29] + 14 A[8]], [A[39] + 7 A[9] + 2 A[35] + 3 A[14] + 3 A[28] + 10 A[6] + 4 A[32] + 2 A[26] + 2 A[7] + 3 A[12] + 12 A[11] + 3 A[25] + 5 A[24] + 2 A[34] + 2 A[30] + 6 A[27] + 6 A[10] + 2 A[23] + 13 A[8] + A[33], A[38] + 10 A[9] + 2 A[35] + 3 A[14] + 4 A[28] + 10 A[6] + 3 A[32] + 2 A[26] + 2 A[7] + 6 A[12] + 9 A[11] + 6 A[25] + 5 A[24] + 2 A[34] + 2 A[30] + 5 A[27] + 6 A[10] + 2 A[23] + 13 A[8] + 2 A[33], A[36] + A[38] + 3 A[9] + 2 A[35] + A[15] + 2 A[14] + 4 A[28] + 4 A[6] + A[21] + 3 A[32] + 5 A[26] + 10 A[7] + 2 A[31] + 6 A[12] + 9 A[11] + 3 A[25] + 5 A[24] + 5 A[27] + 5 A[10] + 4 A[23] + 9 A[13] + 5 A[29] + 13 A[8]]], [1, 1, 2, 16, 1, 26, 1, 2, 7, 11, 16, 20, 25, 26, 1, 26, 1, 26, 1, 26, 1, 25, 2, 7, 20, 7, 11, 16, 11, 11, 16, 20, 7, 20, 25, 2, 25, 26, 1, 26, 1, 26, 1, 26, 1, 26]] For example, C(100000), mudolo , 27, equals , 25 The congruence classes mod, 27, in the following set , {0, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24}, never show up! Theorem Number, 27, : Let C(n) be the constant term, in x, of n (2/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], 0, 0, [3 A[5], 3 A[6], 3 A[7]], 0, 0, [9 A[5], 9 A[6], 9 A[7]], 0, 0, 0, 0], [1, 1, 3, 9, 1, 0, 0, 3, 0, 0, 9, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 28, : Let C(n) be the constant term, in x, of n (2/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 43, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [3 A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [13 A[5], A[30], A[31]], [A[32], A[33], A[18]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [9 A[5], 9 A[6] + 18 A[11], 9 A[7] + 9 A[12]], [6 A[5] + 9 A[8], 18 A[6] + 9 A[9] + 6 A[14], 15 A[7] + 9 A[10]], [ 18 A[8] + 9 A[11] + 3 A[16], 9 A[9] + 9 A[12] + 3 A[17], 9 A[13] + 3 A[18]] , [25 A[5], 12 A[6] + 4 A[14], 7 A[15]], [15 A[8] + 7 A[16], 6 A[9] + 7 A[17], 24 A[10] + 7 A[18]], [ 18 A[5] + 15 A[11] + 4 A[19], 18 A[6] + 6 A[12] + 4 A[20], 18 A[7] + 24 A[13] + 4 A[21]], [18 A[5] + 3 A[16], 3 A[17] + 6 A[22], 18 A[7] + 3 A[18]], [ 18 A[6] + 9 A[8] + 9 A[11] + 6 A[14] + 3 A[19] + A[22], 3 A[7] + 9 A[9] + 9 A[12] + 3 A[20] + 8 A[23], 9 A[10] + 3 A[21]], [24 A[5], 18 A[6] + 6 A[14], 6 A[7] + 6 A[23]], [15 A[5] + 8 A[24], 18 A[6] + 6 A[14] + 3 A[22] + 8 A[25], 9 A[10] + 6 A[18] + 6 A[23]], [ 9 A[5] + 15 A[6] + 18 A[11] + 6 A[14] + 6 A[19] + 2 A[22] + 3 A[24], 3 A[7] + 18 A[12] + 3 A[18] + 6 A[20] + 3 A[22] + 4 A[23] + 3 A[25] + 2 A[26], 6 A[7] + 3 A[18] + 6 A[21] + 6 A[23] + 5 A[26]], [ 9 A[5] + 18 A[6] + 6 A[14] + 6 A[19] + A[22] + 8 A[24] + 4 A[27], 9 A[6] + 3 A[14], 3 A[7] + 3 A[23]], [ 9 A[5] + 18 A[6] + 6 A[14] + 6 A[19] + A[22] + 6 A[24] + 4 A[27], 9 A[6] + 3 A[14] + 3 A[22] + 7 A[25], 6 A[7] + 6 A[18] + 9 A[23] + 11 A[26] ], [6 A[14] + 18 A[6] + A[22] + 4 A[24] + 5 A[27] + 6 A[19] + 12 A[5], 6 A[14] + A[28] + 18 A[6] + 6 A[22] + 8 A[25], 6 A[26] + 6 A[23] + A[29] + 6 A[18]], [2 A[24] + A[16] + 6 A[8] + 6 A[5], 6 A[9] + 4 A[14] + 12 A[6] + 2 A[25] + 2 A[30] + A[17], 6 A[26] + 6 A[10] + 6 A[23] + 7 A[18]], [4 A[14] + 12 A[6] + A[22] + 6 A[11] + 3 A[24] + 2 A[30] + 6 A[27] + 7 A[19] + 9 A[5], 2 A[14] + 2 A[28] + 6 A[6] + 6 A[12] + A[25] + A[30] + A[20], 7 A[21] + 15 A[13]], [4 A[32] + 10 A[24] + 2 A[16] + 18 A[8] + 12 A[5], 4 A[14] + 12 A[6] + 4 A[22] + 2 A[30], A[15] + 6 A[7] + 2 A[31] + 4 A[23]], [4 A[32] + 4 A[24] + 2 A[16] + 18 A[8] + 18 A[5], 4 A[14] + 12 A[6] + 3 A[22] + 3 A[25] + 2 A[30], 2 A[15] + 6 A[21] + 2 A[26] + 6 A[7] + A[31] + 9 A[23] + A[29]], [5 A[14] + 12 A[6] + 4 A[32] + 2 A[22] + 5 A[24] + A[30] + 2 A[16] + 18 A[8] + 15 A[5], 2 A[15] + 2 A[14] + 6 A[6] + 9 A[7] + A[31] + A[25] + A[30] + 7 A[23], A[15] + 5 A[26] + 6 A[7] + 2 A[31] + 4 A[34] + 18 A[10] + 9 A[23] + 9 A[13] + 2 A[18]], [6 A[35] + 5 A[14] + 12 A[6] + 4 A[32] + 2 A[22] + 2 A[24] + A[30] + 2 A[16] + 5 A[27] + 6 A[19] + 18 A[8] + A[5], A[14] + 3 A[6] + 3 A[22], 2 A[15] + 12 A[7] + 2 A[31] + 9 A[23]], [6 A[35] + 5 A[32] + 18 A[11] + 2 A[24] + 2 A[16] + 6 A[27] + 3 A[19] + 15 A[8] + 6 A[5], 15 A[9] + 4 A[14] + 12 A[6] + 2 A[25] + 2 A[30] + 2 A[17] + 5 A[33], 2 A[15] + 4 A[26] + 6 A[7] + A[31] + A[34] + 6 A[10] + 9 A[23]], [4 A[35] + 4 A[14] + 12 A[6] + 2 A[32] + A[22] + 15 A[11] + 2 A[30] + A[16] + 2 A[27] + 3 A[19] + 9 A[8], A[36] + 2 A[15] + 4 A[14] + 4 A[28] + 12 A[6] + 9 A[7] + A[31] + 6 A[12] + 2 A[25] + 2 A[30] + 7 A[23], 5 A[37] + 2 A[15] + 2 A[21] + 12 A[7] + A[31] + 9 A[23] + 15 A[13]], [ 0, 4 A[14] + 12 A[6] + A[22] + 2 A[30], 2 A[15] + 9 A[7] + A[31] + 7 A[23]] , [8 A[35] + 5 A[14] + 12 A[6] + 2 A[22] + 9 A[11] + 3 A[24] + A[30] + 6 A[27] + 7 A[19] + 6 A[5], 4 A[14] + 12 A[6] + 2 A[30], A[15] + 6 A[7] + 2 A[31] + 6 A[23]], [A[38] + 6 A[35] + 5 A[14] + 12 A[6] + 4 A[32] + 2 A[22] + 2 A[24] + A[30] + A[16] + 5 A[27] + 6 A[19] + 12 A[8] + 6 A[5], 2 A[14] + 6 A[6] + A[25] + A[30], 2 A[15] + A[26] + 6 A[7] + A[31] + 4 A[34] + 12 A[10] + 6 A[23] + A[40] + A[18]], [2 A[38] + 5 A[35] + 8 A[32] + 2 A[24] + 2 A[16] + 4 A[27] + 6 A[41] + A[19] + 24 A[8] + 4 A[5], 2 A[36] + 4 A[37] + 4 A[14] + 4 A[28] + 9 A[6] + 2 A[21] + 4 A[26] + 9 A[12] + 2 A[25] + 8 A[34] + 24 A[10] + 4 A[23] + 9 A[13] + 2 A[29] + 2 A[40] + 2 A[18] + A[20], 2 A[37] + A[15] + A[21] + 2 A[26] + 3 A[7] + 4 A[34] + 12 A[10] + 6 A[23] + 18 A[13] + A[29] + A[40] + A[18]], [5 A[32] + 2 A[16] + 15 A[8], 8 A[36] + A[39] + 9 A[9] + 2 A[42] + 5 A[14] + 8 A[28] + 12 A[6] + 24 A[12] + 7 A[25] + A[30] + 3 A[33] + 2 A[20], 3 A[34] + 9 A[10] + A[40]], [2 A[38] + 7 A[35] + 8 A[32] + 6 A[11] + 2 A[24] + 2 A[16] + 4 A[27] + 6 A[41] + 24 A[8] + 6 A[5], 7 A[36] + 2 A[42] + 3 A[14] + 6 A[28] + 6 A[6] + 21 A[12] + 3 A[25] + A[20] , A[37] + 2 A[15] + 4 A[26] + 6 A[7] + A[31] + 8 A[34] + 24 A[10] + 9 A[23] + 15 A[13] + 2 A[29] + 2 A[40] + 2 A[18]]], [1, 1, 3, 13, 1, 9, 25, 3, 24, 12, 13, 9, 22, 0, 7, 9, 6, 9, 25, 0, 16, 18, 21, 24, 18, 15, 12, 18, 12, 0, 19, 9, 6, 9, 22, 0, 22, 0, 6, 0, 7, 0, 25]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 8, 10, 11, 14, 17, 20, 23, 26}, never show up! Theorem Number, 29, : Let C(n) be the constant term, in x, of n (2/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 28, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [3 A[5], A[22], A[23]], [6 A[5] + 21 A[8], 24 A[6] + 21 A[9], A[24]], [A[25], A[26], A[27]], [17 A[5], 17 A[6] + 9 A[11], A[28]], [12 A[5] + 8 A[8], 21 A[6] + 8 A[9], 3 A[7] + 8 A[10]], [ 18 A[5] + 24 A[8] + 8 A[11], 18 A[6] + 6 A[9] + 8 A[12], 18 A[7] + 15 A[10] + 8 A[13]], [15 A[5] + 13 A[8], 6 A[6] + 22 A[9], 24 A[7] + 4 A[10]], [12 A[8] + 7 A[11], 3 A[9] + 16 A[12], 21 A[10] + 25 A[13]], [18 A[5], 18 A[6] + 18 A[11], 18 A[7] + 9 A[12]], [6 A[5] + 18 A[8], 24 A[6] + 18 A[9], 15 A[7] + 18 A[10]], [ 18 A[5] + 3 A[8] + 18 A[11], 18 A[6] + 21 A[9] + 18 A[12], 18 A[7] + 12 A[10] + 18 A[13]], [11 A[5], 11 A[6] + 9 A[11], 11 A[7] + 18 A[12]], [21 A[5] + 20 A[8], 3 A[6] + 20 A[9], 12 A[7] + 20 A[10]], [ 18 A[5] + 15 A[8] + 11 A[11], 18 A[6] + 24 A[9] + 11 A[12], 18 A[7] + 6 A[10] + 11 A[13]], [12 A[8], 12 A[9], 12 A[10]], [21 A[11], 21 A[12], 21 A[13]], [6 A[11], 6 A[12], 6 A[13]], [15 A[5], 15 A[6], 15 A[7]], [15 A[8], 15 A[9], 15 A[10]], [15 A[11], 15 A[12], 15 A[13]], [15 A[8] + 2 A[11], 24 A[9] + 20 A[12], 6 A[10] + 11 A[13]]], [1, 1, 3, 17, 1, 18, 11, 3, 15, 15, 17, 9, 10, 0, 20, 18, 6, 9, 11, 0, 7, 9, 6, 21, 15, 18, 12, 25]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 8, 13, 14, 16, 19, 22, 23, 24, 26}, never show up! Theorem Number, 30, : Let C(n) be the constant term, in x, of n (2/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[9], A[13]], [A[5], A[14], A[15]], 0, 0, [3 A[5], 3 A[6], 3 A[7]], 0, 0, [21 A[5], 21 A[6], 21 A[7]], 0, 0, 0, 0], [1, 1, 3, 21, 1, 0, 0, 3, 0, 0, 21, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26}, never show up! Theorem Number, 31, : Let C(n) be the constant term, in x, of n (3/x + x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 12, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], 0, [A[8], A[9], A[10]], [A[5], A[6], A[7]], 0, 0, [6 A[5], 6 A[6], 6 A[7]], 0, 0, 0, 0], [1, 1, 0, 6, 1, 0, 0, 6, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 32, : Let C(n) be the constant term, in x, of n (3/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 12, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], 0, [A[8], A[9], A[10]], [A[5], A[11], A[12]], 0, 0, [12 A[5], 12 A[6], 12 A[7]], 0, 0, 0, 0], [1, 1, 0, 12, 1, 0, 0, 12, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 33, : Let C(n) be the constant term, in x, of n (3/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 13, states . Here it is: [[[A[2], A[3], A[4]], %1, %1, %1, %1, [ (676 c[5] + 27) A[5] + (26 c[5] + 1) A[6], (676 c[5] + 27) A[5] + (26 c[5] + 1) A[6], (676 c[5] + 27) A[5] + (26 c[5] + 1) A[6]], [ (18928 c[5] + 18252 c[6] + 729) A[5] + (702 c[5] + 676 c[6] + 27) A[6] + (26 c[5] + 26 c[6] + 1) A[7], (18928 c[5] + 18252 c[6] + 729) A[5] + (702 c[5] + 676 c[6] + 27) A[6] + (26 c[5] + 26 c[6] + 1) A[7], (18928 c[5] + 18252 c[6] + 729) A[5] + (702 c[5] + 676 c[6] + 27) A[6] + (26 c[5] + 26 c[6] + 1) A[7]], [ (19683 + 511732 c[5] + 511056 c[6] + 492804 c[7]) A[5] + (18954 c[5] + 18928 c[6] + 18252 c[7] + 729) A[6] + (702 c[5] + 702 c[6] + 676 c[7] + 27) A[7] + (26 c[5] + 26 c[6] + 26 c[7] + 1) A[8], (19683 + 511732 c[5] + 511056 c[6] + 492804 c[7]) A[5] + (18954 c[5] + 18928 c[6] + 18252 c[7] + 729) A[6] + (702 c[5] + 702 c[6] + 676 c[7] + 27) A[7] + (26 c[5] + 26 c[6] + 26 c[7] + 1) A[8], (19683 + 511732 c[5] + 511056 c[6] + 492804 c[7]) A[5] + (18954 c[5] + 18928 c[6] + 18252 c[7] + 729) A[6] + (702 c[5] + 702 c[6] + 676 c[7] + 27) A[7] + (26 c[5] + 26 c[6] + 26 c[7] + 1) A[8]], [ (13817440 c[5] + 13816764 c[6] + 13798512 c[7] + 13305708 c[8] + 531441) A[5] + (511758 c[5] + 19683 + 511732 c[6] + 511056 c[7] + 492804 c[8]) A[6] + (18954 c[5] + 18954 c[6] + 18928 c[7] + 18252 c[8] + 729) A[7] + (702 c[5] + 702 c[6] + 702 c[7] + 676 c[8] + 27) A[8] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 1) A[9], (13817440 c[5] + 13816764 c[6] + 13798512 c[7] + 13305708 c[8] + 531441) A[5] + (511758 c[5] + 19683 + 511732 c[6] + 511056 c[7] + 492804 c[8]) A[6] + (18954 c[5] + 18954 c[6] + 18928 c[7] + 18252 c[8] + 729) A[7] + (702 c[5] + 702 c[6] + 702 c[7] + 676 c[8] + 27) A[8] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 1) A[9], (13817440 c[5] + 13816764 c[6] + 13798512 c[7] + 13305708 c[8] + 531441) A[5] + (511758 c[5] + 19683 + 511732 c[6] + 511056 c[7] + 492804 c[8]) A[6] + (18954 c[5] + 18954 c[6] + 18928 c[7] + 18252 c[8] + 729) A[7] + (702 c[5] + 702 c[6] + 702 c[7] + 676 c[8] + 27) A[8] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 1) A[9]], [(373071556 c[5] + 373070880 c[6] + 373052628 c[7] + 372559824 c[8] + 359254116 c[9] + 14348907) A[5] + (13817466 c[5] + 13817440 c[6] + 13816764 c[7] + 13798512 c[8] + 13305708 c[9] + 531441) A[6] + ( 511758 c[5] + 511758 c[6] + 19683 + 511732 c[7] + 511056 c[8] + 492804 c[9] ) A[7] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18928 c[8] + 18252 c[9] + 729) A[8] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 676 c[9] + 27) A[9] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 1) A[10], ( 373071556 c[5] + 373070880 c[6] + 373052628 c[7] + 372559824 c[8] + 359254116 c[9] + 14348907) A[5] + (13817466 c[5] + 13817440 c[6] + 13816764 c[7] + 13798512 c[8] + 13305708 c[9] + 531441) A[6] + ( 511758 c[5] + 511758 c[6] + 19683 + 511732 c[7] + 511056 c[8] + 492804 c[9] ) A[7] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18928 c[8] + 18252 c[9] + 729) A[8] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 676 c[9] + 27) A[9] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 1) A[10], ( 373071556 c[5] + 373070880 c[6] + 373052628 c[7] + 372559824 c[8] + 359254116 c[9] + 14348907) A[5] + (13817466 c[5] + 13817440 c[6] + 13816764 c[7] + 13798512 c[8] + 13305708 c[9] + 531441) A[6] + ( 511758 c[5] + 511758 c[6] + 19683 + 511732 c[7] + 511056 c[8] + 492804 c[9] ) A[7] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18928 c[8] + 18252 c[9] + 729) A[8] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 676 c[9] + 27) A[9] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 1) A[10]], [( 10072932688 c[5] + 10072932012 c[6] + 10072913760 c[7] + 10072420956 c[8] + 10059115248 c[9] + 9699861132 c[10] + 387420489) A[5] + (373071582 c[5] + 373071556 c[6] + 373070880 c[7] + 373052628 c[8] + 372559824 c[9] + 359254116 c[10] + 14348907) A[6] + (13817466 c[5] + 13817466 c[6] + 13817440 c[7] + 13816764 c[8] + 13798512 c[9] + 13305708 c[10] + 531441) A[7] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 19683 + 511732 c[8] + 511056 c[9] + 492804 c[10]) A[8] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18928 c[9] + 18252 c[10] + 729) A[9] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 676 c[10] + 27) A[10] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 1) A[11], (10072932688 c[5] + 10072932012 c[6] + 10072913760 c[7] + 10072420956 c[8] + 10059115248 c[9] + 9699861132 c[10] + 387420489) A[5] + (373071582 c[5] + 373071556 c[6] + 373070880 c[7] + 373052628 c[8] + 372559824 c[9] + 359254116 c[10] + 14348907) A[6] + (13817466 c[5] + 13817466 c[6] + 13817440 c[7] + 13816764 c[8] + 13798512 c[9] + 13305708 c[10] + 531441) A[7] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 19683 + 511732 c[8] + 511056 c[9] + 492804 c[10]) A[8] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18928 c[9] + 18252 c[10] + 729) A[9] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 676 c[10] + 27) A[10] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 1) A[11], (10072932688 c[5] + 10072932012 c[6] + 10072913760 c[7] + 10072420956 c[8] + 10059115248 c[9] + 9699861132 c[10] + 387420489) A[5] + (373071582 c[5] + 373071556 c[6] + 373070880 c[7] + 373052628 c[8] + 372559824 c[9] + 359254116 c[10] + 14348907) A[6] + (13817466 c[5] + 13817466 c[6] + 13817440 c[7] + 13816764 c[8] + 13798512 c[9] + 13305708 c[10] + 531441) A[7] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 19683 + 511732 c[8] + 511056 c[9] + 492804 c[10]) A[8] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18928 c[9] + 18252 c[10] + 729) A[9] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 676 c[10] + 27) A[10] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 1) A[11]], [(271969183252 c[5] + 271969182576 c[6] + 271969164324 c[7] + 271968671520 c[8] + 10460353203 + 271955365812 c[9] + 271596111696 c[10] + 261896250564 c[11]) A[5] + (10072932714 c[5] + 10072932688 c[6] + 10072932012 c[7] + 10072913760 c[8] + 10072420956 c[9] + 10059115248 c[10] + 9699861132 c[11] + 387420489) A[6] + (373071582 c[5] + 373071582 c[6] + 373071556 c[7] + 373070880 c[8] + 373052628 c[9] + 372559824 c[10] + 359254116 c[11] + 14348907) A[7] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817440 c[8] + 13816764 c[9] + 13798512 c[10] + 13305708 c[11] + 531441) A[8] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 19683 + 511732 c[9] + 511056 c[10] + 492804 c[11]) A[9] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18928 c[10] + 18252 c[11] + 729) A[10] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 676 c[11] + 27) A[11] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 1) A[12], (271969183252 c[5] + 271969182576 c[6] + 271969164324 c[7] + 271968671520 c[8] + 10460353203 + 271955365812 c[9] + 271596111696 c[10] + 261896250564 c[11]) A[5] + (10072932714 c[5] + 10072932688 c[6] + 10072932012 c[7] + 10072913760 c[8] + 10072420956 c[9] + 10059115248 c[10] + 9699861132 c[11] + 387420489) A[6] + (373071582 c[5] + 373071582 c[6] + 373071556 c[7] + 373070880 c[8] + 373052628 c[9] + 372559824 c[10] + 359254116 c[11] + 14348907) A[7] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817440 c[8] + 13816764 c[9] + 13798512 c[10] + 13305708 c[11] + 531441) A[8] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 19683 + 511732 c[9] + 511056 c[10] + 492804 c[11]) A[9] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18928 c[10] + 18252 c[11] + 729) A[10] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 676 c[11] + 27) A[11] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 1) A[12], (271969183252 c[5] + 271969182576 c[6] + 271969164324 c[7] + 271968671520 c[8] + 10460353203 + 271955365812 c[9] + 271596111696 c[10] + 261896250564 c[11]) A[5] + (10072932714 c[5] + 10072932688 c[6] + 10072932012 c[7] + 10072913760 c[8] + 10072420956 c[9] + 10059115248 c[10] + 9699861132 c[11] + 387420489) A[6] + (373071582 c[5] + 373071582 c[6] + 373071556 c[7] + 373070880 c[8] + 373052628 c[9] + 372559824 c[10] + 359254116 c[11] + 14348907) A[7] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817440 c[8] + 13816764 c[9] + 13798512 c[10] + 13305708 c[11] + 531441) A[8] + (511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 19683 + 511732 c[9] + 511056 c[10] + 492804 c[11]) A[9] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18928 c[10] + 18252 c[11] + 729) A[10] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 676 c[11] + 27) A[11] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 1) A[12]], [(7343167948480 c[5] + 7343167947804 c[6] + 7343167929552 c[7] + 7343167436748 c[8] + 7343154131040 c[9] + 7342794876924 c[10] + 7333095015792 c[11] + 7071198765228 c[12] + 282429536481) A[5] + ( 271969183278 c[5] + 271969183252 c[6] + 271969182576 c[7] + 271969164324 c[8] + 271968671520 c[9] + 10460353203 + 271955365812 c[10] + 271596111696 c[11] + 261896250564 c[12]) A[6] + (10072932714 c[5] + 10072932714 c[6] + 10072932688 c[7] + 10072932012 c[8] + 10072913760 c[9] + 10072420956 c[10] + 10059115248 c[11] + 9699861132 c[12] + 387420489) A[7] + (373071582 c[5] + 373071582 c[6] + 373071582 c[7] + 373071556 c[8] + 373070880 c[9] + 373052628 c[10] + 372559824 c[11] + 359254116 c[12] + 14348907) A[8] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817466 c[8] + 13817440 c[9] + 13816764 c[10] + 13798512 c[11] + 13305708 c[12] + 531441) A[9] + ( 511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 511758 c[9] + 19683 + 511732 c[10] + 511056 c[11] + 492804 c[12]) A[10] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18954 c[10] + 18928 c[11] + 18252 c[12] + 729) A[11] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 702 c[11] + 676 c[12] + 27) A[12] + ( 26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 26 c[12] + 1) A[13], (7343167948480 c[5] + 7343167947804 c[6] + 7343167929552 c[7] + 7343167436748 c[8] + 7343154131040 c[9] + 7342794876924 c[10] + 7333095015792 c[11] + 7071198765228 c[12] + 282429536481) A[5] + (271969183278 c[5] + 271969183252 c[6] + 271969182576 c[7] + 271969164324 c[8] + 271968671520 c[9] + 10460353203 + 271955365812 c[10] + 271596111696 c[11] + 261896250564 c[12]) A[6] + ( 10072932714 c[5] + 10072932714 c[6] + 10072932688 c[7] + 10072932012 c[8] + 10072913760 c[9] + 10072420956 c[10] + 10059115248 c[11] + 9699861132 c[12] + 387420489) A[7] + (373071582 c[5] + 373071582 c[6] + 373071582 c[7] + 373071556 c[8] + 373070880 c[9] + 373052628 c[10] + 372559824 c[11] + 359254116 c[12] + 14348907) A[8] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817466 c[8] + 13817440 c[9] + 13816764 c[10] + 13798512 c[11] + 13305708 c[12] + 531441) A[9] + ( 511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 511758 c[9] + 19683 + 511732 c[10] + 511056 c[11] + 492804 c[12]) A[10] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18954 c[10] + 18928 c[11] + 18252 c[12] + 729) A[11] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 702 c[11] + 676 c[12] + 27) A[12] + ( 26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 26 c[12] + 1) A[13], (7343167948480 c[5] + 7343167947804 c[6] + 7343167929552 c[7] + 7343167436748 c[8] + 7343154131040 c[9] + 7342794876924 c[10] + 7333095015792 c[11] + 7071198765228 c[12] + 282429536481) A[5] + (271969183278 c[5] + 271969183252 c[6] + 271969182576 c[7] + 271969164324 c[8] + 271968671520 c[9] + 10460353203 + 271955365812 c[10] + 271596111696 c[11] + 261896250564 c[12]) A[6] + ( 10072932714 c[5] + 10072932714 c[6] + 10072932688 c[7] + 10072932012 c[8] + 10072913760 c[9] + 10072420956 c[10] + 10059115248 c[11] + 9699861132 c[12] + 387420489) A[7] + (373071582 c[5] + 373071582 c[6] + 373071582 c[7] + 373071556 c[8] + 373070880 c[9] + 373052628 c[10] + 372559824 c[11] + 359254116 c[12] + 14348907) A[8] + (13817466 c[5] + 13817466 c[6] + 13817466 c[7] + 13817466 c[8] + 13817440 c[9] + 13816764 c[10] + 13798512 c[11] + 13305708 c[12] + 531441) A[9] + ( 511758 c[5] + 511758 c[6] + 511758 c[7] + 511758 c[8] + 511758 c[9] + 19683 + 511732 c[10] + 511056 c[11] + 492804 c[12]) A[10] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18954 c[8] + 18954 c[9] + 18954 c[10] + 18928 c[11] + 18252 c[12] + 729) A[11] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 702 c[9] + 702 c[10] + 702 c[11] + 676 c[12] + 27) A[12] + ( 26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 26 c[10] + 26 c[11] + 26 c[12] + 1) A[13]]], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] %1 := [A[5], A[5], A[5]] For example, C(100000), mudolo , 27, equals , 1 The congruence classes mod, 27, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 34, : Let C(n) be the constant term, in x, of n (3/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 46, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [7 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [19 A[5], 10 A[6] + 25 A[7] + 9 A[9] + 2 A[15], 6 A[7] + 9 A[12] + 4 A[15]] , [21 A[5] + 14 A[6] + 2 A[8] + 22 A[9] + 6 A[11] + 4 A[14] + 8 A[16] + 5 A[17], 3 A[5] + 18 A[6] + 26 A[7] + 22 A[8] + 17 A[9] + 3 A[11] + 6 A[14] + A[15] + 8 A[16] + 5 A[17], 18 A[6] + 17 A[9] + 13 A[10] + 24 A[12] + 6 A[14] + 6 A[15] + 7 A[17]], [A[44], A[45], A[46]], [10 A[5], 5 A[6] + 15 A[11] + 5 A[14] + 9 A[16] + 12 A[19], 21 A[5] + 5 A[7] + 19 A[8] + 11 A[9] + 12 A[11] + 5 A[15] + 2 A[16] + 7 A[17] + 9 A[19]], [ 6 A[5] + 8 A[6] + 17 A[8] + 16 A[9] + 18 A[10] + 15 A[11] + 18 A[12] + 10 A[14] + 8 A[16] + 11 A[17] + 9 A[18] + 9 A[19] + 9 A[20], 6 A[5] + 4 A[6] + 4 A[7] + 8 A[8] + 11 A[9] + 17 A[10] + 10 A[11] + 14 A[12] + 5 A[14] + 5 A[15] + 4 A[16] + 11 A[17] + 10 A[18] + 2 A[19] + 7 A[20], 18 A[8] + 12 A[9] + 12 A[10] + 12 A[11] + 12 A[12] + 6 A[17] + 7 A[18] + 6 A[19] + 6 A[20]], [15 A[5] + 14 A[6] + 23 A[8] + 25 A[9] + 15 A[10] + 21 A[11] + 24 A[12] + 4 A[14] + 10 A[16] + 11 A[17] + 12 A[18] + 13 A[19] + 12 A[20], 6 A[5] + 9 A[6] + 7 A[7] + 8 A[8] + 20 A[9] + 17 A[10] + 12 A[11] + 21 A[12] + 3 A[14] + 2 A[15] + 7 A[16] + 10 A[17] + 10 A[18] + 3 A[19] + 10 A[20], 6 A[6] + 4 A[7] + 18 A[8] + 24 A[9] + 18 A[10] + 3 A[11] + 7 A[12] + 8 A[13] + 3 A[14] + 2 A[15] + 18 A[17] + 9 A[18] + 15 A[19] + 17 A[20] + 2 A[21]], [9 A[5] + 7 A[6] + 22 A[7] + 21 A[8] + 21 A[9] + 15 A[10] + 15 A[11] + 20 A[12] + 25 A[13] + 2 A[14] + 8 A[15] + 7 A[16] + 9 A[17] + 12 A[18] + 12 A[19] + 10 A[20] + 2 A[21], 3 A[5] + 13 A[6] + 15 A[7] + 4 A[8] + 7 A[9] + 16 A[10] + 5 A[11] + 6 A[12] + 25 A[13] + 4 A[14] + 6 A[15] + 2 A[16] + 3 A[17] + 11 A[18] + A[19] + 3 A[20] + 2 A[21] + A[22], 9 A[10] + A[18]], [3 A[5] + 7 A[6] + 4 A[8] + 14 A[9] + 21 A[10] + 9 A[11] + 12 A[12] + 2 A[14] + 8 A[16] + 4 A[17] + 6 A[18] + 4 A[19] + 6 A[20], 22 A[6] + 23 A[7] + 12 A[9] + 21 A[10] + 2 A[11] + 11 A[12] + 26 A[13] + 7 A[14] + 2 A[15] + 3 A[16] + 3 A[17] + 6 A[18] + A[19] + 5 A[20] + A[21] + A[22] + 8 A[23], 20 A[7] + 9 A[8] + 6 A[9] + 6 A[11] + 6 A[12] + 15 A[13] + 3 A[17] + 3 A[19] + 3 A[20] + 4 A[21] + 7 A[23]], [7 A[5] + 14 A[6] + 10 A[7] + 22 A[8] + 16 A[9] + 19 A[10] + 11 A[11] + 14 A[12] + 26 A[13] + 4 A[14] + A[15] + 4 A[16] + 5 A[17] + 8 A[18] + 4 A[19] + 7 A[20] + A[21] + 4 A[23] + 4 A[24], 3 A[5] + 21 A[6] + 10 A[7] + 10 A[8] + 10 A[9] + 22 A[10] + 5 A[11] + 8 A[12] + 26 A[13] + 6 A[14] + A[15] + 2 A[16] + 2 A[17] + 5 A[18] + A[19] + 4 A[20] + A[21] + A[22] + 4 A[23] + 3 A[24], 3 A[5] + 8 A[7] + 10 A[8] + 4 A[9] + 25 A[10] + 5 A[11] + 4 A[12] + 2 A[16] + 2 A[17] + 2 A[18] + A[19] + 2 A[20] + 2 A[23] + 3 A[24]], [6 A[5] + 3 A[6] + 10 A[7] + 18 A[8] + 11 A[9] + 25 A[10] + 8 A[11] + 2 A[12] + 26 A[13] + 13 A[14] + A[15] + 4 A[16] + 10 A[17] + 2 A[18] + 4 A[19] + A[20] + A[21] + 2 A[22] + 4 A[23] + 3 A[24] + 9 A[25], 4 A[6] + 20 A[7] + 21 A[9] + 12 A[14] + A[15] + 3 A[17] + 2 A[22] + 6 A[23] + 4 A[25], 26 A[6] + 11 A[7] + 15 A[9] + 9 A[10] + 2 A[12] + 25 A[13] + 9 A[14] + 2 A[15] + 3 A[17] + 4 A[18] + A[20] + 2 A[21] + A[22] + 14 A[23] + 3 A[25]], [24 A[6] + 23 A[7] + 5 A[8] + A[9] + 24 A[10] + 3 A[11] + 4 A[12] + 26 A[13] + 10 A[14] + A[15] + 8 A[17] + 6 A[18] + A[19] + 2 A[20] + A[21] + 2 A[22] + 9 A[23] + A[24] + 6 A[25] + 6 A[26], 17 A[6] + 24 A[7] + 14 A[9] + 4 A[10] + 5 A[12] + 25 A[13] + 6 A[14] + 4 A[17] + 12 A[18] + 2 A[20] + 2 A[21] + A[22] + 12 A[23] + 3 A[25] + 14 A[26], 7 A[7] + 24 A[13] + A[21] + 2 A[23] + 3 A[26]], [4 A[5] + 11 A[6] + 16 A[7] + 8 A[8] + 8 A[9] + 8 A[10] + 4 A[11] + 26 A[13] + 4 A[14] + 2 A[16] + 2 A[17] + 2 A[18] + A[19] + A[21] + 5 A[23] + 2 A[24] + 2 A[25] + 2 A[26] + A[27], 15 A[6] + 4 A[14], 18 A[7] + 9 A[10] + A[15] + A[18] + 6 A[23] + 2 A[26]], [3 A[5] + 18 A[6] + 16 A[7] + 9 A[8] + A[9] + 20 A[10] + 4 A[11] + 4 A[12] + 25 A[13] + 8 A[14] + 2 A[15] + 3 A[16] + 8 A[17] + 2 A[19] + 2 A[20] + 2 A[21] + A[22] + 6 A[23] + A[24] + 6 A[25] + 4 A[26], 14 A[6] + 23 A[9] + 19 A[10] + 6 A[12] + 6 A[14] + 2 A[15] + 6 A[17] + A[18] + 2 A[20] + A[22] + 10 A[23] + 5 A[25] + 4 A[26] + A[28], A[6] + 13 A[7] + 8 A[9] + 4 A[10] + 4 A[12] + 26 A[13] + 9 A[14] + 2 A[15] + 2 A[17] + 7 A[18] + A[20] + A[21] + 2 A[22] + 15 A[23] + 2 A[25] + 11 A[26] + A[28]], [ 13 A[9] + 11 A[14] + 25 A[6] + 2 A[26] + 4 A[7] + 2 A[12] + 7 A[11] + 9 A[25] + 4 A[24] + 2 A[16] + A[27] + 8 A[17] + 8 A[10] + 2 A[23] + 2 A[19] + 12 A[8] + 3 A[5] + 2 A[18] + A[20], 11 A[9] + A[15] + 12 A[14] + 2 A[28] + 3 A[6] + 4 A[26] + A[7] + 9 A[12] + 9 A[25] + 7 A[17] + 16 A[10] + 10 A[23] + 4 A[18] + 2 A[20], 23 A[9] + A[15] + 6 A[14] + A[28] + 15 A[6] + 2 A[21] + 8 A[26] + 9 A[7] + 6 A[12] + 5 A[25] + 5 A[17] + A[10] + 14 A[23] + 12 A[13] + 2 A[29] + 6 A[18] + 2 A[20]], [ 26 A[9] + 10 A[14] + 25 A[6] + 2 A[26] + 4 A[7] + A[22] + 2 A[12] + 10 A[11] + 6 A[25] + 3 A[24] + 5 A[16] + A[27] + 7 A[17] + 8 A[10] + 2 A[23] + 4 A[19] + 17 A[8] + 6 A[5] + 2 A[18] + A[20], 3 A[9] + 2 A[15] + 3 A[14] + 9 A[6] + 19 A[7] + A[25] + 6 A[23], 2 A[9] + 2 A[15] + 4 A[14] + A[28] + 18 A[6] + 3 A[26] + 21 A[7] + 2 A[12] + A[25] + 2 A[30] + 7 A[10] + 7 A[23] + 4 A[13] + 2 A[29]], [26 A[9] + 10 A[14] + A[6] + 2 A[26] + 4 A[7] + 2 A[22] + 2 A[12] + 7 A[11] + 6 A[25] + 3 A[24] + 2 A[30] + 4 A[16] + 7 A[17] + 8 A[10] + 2 A[23] + 3 A[19] + 14 A[8] + 6 A[5] + 2 A[18] + A[20], 18 A[9] + 8 A[14] + 2 A[28] + 22 A[6] + 6 A[26] + 12 A[7] + 2 A[22] + 9 A[12] + 4 A[25] + A[30] + 5 A[17] + 24 A[10] + 6 A[23] + 6 A[18] + 2 A[20], 2 A[9] + 4 A[14] + A[28] + 18 A[6] + 2 A[21] + 3 A[26] + 6 A[7] + 2 A[31] + 2 A[12] + A[25] + 2 A[30] + 10 A[10] + A[23] + 7 A[13] + A[29] + 2 A[18] ], [8 A[9] + 2 A[15] + 5 A[14] + A[28] + 16 A[6] + A[32] + 2 A[26] + 20 A[7] + 2 A[31] + 2 A[22] + 4 A[12] + 2 A[11] + 2 A[25] + 2 A[24] + A[30] + A[16] + A[27] + 2 A[17] + 8 A[10] + 6 A[23] + 8 A[8] + A[5] + 2 A[18] + A[20], 4 A[14] + 15 A[6] + A[22] + 2 A[30], A[15] + 15 A[7] + 2 A[31] + 4 A[23]], [24 A[9] + 2 A[15] + 5 A[14] + A[28] + 12 A[6] + 7 A[32] + 4 A[26] + 24 A[7] + 2 A[31] + 6 A[12] + 6 A[11] + 6 A[25] + 3 A[24] + A[30] + 4 A[16] + A[27] + A[17] + 16 A[10] + 8 A[23] + 2 A[19] + 2 A[8] + 3 A[5] + 5 A[33] + 4 A[18] + 2 A[20], 19 A[9] + A[15] + 5 A[14] + 2 A[28] + 12 A[6] + 2 A[26] + 12 A[7] + A[31] + 6 A[12] + 5 A[25] + A[30] + A[17] + 8 A[10] + 4 A[23] + 3 A[33] + 2 A[18] + A[20], 16 A[9] + A[15] + 3 A[14] + A[28] + 10 A[6] + 2 A[32] + 5 A[26] + 14 A[7] + A[31] + 4 A[12] + 6 A[11] + 4 A[25] + 2 A[24] + 2 A[30] + A[16] + 2 A[27] + A[17] + 19 A[10] + 5 A[23] + A[19] + 2 A[13] + A[29] + 10 A[8] + 3 A[33] + 4 A[18] + A[20]], [22 A[9] + A[15] + 3 A[14] + A[28] + 8 A[6] + 6 A[32] + 6 A[7] + 4 A[12] + 5 A[11] + 4 A[25] + 2 A[24] + A[34] + A[30] + 4 A[16] + A[17] + 2 A[10] + 2 A[23] + 2 A[19] + 24 A[8] + 6 A[5] + 6 A[33] + A[20], 14 A[9] + 2 A[15] + 2 A[14] + A[28] + 6 A[6] + 12 A[7] + 3 A[12] + 3 A[25] + 2 A[34] + A[30] + 2 A[17] + 4 A[10] + 4 A[23] + 2 A[33], 20 A[9] + A[15] + 4 A[14] + 2 A[28] + 8 A[6] + 3 A[21] + 4 A[32] + 2 A[26] + 14 A[7] + A[31] + 8 A[12] + 6 A[11] + 6 A[25] + A[24] + 5 A[34] + 2 A[16] + A[27] + 14 A[10] + 5 A[23] + 2 A[19] + 10 A[13] + 14 A[8] + 4 A[33] + 2 A[20]], [14 A[9] + 4 A[35] + A[15] + 4 A[14] + 10 A[6] + 2 A[32] + 10 A[7] + 2 A[31] + 4 A[12] + 14 A[11] + 4 A[25] + 4 A[24] + 2 A[34] + A[30] + 2 A[27] + 4 A[10] + 2 A[23] + A[19] + 12 A[8] + A[5] + 3 A[33] + 2 A[20], 12 A[9] + 2 A[35] + A[15] + 6 A[14] + 17 A[6] + 4 A[32] + 10 A[7] + 2 A[31] + 4 A[12] + 8 A[11] + 4 A[25] + 4 A[24] + 2 A[34] + 2 A[30] + 2 A[16] + 2 A[27] + 4 A[10] + 2 A[23] + 20 A[8] + 2 A[33] + 2 A[20], 12 A[9] + 2 A[15] + 2 A[14] + 6 A[6] + 19 A[7] + A[31] + 2 A[25] + A[30] + 2 A[17] + 6 A[23] + 2 A[33] ], [A[36] + 10 A[9] + 3 A[35] + 3 A[14] + 6 A[6] + A[32] + 2 A[12] + 10 A[11] + A[25] + 4 A[24] + A[27] + A[17] + A[19] + 11 A[8] + 3 A[33], 11 A[9] + A[35] + 2 A[15] + 2 A[14] + A[28] + 4 A[6] + 2 A[32] + 16 A[7] + 2 A[31] + 2 A[12] + 4 A[11] + 3 A[25] + 2 A[24] + A[16] + A[27] + A[17] + 4 A[23] + 10 A[8] + A[33], 2 A[36] + 8 A[9] + 2 A[35] + A[15] + 4 A[14] + 2 A[28] + 8 A[6] + 4 A[32] + 2 A[26] + 10 A[7] + A[31] + 8 A[12] + 8 A[11] + 2 A[25] + 4 A[24] + 2 A[16] + 2 A[27] + 5 A[10] + 3 A[23] + 2 A[13] + A[29] + 20 A[8] + 2 A[33]], [2 A[36] + A[37] + 12 A[9] + 6 A[35] + 4 A[14] + A[28] + 12 A[6] + 2 A[32] + A[26] + 4 A[7] + 6 A[12] + 17 A[11] + A[25] + 7 A[24] + A[34] + 2 A[30] + A[16] + 4 A[10] + 2 A[23] + 2 A[19] + 4 A[13] + A[29] + 20 A[8] + 5 A[33], A[36] + A[37] + 10 A[9] + 2 A[35] + 2 A[15] + 3 A[14] + 8 A[6] + 4 A[32] + A[26] + 18 A[7] + A[31] + 3 A[12] + 8 A[11] + 2 A[25] + 4 A[24] + 2 A[34] + A[30] + 2 A[16] + 2 A[27] + 6 A[10] + 6 A[23] + 4 A[13] + A[29] + 20 A[8] + 3 A[33], A[37] + 2 A[21] + A[26] + 8 A[7] + 2 A[34] + 6 A[10] + 4 A[23] + 9 A[13] + A[29] ], [A[5], 2 A[37] + A[35] + 2 A[15] + 5 A[6] + 2 A[32] + 2 A[26] + 24 A[7] + 2 A[31] + 4 A[11] + 2 A[24] + 2 A[34] + 2 A[30] + A[16] + A[27] + 8 A[10] + 8 A[23] + 8 A[13] + 2 A[29] + 10 A[8], 2 A[15] + 15 A[7] + A[31] + 4 A[23]], [2 A[37] + A[38] + 3 A[35] + 2 A[15] + 2 A[6] + 4 A[32] + 2 A[26] + 24 A[7] + 2 A[31] + 10 A[11] + 6 A[24] + 2 A[34] + A[30] + A[16] + 2 A[27] + 8 A[10] + 8 A[23] + 8 A[13] + 2 A[29] + 25 A[8], 11 A[9] + A[35] + 2 A[15] + A[14] + A[28] + 4 A[6] + 2 A[32] + 16 A[7] + 2 A[31] + 2 A[12] + 4 A[11] + 3 A[25] + 2 A[24] + A[30] + A[16] + A[27] + 4 A[23] + 10 A[8] + 2 A[33], A[36] + A[37] + 4 A[9] + A[35] + A[15] + 2 A[14] + A[28] + 4 A[6] + 2 A[32] + 3 A[26] + 12 A[7] + 4 A[12] + 4 A[11] + A[25] + 2 A[24] + 3 A[34] + A[16] + A[27] + 13 A[10] + 5 A[23] + 4 A[13] + A[29] + 10 A[8] + A[33]], [A[37] + 4 A[9] + A[28] + 3 A[26] + 12 A[7] + 2 A[31] + 2 A[12] + 3 A[11] + A[25] + 2 A[24] + 3 A[34] + A[27] + 16 A[10] + 4 A[23] + 4 A[13] + A[29] + 4 A[8] + A[33] + 2 A[40], A[36] + 4 A[9] + 2 A[14] + 6 A[6] + A[26] + 2 A[7] + 3 A[12] + A[25] + 2 A[34] + A[30] + 8 A[10] + A[23] + A[33] + A[40], A[37] + A[15] + 6 A[21] + 3 A[26] + 22 A[7] + 2 A[31] + 8 A[34] + 24 A[10] + 8 A[23] + 21 A[13] + A[40]], [A[5], A[37] + 2 A[9] + A[28] + 3 A[6] + 3 A[26] + 12 A[7] + 2 A[31] + 2 A[12] + A[25] + 3 A[34] + A[30] + 16 A[10] + 4 A[23] + 4 A[13] + A[29] + 2 A[40], A[26] + 5 A[7] + A[31] + A[34] + 6 A[10] + A[23] + A[40]], [A[36] + 2 A[37] + A[38] + 2 A[9] + A[35] + 2 A[15] + 2 A[14] + 4 A[6] + 3 A[32] + 4 A[26] + 26 A[7] + A[31] + 2 A[12] + 2 A[11] + A[24] + 4 A[34] + 20 A[10] + 10 A[23] + 8 A[13] + 2 A[29] + 11 A[8] + A[33] + 2 A[40], A[36] + A[37] + 5 A[9] + A[15] + 2 A[14] + 6 A[6] + A[26] + 12 A[7] + A[31] + 2 A[12] + A[34] + A[30] + 4 A[10] + 4 A[23] + 4 A[13] + A[29] + 2 A[33], 2 A[37] + 2 A[15] + A[26] + 22 A[7] + A[31] + 3 A[34] + 9 A[10] + 8 A[23] + 6 A[13] + A[29]], [A[36] + A[38] + 6 A[9] + 2 A[35] + A[15] + 2 A[14] + 2 A[28] + 4 A[6] + 2 A[32] + 8 A[7] + A[31] + 6 A[12] + 9 A[11] + 2 A[25] + 3 A[24] + 2 A[27] + 2 A[23] + 12 A[8] + A[33], A[36] + 2 A[37] + 10 A[9] + 3 A[14] + 2 A[28] + 6 A[6] + 2 A[26] + 8 A[7] + 7 A[12] + 2 A[25] + 2 A[34] + 8 A[10] + 4 A[23] + 8 A[13] + 2 A[29] + 3 A[33], 2 A[37] + 3 A[26] + 12 A[7] + 5 A[34] + 18 A[10] + 6 A[23] + 9 A[13] + 2 A[29] + A[40]], [2 A[36] + 2 A[37] + 2 A[38] + 8 A[9] + 2 A[35] + 4 A[14] + 2 A[28] + 8 A[6] + 4 A[32] + 4 A[26] + 16 A[7] + 2 A[31] + 8 A[12] + 8 A[11] + 2 A[25] + 4 A[24] + 4 A[34] + 2 A[27] + 20 A[10] + 6 A[23] + 8 A[13] + 2 A[29] + 20 A[8] + A[5] + 2 A[33] + 2 A[40], 2 A[37] + 4 A[9] + 2 A[28] + 3 A[6] + 3 A[26] + 12 A[7] + A[31] + 4 A[12] + 2 A[25] + 3 A[34] + A[30] + 14 A[10] + 5 A[23] + 8 A[13] + 2 A[29] + A[40], 2 A[26] + 7 A[7] + A[31] + 2 A[34] + 12 A[10] + 2 A[23] + 2 A[40] ], [A[37] + 4 A[9] + 2 A[15] + 2 A[28] + A[32] + 2 A[26] + 18 A[7] + 4 A[12] + 4 A[11] + 2 A[25] + 2 A[24] + 2 A[34] + 2 A[27] + 10 A[10] + 7 A[23] + 4 A[13] + A[29] + 7 A[8] + A[40], A[36] + 2 A[37] + 7 A[9] + A[15] + 2 A[14] + A[28] + 6 A[6] + 4 A[26] + 18 A[7] + 4 A[12] + A[25] + 4 A[34] + A[30] + 20 A[10] + 8 A[23] + 8 A[13] + 2 A[29] + 2 A[33] + 2 A[40], 2 A[37] + 2 A[15] + 2 A[26] + 24 A[7] + A[31] + 4 A[34] + 15 A[10] + 9 A[23] + 6 A[13] + A[29] + A[40]], [ 2 A[38] + 3 A[35] + 4 A[32] + 9 A[11] + 3 A[24] + A[27] + 18 A[8], 2 A[37] + 4 A[9] + A[15] + A[14] + 6 A[6] + 4 A[26] + 18 A[7] + A[12] + 4 A[34] + 2 A[30] + 20 A[10] + 8 A[23] + 8 A[13] + 2 A[29] + 2 A[33] + 2 A[40], 2 A[15] + 5 A[26] + 18 A[7] + A[31] + 3 A[34] + 18 A[10] + 6 A[23] + 9 A[13] + 2 A[29] + 2 A[46] + A[40]]], [1, 1, 1, 7, 1, 19, 10, 1, 10, 19, 7, 7, 7, 1, 1, 19, 19, 19, 10, 10, 10, 1, 1, 10, 10, 10, 19, 19, 19, 7, 7, 7, 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 19, 19, 19]] For example, C(100000), mudolo , 27, equals , 1 The congruence classes mod, 27, in the following set , {0, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 35, : Let C(n) be the constant term, in x, of n (3/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 46, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [13 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [ 10 A[5], 9 A[5] + 8 A[6] + 21 A[8] + 2 A[14] + 6 A[16], 9 A[5] + 16 A[6] + 8 A[7] + 21 A[8] + 9 A[9] + 2 A[14] + 2 A[15] + 6 A[16]] , [18 A[5] + 4 A[6] + 19 A[8] + 5 A[9] + 6 A[11] + 8 A[14] + 3 A[16] + 10 A[17], 24 A[5] + 22 A[6] + 20 A[8] + 13 A[9] + 3 A[11] + 8 A[14] + 7 A[16] + 3 A[17], 9 A[6] + 18 A[7] + 20 A[9] + 16 A[10] + 3 A[12] + 6 A[14] + 3 A[15] + 7 A[17]], [A[27], A[28], A[29]], [7 A[5] + 5 A[6] + 8 A[7] + 17 A[8] + 2 A[9] + 8 A[10] + 3 A[11] + 6 A[12] + 22 A[14] + 4 A[15] + 10 A[16] + 10 A[17] + 7 A[18] + 18 A[19], 9 A[6] + 10 A[14], 9 A[7] + 10 A[15]], [9 A[8] + A[16], 9 A[9] + A[17], 9 A[10] + A[18]], [ 20 A[6] + 15 A[7] + 3 A[8] + 19 A[9] + 20 A[10] + 15 A[11] + 3 A[12] + 10 A[14] + 6 A[15] + 6 A[16] + 8 A[17] + 4 A[18] + 4 A[19] + 3 A[20], 6 A[5] + 22 A[6] + 26 A[7] + 14 A[8] + 3 A[9] + 4 A[10] + 15 A[11] + 4 A[12] + 20 A[13] + 8 A[14] + 10 A[15] + 4 A[16] + 6 A[17] + 5 A[18] + 6 A[19] + A[21], 3 A[5] + 26 A[6] + 17 A[7] + 16 A[8] + 13 A[9] + A[10] + 18 A[11] + 3 A[12] + 9 A[13] + 10 A[14] + 7 A[15] + 2 A[16] + 11 A[17] + 11 A[18] + 6 A[19] + A[21]], [15 A[7] + 19 A[9] + 20 A[10] + 15 A[11] + 3 A[12] + 12 A[14] + 6 A[15] + A[16] + 8 A[17] + 4 A[18] + 12 A[19] + 3 A[20], 6 A[5] + 23 A[6] + 14 A[8] + 19 A[9] + 15 A[11] + 8 A[14] + 4 A[16] + 3 A[17] + 6 A[19] + 2 A[22], 6 A[5] + 23 A[6] + 4 A[7] + 14 A[8] + 17 A[9] + 26 A[10] + 15 A[11] + 8 A[12] + 13 A[13] + 20 A[14] + 14 A[15] + 4 A[16] + 7 A[17] + 11 A[18] + 6 A[19] + 7 A[20] + 2 A[21] + 2 A[22]], [3 A[5] + 19 A[6] + 7 A[7] + 16 A[8] + 5 A[9] + 21 A[10] + 9 A[11] + 7 A[12] + 20 A[13] + 18 A[14] + 2 A[15] + 2 A[16] + 7 A[17] + 3 A[18] + 7 A[19] + 2 A[20] + A[21] + 2 A[22] + 12 A[23], 17 A[6] + 11 A[7] + 8 A[9] + 10 A[10] + 3 A[12] + 6 A[14] + A[17] + 2 A[18] + A[20] + A[22] + 4 A[23], 17 A[6] + 23 A[7] + 8 A[9] + A[10] + 3 A[12] + 9 A[13] + 6 A[14] + A[15] + A[17] + 11 A[18] + A[21] + A[22] + 9 A[23]], [7 A[5] + 15 A[6] + 3 A[8] + 24 A[9] + 26 A[11] + 4 A[12] + 16 A[14] + 4 A[16] + 6 A[17] + 10 A[19] + 2 A[20] + 2 A[22] + 8 A[24], 6 A[5] + 10 A[6] + 24 A[9] + 26 A[11] + 4 A[12] + 14 A[14] + 7 A[16] + 6 A[17] + 10 A[19] + 2 A[20] + A[22] + 8 A[24], 15 A[7] + 4 A[23]], [3 A[5] + 16 A[6] + 3 A[7] + 5 A[8] + 2 A[9] + 11 A[10] + 8 A[11] + 2 A[12] + 20 A[13] + 6 A[14] + 2 A[15] + 2 A[16] + A[17] + A[18] + 4 A[19] + A[20] + A[21] + 2 A[22] + 10 A[23], A[6] + 3 A[7] + 16 A[9] + 11 A[10] + 4 A[12] + 20 A[13] + 12 A[14] + 2 A[15] + 4 A[17] + A[18] + 2 A[20] + A[21] + 2 A[22] + 10 A[23] + 2 A[25], 6 A[5] + 6 A[6] + 25 A[7] + 26 A[8] + 6 A[9] + 7 A[10] + 21 A[11] + 2 A[12] + 13 A[13] + 3 A[14] + A[15] + 4 A[16] + A[17] + 3 A[18] + 9 A[19] + A[20] + 2 A[21] + 10 A[23] + 6 A[24] + 2 A[25]], [6 A[5] + 5 A[6] + 3 A[8] + 6 A[9] + 24 A[11] + 11 A[14] + 4 A[16] + 10 A[19] + 2 A[22] + 8 A[24] + 3 A[25], 26 A[6] + 16 A[7] + 10 A[9] + 2 A[10] + 3 A[12] + 9 A[14] + 2 A[15] + 3 A[17] + A[20] + A[22] + 6 A[23] + 2 A[25] + A[26], 7 A[6] + 2 A[7] + 18 A[9] + 18 A[10] + 6 A[13] + 12 A[14] + A[15] + 2 A[17] + 10 A[18] + A[21] + 2 A[22] + 12 A[23] + 4 A[25] + 5 A[26]], [A[5] + 22 A[6] + 14 A[8] + 4 A[9] + 10 A[11] + 2 A[12] + 8 A[14] + A[17] + 4 A[19] + A[20] + 4 A[24] + A[25] + A[27], 21 A[6] + 18 A[8] + 6 A[9] + 15 A[11] + 7 A[14] + 5 A[19] + 6 A[24] + 3 A[25] + A[27], 3 A[5] + 18 A[6] + 8 A[7] + 4 A[8] + 2 A[9] + 6 A[11] + 6 A[14] + 2 A[16] + 2 A[19] + 2 A[23] + A[25] + A[27] ], [6 A[5] + 22 A[6] + 19 A[8] + 8 A[9] + 17 A[11] + 4 A[12] + 9 A[14] + 5 A[16] + 2 A[17] + 7 A[19] + 2 A[20] + 2 A[22] + 4 A[24] + 2 A[25], 18 A[6] + 9 A[9] + 6 A[14] + 2 A[17] + 2 A[25], 20 A[6] + 20 A[7] + 18 A[8] + 10 A[9] + 17 A[10] + 15 A[11] + 8 A[12] + 13 A[13] + 9 A[14] + 2 A[17] + 4 A[18] + 5 A[19] + 2 A[20] + 2 A[21] + A[22] + 10 A[23] + 6 A[24] + 3 A[25] + 4 A[26] + A[27] + 2 A[28]], [20 A[9] + 11 A[14] + 22 A[6] + 4 A[12] + 8 A[11] + 8 A[25] + 10 A[24] + 4 A[16] + 2 A[27] + 2 A[17] + 12 A[19] + 7 A[8] + 6 A[5] + 2 A[20], 14 A[9] + 2 A[15] + 12 A[14] + A[28] + 7 A[6] + 3 A[26] + 18 A[7] + 2 A[22] + 3 A[12] + 18 A[11] + 6 A[25] + 6 A[24] + 2 A[16] + A[17] + 6 A[10] + 7 A[23] + 6 A[19] + 19 A[8] + 3 A[5], 8 A[9] + 9 A[14] + 24 A[6] + 6 A[26] + 8 A[7] + 2 A[12] + 15 A[11] + 3 A[25] + 6 A[24] + A[27] + A[17] + 22 A[10] + 4 A[23] + 5 A[19] + 13 A[13] + 3 A[29] + 18 A[8] + 5 A[18] + A[20]], [ 8 A[9] + 10 A[14] + 23 A[6] + 4 A[12] + 8 A[11] + 2 A[25] + 3 A[16] + 2 A[27] + 2 A[17] + 2 A[19] + 7 A[8] + 3 A[5] + 2 A[20], 3 A[9] + 5 A[14] + 17 A[6] + A[22] + A[30] + A[17], 6 A[9] + A[15] + 12 A[14] + 2 A[28] + 5 A[6] + 2 A[26] + 14 A[7] + A[22] + 6 A[12] + 12 A[11] + 2 A[25] + 4 A[16] + 2 A[27] + A[17] + 7 A[10] + 6 A[23] + 4 A[19] + 8 A[8] + 6 A[5] + A[18] + A[20]], [6 A[9] + A[15] + 7 A[14] + A[28] + 16 A[6] + A[26] + 14 A[7] + A[31] + 6 A[12] + 9 A[11] + A[25] + 2 A[24] + A[30] + A[27] + 2 A[17] + 2 A[10] + 5 A[23] + 3 A[19] + 7 A[8] + 2 A[20], 10 A[9] + A[15] + 8 A[14] + 18 A[6] + 14 A[7] + 2 A[31] + 3 A[12] + 12 A[11] + 2 A[25] + A[30] + 4 A[16] + 2 A[27] + 3 A[17] + 4 A[23] + 4 A[19] + 8 A[8] + 6 A[5] + A[20], 4 A[9] + 10 A[14] + A[28] + A[6] + A[26] + 10 A[7] + A[31] + 2 A[22] + 2 A[12] + 12 A[11] + 2 A[25] + 2 A[30] + 4 A[16] + 2 A[27] + 6 A[10] + 4 A[23] + 4 A[19] + A[13] + 8 A[8] + 6 A[5] + 2 A[18]], [ 3 A[14] + 10 A[6] + A[32] + A[22] + 2 A[24] + A[30] + 6 A[8] + A[5], 3 A[14] + 9 A[6] + A[22], 4 A[9] + 10 A[14] + A[6] + 2 A[32] + 11 A[7] + 2 A[31] + 2 A[22] + 4 A[12] + 2 A[11] + A[24] + 2 A[30] + A[16] + 2 A[17] + 3 A[23] + A[19] + 8 A[8] + 2 A[20]], [ 6 A[14] + 20 A[6] + 2 A[22] + A[24] + 2 A[30] + 3 A[8], 9 A[9] + A[15] + 6 A[14] + A[28] + 16 A[6] + 4 A[32] + A[26] + 14 A[7] + A[31] + 2 A[22] + 2 A[12] + 4 A[11] + 3 A[25] + 2 A[24] + 2 A[16] + 2 A[10] + 5 A[23] + 2 A[19] + 16 A[8] + A[33], 6 A[9] + 5 A[14] + A[28] + 14 A[6] + A[21] + 4 A[26] + 10 A[7] + A[22] + 2 A[12] + 2 A[25] + A[30] + 15 A[10] + 5 A[23] + 7 A[13] + A[29] + A[33] + 3 A[18]], [12 A[9] + 9 A[14] + 24 A[6] + A[22] + 3 A[11] + 5 A[25] + 2 A[30] + A[27] + A[33], 12 A[9] + 2 A[15] + 6 A[14] + 14 A[6] + 2 A[32] + 2 A[26] + 18 A[7] + A[31] + A[22] + A[12] + 2 A[11] + 4 A[25] + A[24] + A[16] + 4 A[10] + 6 A[23] + A[19] + 8 A[8] + 2 A[33], 6 A[9] + 2 A[15] + 4 A[14] + A[28] + 12 A[6] + A[21] + 2 A[32] + A[26] + 20 A[7] + 2 A[12] + 2 A[11] + 2 A[25] + A[24] + 2 A[30] + A[16] + 4 A[10] + 8 A[23] + A[19] + 5 A[13] + A[29] + 8 A[8] + A[33] + A[18]], [ 3 A[14] + 10 A[6] + A[32] + A[22] + 2 A[24] + A[30] + 6 A[8] + A[5], 6 A[9] + 7 A[14] + 19 A[6] + 2 A[22] + 2 A[25] + A[33], A[15] + 9 A[7] + 3 A[23]], [A[36] + 14 A[9] + A[35] + 2 A[15] + 5 A[14] + A[28] + 14 A[6] + 3 A[26] + 18 A[7] + 4 A[12] + 2 A[11] + 6 A[25] + 2 A[24] + 2 A[30] + A[16] + 6 A[10] + 7 A[23] + 7 A[8] + A[33], A[36] + 13 A[9] + 2 A[15] + 4 A[14] + A[28] + 12 A[6] + 2 A[32] + 3 A[26] + 18 A[7] + 4 A[12] + 2 A[11] + 4 A[25] + A[24] + 2 A[30] + A[16] + A[17] + 6 A[10] + 7 A[23] + A[19] + 8 A[8] + A[33], 4 A[26] + 8 A[7] + 2 A[34] + 15 A[10] + 4 A[23] + A[18]], [A[36] + 4 A[9] + A[15] + A[32] + 2 A[26] + 16 A[7] + 2 A[31] + 2 A[12] + A[11] + 2 A[25] + 2 A[24] + 2 A[16] + 4 A[10] + 5 A[23] + 10 A[8], 2 A[36] + 12 A[9] + 2 A[15] + 2 A[14] + 2 A[28] + 6 A[6] + 6 A[26] + 5 A[7] + 2 A[31] + 9 A[12] + 4 A[25] + A[30] + 2 A[17] + 12 A[10] + 12 A[23], 2 A[36] + 14 A[9] + 2 A[15] + 4 A[14] + 12 A[6] + 4 A[32] + 8 A[26] + 26 A[7] + 4 A[12] + 4 A[11] + 6 A[25] + 2 A[24] + 2 A[34] + 2 A[30] + 2 A[16] + 24 A[10] + 11 A[23] + 2 A[19] + A[13] + 16 A[8] + A[33] + 2 A[18]], [ A[5], 6 A[9] + 4 A[14] + 13 A[6] + 2 A[25] + 2 A[30] + A[33], 2 A[36] + 20 A[9] + A[15] + 5 A[14] + A[28] + 12 A[6] + 5 A[26] + 15 A[7] + 6 A[12] + 8 A[25] + A[30] + 10 A[10] + 6 A[23] + 2 A[33]], [2 A[36] + 5 A[38] + 3 A[9] + 2 A[35] + 8 A[14] + A[28] + 16 A[6] + 2 A[32] + 5 A[26] + 16 A[7] + 2 A[31] + 6 A[12] + 6 A[11] + 13 A[25] + 5 A[24] + 2 A[16] + A[27] + 10 A[10] + 6 A[23] + 2 A[8] + 2 A[33], A[36] + A[39] + 4 A[38] + 19 A[9] + 7 A[14] + 2 A[28] + 16 A[6] + 4 A[26] + 14 A[7] + A[31] + 6 A[12] + 6 A[25] + A[30] + 2 A[16] + 8 A[10] + 6 A[23] + 12 A[8] + 2 A[33], A[36] + 10 A[9] + A[15] + 4 A[14] + 2 A[28] + 12 A[6] + 8 A[26] + 5 A[7] + A[31] + 6 A[12] + 4 A[25] + 2 A[34] + 2 A[30] + 23 A[10] + 14 A[23] + A[33] + A[18]], [A[36] + 6 A[38] + 20 A[9] + 2 A[35] + 2 A[15] + 6 A[14] + A[28] + 12 A[6] + A[32] + 3 A[26] + 18 A[7] + 4 A[12] + 5 A[11] + 9 A[25] + 5 A[24] + 2 A[16] + 6 A[10] + 7 A[23] + A[8] + A[33], A[36] + 2 A[39] + 18 A[9] + A[15] + 3 A[14] + 6 A[6] + 3 A[26] + 16 A[7] + A[31] + 3 A[12] + 6 A[25] + 6 A[10] + 6 A[23] + A[33], A[36] + 2 A[38] + 4 A[9] + A[15] + 2 A[14] + A[28] + 8 A[6] + 7 A[26] + 22 A[7] + A[31] + 4 A[12] + 2 A[25] + A[34] + 2 A[30] + A[16] + 20 A[10] + 9 A[23] + 5 A[13] + 2 A[29] + 6 A[8] + 2 A[40]], [ A[5], 4 A[38] + 12 A[9] + 5 A[14] + 13 A[6] + 4 A[25] + A[30] + 2 A[16] + 12 A[8] + 2 A[33], 2 A[15] + 21 A[7] + 2 A[31] + 6 A[23]], [A[36] + 2 A[38] + 16 A[9] + A[35] + A[15] + 6 A[14] + 14 A[6] + 2 A[32] + 2 A[26] + 16 A[7] + 2 A[31] + 2 A[12] + 4 A[11] + 7 A[25] + 2 A[24] + A[30] + A[27] + 4 A[10] + 5 A[23] + 13 A[8] + A[33], A[36] + 2 A[39] + 4 A[38] + 19 A[9] + 6 A[14] + 2 A[28] + 16 A[6] + 4 A[26] + 14 A[7] + A[31] + 6 A[12] + 6 A[25] + 2 A[30] + 2 A[16] + 8 A[10] + 6 A[23] + 12 A[8] + A[33], 2 A[36] + 20 A[9] + A[15] + 5 A[14] + A[28] + 12 A[6] + 7 A[26] + 18 A[7] + 6 A[12] + 8 A[25] + A[30] + 19 A[10] + 8 A[23] + 2 A[33] + 2 A[40]], [2 A[36] + 3 A[38] + 26 A[9] + 2 A[35] + 2 A[15] + 5 A[14] + 10 A[6] + 2 A[32] + 4 A[26] + 20 A[7] + A[31] + 4 A[12] + 5 A[11] + 12 A[25] + 4 A[24] + A[16] + 8 A[10] + 7 A[23] + 20 A[8] + A[33], A[36] + A[39] + 4 A[38] + 16 A[9] + 2 A[15] + 6 A[14] + 16 A[6] + 4 A[26] + 20 A[7] + A[31] + 3 A[12] + 6 A[25] + 2 A[30] + 2 A[16] + 8 A[10] + 7 A[23] + 12 A[8] + A[33], 2 A[36] + A[37] + 4 A[38] + 8 A[9] + A[14] + 2 A[28] + 4 A[6] + 14 A[26] + A[7] + A[31] + 8 A[12] + 4 A[25] + 2 A[34] + A[30] + 2 A[16] + 7 A[10] + 13 A[23] + 7 A[13] + 2 A[29] + 12 A[8] + A[40]], [A[36] + 4 A[38] + 14 A[9] + A[35] + 2 A[15] + 5 A[14] + A[28] + 14 A[6] + 3 A[26] + 18 A[7] + 4 A[12] + 2 A[11] + 6 A[25] + 2 A[24] + 2 A[30] + 2 A[16] + 6 A[10] + 7 A[23] + 16 A[8] + A[5] + A[33] , 2 A[14] + 9 A[6] + 2 A[30], 2 A[36] + 20 A[9] + 2 A[15] + 5 A[14] + A[28] + 12 A[6] + 5 A[26] + 21 A[7] + 6 A[12] + 8 A[25] + A[30] + 10 A[10] + 8 A[23] + 2 A[33]], [A[36] + 4 A[38] + 18 A[9] + A[35] + 7 A[14] + 2 A[28] + 14 A[6] + 2 A[32] + 4 A[26] + 14 A[7] + A[31] + 6 A[12] + 6 A[11] + 8 A[25] + 2 A[24] + A[16] + 2 A[27] + 8 A[10] + 6 A[23] + 19 A[8] + A[33], 2 A[36] + 2 A[39] + 23 A[9] + 5 A[14] + A[28] + 12 A[6] + 5 A[26] + 16 A[7] + 2 A[31] + 6 A[12] + 8 A[25] + A[30] + 10 A[10] + 6 A[23] + A[33], A[36] + 10 A[9] + 4 A[14] + 2 A[28] + 12 A[6] + 8 A[26] + 26 A[7] + 2 A[31] + 6 A[12] + 4 A[25] + A[34] + 2 A[30] + 23 A[10] + 11 A[23] + A[33] + 2 A[40]], [A[36] + 3 A[38] + 16 A[9] + A[35] + A[15] + 8 A[14] + 20 A[6] + 2 A[32] + 2 A[26] + 16 A[7] + 2 A[31] + 2 A[12] + 7 A[11] + 8 A[25] + 2 A[24] + 2 A[30] + A[16] + 2 A[27] + 4 A[10] + 5 A[23] + 16 A[8], A[39] + 6 A[9] + 2 A[15] + 4 A[14] + A[28] + 12 A[6] + 3 A[26] + 18 A[7] + 3 A[12] + 2 A[25] + 2 A[30] + 6 A[10] + 7 A[23], A[36] + A[37] + 4 A[38] + 16 A[9] + A[15] + 6 A[14] + 16 A[6] + 8 A[26] + 20 A[7] + A[31] + 2 A[12] + 6 A[25] + A[34] + 2 A[30] + 2 A[16] + 20 A[10] + 8 A[23] + 7 A[13] + 2 A[29] + 12 A[8] + 2 A[33] + A[40]]], [ 1, 1, 1, 13, 1, 10, 19, 1, 19, 10, 13, 13, 13, 1, 1, 10, 10, 10, 19, 19, 19, 1, 1, 19, 19, 19, 10, 10, 10, 13, 13, 13, 13, 13, 13, 13, 13, 1, 1, 1, 1, 1, 1, 10, 10, 10]] For example, C(100000), mudolo , 27, equals , 1 The congruence classes mod, 27, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 36, : Let C(n) be the constant term, in x, of n (3/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 19, 55, 595, 2611, 22141, 119449, 902035, 5420035, 38712169, 246360709, 1714206781, 11255897485, 77419522675, 517370395015 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 13, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[5], A[5]], [A[5], A[5], A[5]], [A[11], A[11], A[11]], [A[5], A[5], A[5]], [ (676 c[5] + 27) A[5] + (26 c[5] + 1) A[6], (676 c[5] + 27) A[5] + (26 c[5] + 1) A[6], (676 c[5] + 27) A[5] + (26 c[5] + 1) A[6]], [ (18928 c[5] + 18252 c[6] + 729) A[5] + (702 c[5] + 676 c[6] + 27) A[6] + (26 c[5] + 26 c[6] + 1) A[7], (18928 c[5] + 18252 c[6] + 729) A[5] + (702 c[5] + 676 c[6] + 27) A[6] + (26 c[5] + 26 c[6] + 1) A[7], (18928 c[5] + 18252 c[6] + 729) A[5] + (702 c[5] + 676 c[6] + 27) A[6] + (26 c[5] + 26 c[6] + 1) A[7]], [ (19683 + 511732 c[5] + 511056 c[6] + 492804 c[7]) A[5] + (18954 c[5] + 18928 c[6] + 18252 c[7] + 729) A[6] + (702 c[5] + 702 c[6] + 676 c[7] + 27) A[7] + (26 c[5] + 26 c[6] + 26 c[7] + 1) A[8], (19683 + 511732 c[5] + 511056 c[6] + 492804 c[7]) A[5] + (18954 c[5] + 18928 c[6] + 18252 c[7] + 729) A[6] + (702 c[5] + 702 c[6] + 676 c[7] + 27) A[7] + (26 c[5] + 26 c[6] + 26 c[7] + 1) A[8], (19683 + 511732 c[5] + 511056 c[6] + 492804 c[7]) A[5] + (18954 c[5] + 18928 c[6] + 18252 c[7] + 729) A[6] + (702 c[5] + 702 c[6] + 676 c[7] + 27) A[7] + (26 c[5] + 26 c[6] + 26 c[7] + 1) A[8]], [ (13817440 c[5] + 13816764 c[6] + 13798512 c[7] + 13305708 c[8] + 531441) A[5] + (511758 c[5] + 19683 + 511732 c[6] + 511056 c[7] + 492804 c[8]) A[6] + (18954 c[5] + 18954 c[6] + 18928 c[7] + 18252 c[8] + 729) A[7] + (702 c[5] + 702 c[6] + 702 c[7] + 676 c[8] + 27) A[8] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 1) A[9], (13817440 c[5] + 13816764 c[6] + 13798512 c[7] + 13305708 c[8] + 531441) A[5] + (511758 c[5] + 19683 + 511732 c[6] + 511056 c[7] + 492804 c[8]) A[6] + (18954 c[5] + 18954 c[6] + 18928 c[7] + 18252 c[8] + 729) A[7] + (702 c[5] + 702 c[6] + 702 c[7] + 676 c[8] + 27) A[8] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 1) A[9], (13817440 c[5] + 13816764 c[6] + 13798512 c[7] + 13305708 c[8] + 531441) A[5] + (511758 c[5] + 19683 + 511732 c[6] + 511056 c[7] + 492804 c[8]) A[6] + (18954 c[5] + 18954 c[6] + 18928 c[7] + 18252 c[8] + 729) A[7] + (702 c[5] + 702 c[6] + 702 c[7] + 676 c[8] + 27) A[8] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 1) A[9]], [(373071556 c[5] + 373070880 c[6] + 373052628 c[7] + 372559824 c[8] + 359254116 c[9] + 14348907) A[5] + (13817466 c[5] + 13817440 c[6] + 13816764 c[7] + 13798512 c[8] + 13305708 c[9] + 531441) A[6] + ( 511758 c[5] + 511758 c[6] + 19683 + 511732 c[7] + 511056 c[8] + 492804 c[9] ) A[7] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18928 c[8] + 18252 c[9] + 729) A[8] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 676 c[9] + 27) A[9] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 1) A[10], ( 373071556 c[5] + 373070880 c[6] + 373052628 c[7] + 372559824 c[8] + 359254116 c[9] + 14348907) A[5] + (13817466 c[5] + 13817440 c[6] + 13816764 c[7] + 13798512 c[8] + 13305708 c[9] + 531441) A[6] + ( 511758 c[5] + 511758 c[6] + 19683 + 511732 c[7] + 511056 c[8] + 492804 c[9] ) A[7] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18928 c[8] + 18252 c[9] + 729) A[8] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 676 c[9] + 27) A[9] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 1) A[10], ( 373071556 c[5] + 373070880 c[6] + 373052628 c[7] + 372559824 c[8] + 359254116 c[9] + 14348907) A[5] + (13817466 c[5] + 13817440 c[6] + 13816764 c[7] + 13798512 c[8] + 13305708 c[9] + 531441) A[6] + ( 511758 c[5] + 511758 c[6] + 19683 + 511732 c[7] + 511056 c[8] + 492804 c[9] ) A[7] + (18954 c[5] + 18954 c[6] + 18954 c[7] + 18928 c[8] + 18252 c[9] + 729) A[8] + (702 c[5] + 702 c[6] + 702 c[7] + 702 c[8] + 676 c[9] + 27) A[9] + (26 c[5] + 26 c[6] + 26 c[7] + 26 c[8] + 26 c[9] + 1) A[10]], [( 2324522908 c[5] + 2324522232 c[6] + 2324503980 c[7] + 2324011176 c[8] + 2310705468 c[9] + 1951451352 c[10] + 387420489) A[5] + (86093442 c[5] + 86093416 c[6] + 86092740 c[7] + 86074488 c[8] + 85581684 c[9] + 72275976 c[10] + 14348907) A[6] + (3188646 c[5] + 3188646 c[6] + 3188620 c[7] + 3187944 c[8] + 3169692 c[9] + 2676888 c[10] + 531441) A[7] + (118098 c[5] + 118098 c[6] + 118098 c[7] + 19683 + 118072 c[8] + 117396 c[9] + 99144 c[10]) A[8] + (4374 c[5] + 4374 c[6] + 4374 c[7] + 4374 c[8] + 4348 c[9] + 3672 c[10] + 729) A[9] + (162 c[5] + 162 c[6] + 162 c[7] + 162 c[8] + 162 c[9] + 136 c[10] + 27) A[10] + (17 c[5] + 17 c[6] + 17 c[7] + 17 c[8] + 17 c[9] + 17 c[10] + 1) A[11], (2324522908 c[5] + 2324522232 c[6] + 2324503980 c[7] + 2324011176 c[8] + 2310705468 c[9] + 1951451352 c[10] + 387420489) A[5] + (86093442 c[5] + 86093416 c[6] + 86092740 c[7] + 86074488 c[8] + 85581684 c[9] + 72275976 c[10] + 14348907) A[6] + (3188646 c[5] + 3188646 c[6] + 3188620 c[7] + 3187944 c[8] + 3169692 c[9] + 2676888 c[10] + 531441) A[7] + (118098 c[5] + 118098 c[6] + 118098 c[7] + 19683 + 118072 c[8] + 117396 c[9] + 99144 c[10]) A[8] + (4374 c[5] + 4374 c[6] + 4374 c[7] + 4374 c[8] + 4348 c[9] + 3672 c[10] + 729) A[9] + (162 c[5] + 162 c[6] + 162 c[7] + 162 c[8] + 162 c[9] + 136 c[10] + 27) A[10] + (17 c[5] + 17 c[6] + 17 c[7] + 17 c[8] + 17 c[9] + 17 c[10] + 1) A[11], (2324522908 c[5] + 2324522232 c[6] + 2324503980 c[7] + 2324011176 c[8] + 2310705468 c[9] + 1951451352 c[10] + 387420489) A[5] + (86093442 c[5] + 86093416 c[6] + 86092740 c[7] + 86074488 c[8] + 85581684 c[9] + 72275976 c[10] + 14348907) A[6] + (3188646 c[5] + 3188646 c[6] + 3188620 c[7] + 3187944 c[8] + 3169692 c[9] + 2676888 c[10] + 531441) A[7] + (118098 c[5] + 118098 c[6] + 118098 c[7] + 19683 + 118072 c[8] + 117396 c[9] + 99144 c[10]) A[8] + (4374 c[5] + 4374 c[6] + 4374 c[7] + 4374 c[8] + 4348 c[9] + 3672 c[10] + 729) A[9] + (162 c[5] + 162 c[6] + 162 c[7] + 162 c[8] + 162 c[9] + 136 c[10] + 27) A[10] + (17 c[5] + 17 c[6] + 17 c[7] + 17 c[8] + 17 c[9] + 17 c[10] + 1) A[11]], [(55013709412 c[5] + 55013708736 c[6] + 55013690484 c[7] + 55013197680 c[8] + 3486784401 + 54999891972 c[9] + 54640637856 c[10] + 80583461712 c[11]) A[5] + (2037544794 c[5] + 2037544768 c[6] + 2037544092 c[7] + 2037525840 c[8] + 2037033036 c[9] + 2023727328 c[10] + 2984572656 c[11] + 129140163) A[6] + (75464622 c[5] + 75464622 c[6] + 75464596 c[7] + 75463920 c[8] + 75445668 c[9] + 74952864 c[10] + 110539728 c[11] + 4782969) A[7] + (2794986 c[5] + 2794986 c[6] + 2794986 c[7] + 2794960 c[8] + 2794284 c[9] + 2776032 c[10] + 4094064 c[11] + 177147) A[8] + (103518 c[5] + 103518 c[6] + 103518 c[7] + 103518 c[8] + 6561 + 103492 c[9] + 102816 c[10] + 151632 c[11]) A[9] + (3834 c[5] + 3834 c[6] + 3834 c[7] + 3834 c[8] + 3834 c[9] + 3808 c[10] + 5616 c[11] + 243) A[10] + (459 c[5] + 459 c[6] + 459 c[7] + 459 c[8] + 459 c[9] + 459 c[10] + 676 c[11] + 27) A[11] + (17 c[5] + 17 c[6] + 17 c[7] + 17 c[8] + 17 c[9] + 17 c[10] + 26 c[11] + 1) A[12], (55013709412 c[5] + 55013708736 c[6] + 55013690484 c[7] + 55013197680 c[8] + 3486784401 + 54999891972 c[9] + 54640637856 c[10] + 80583461712 c[11]) A[5] + (2037544794 c[5] + 2037544768 c[6] + 2037544092 c[7] + 2037525840 c[8] + 2037033036 c[9] + 2023727328 c[10] + 2984572656 c[11] + 129140163) A[6] + (75464622 c[5] + 75464622 c[6] + 75464596 c[7] + 75463920 c[8] + 75445668 c[9] + 74952864 c[10] + 110539728 c[11] + 4782969) A[7] + (2794986 c[5] + 2794986 c[6] + 2794986 c[7] + 2794960 c[8] + 2794284 c[9] + 2776032 c[10] + 4094064 c[11] + 177147) A[8] + (103518 c[5] + 103518 c[6] + 103518 c[7] + 103518 c[8] + 6561 + 103492 c[9] + 102816 c[10] + 151632 c[11]) A[9] + ( 3834 c[5] + 3834 c[6] + 3834 c[7] + 3834 c[8] + 3834 c[9] + 3808 c[10] + 5616 c[11] + 243) A[10] + (459 c[5] + 459 c[6] + 459 c[7] + 459 c[8] + 459 c[9] + 459 c[10] + 676 c[11] + 27) A[11] + (17 c[5] + 17 c[6] + 17 c[7] + 17 c[8] + 17 c[9] + 17 c[10] + 26 c[11] + 1) A[12], (55013709412 c[5] + 55013708736 c[6] + 55013690484 c[7] + 55013197680 c[8] + 3486784401 + 54999891972 c[9] + 54640637856 c[10] + 80583461712 c[11]) A[5] + (2037544794 c[5] + 2037544768 c[6] + 2037544092 c[7] + 2037525840 c[8] + 2037033036 c[9] + 2023727328 c[10] + 2984572656 c[11] + 129140163) A[6] + (75464622 c[5] + 75464622 c[6] + 75464596 c[7] + 75463920 c[8] + 75445668 c[9] + 74952864 c[10] + 110539728 c[11] + 4782969) A[7] + (2794986 c[5] + 2794986 c[6] + 2794986 c[7] + 2794960 c[8] + 2794284 c[9] + 2776032 c[10] + 4094064 c[11] + 177147) A[8] + (103518 c[5] + 103518 c[6] + 103518 c[7] + 103518 c[8] + 6561 + 103492 c[9] + 102816 c[10] + 151632 c[11]) A[9] + ( 3834 c[5] + 3834 c[6] + 3834 c[7] + 3834 c[8] + 3834 c[9] + 3808 c[10] + 5616 c[11] + 243) A[10] + (459 c[5] + 459 c[6] + 459 c[7] + 459 c[8] + 459 c[9] + 459 c[10] + 676 c[11] + 27) A[11] + (17 c[5] + 17 c[6] + 17 c[7] + 17 c[8] + 17 c[9] + 17 c[10] + 26 c[11] + 1) A[12]], [(1477621745020 c[5] + 1477621744344 c[6] + 1477621726092 c[7] + 1477621233288 c[8] + 1477607927580 c[9] + 1477248673464 c[10] + 2256336927936 c[11] + 2175753466224 c[12] + 87169610025) A[5] + ( 54726731298 c[5] + 54726731272 c[6] + 54726730596 c[7] + 54726712344 c[8] + 54726219540 c[9] + 3228504075 + 54712913832 c[10] + 83568034368 c[11] + 80583461712 c[12]) A[6] + (2026915974 c[5] + 2026915974 c[6] + 2026915948 c[7] + 2026915272 c[8] + 2026897020 c[9] + 2026404216 c[10] + 3095112384 c[11] + 2984572656 c[12] + 119574225) A[7] + (75070962 c[5] + 75070962 c[6] + 75070962 c[7] + 75070936 c[8] + 75070260 c[9] + 75052008 c[10] + 114633792 c[11] + 110539728 c[12] + 4428675) A[8] + ( 2780406 c[5] + 2780406 c[6] + 2780406 c[7] + 2780406 c[8] + 2780380 c[9] + 2779704 c[10] + 4245696 c[11] + 4094064 c[12] + 164025) A[9] + ( 102978 c[5] + 102978 c[6] + 102978 c[7] + 102978 c[8] + 102978 c[9] + 6075 + 102952 c[10] + 157248 c[11] + 151632 c[12]) A[10] + (12393 c[5] + 12393 c[6] + 12393 c[7] + 12393 c[8] + 12393 c[9] + 12393 c[10] + 18928 c[11] + 18252 c[12] + 729) A[11] + (459 c[5] + 459 c[6] + 459 c[7] + 459 c[8] + 459 c[9] + 459 c[10] + 702 c[11] + 676 c[12] + 27) A[12] + ( 17 c[5] + 17 c[6] + 17 c[7] + 17 c[8] + 17 c[9] + 17 c[10] + 26 c[11] + 26 c[12] + 1) A[13], (1477621745020 c[5] + 1477621744344 c[6] + 1477621726092 c[7] + 1477621233288 c[8] + 1477607927580 c[9] + 1477248673464 c[10] + 2256336927936 c[11] + 2175753466224 c[12] + 87169610025) A[5] + (54726731298 c[5] + 54726731272 c[6] + 54726730596 c[7] + 54726712344 c[8] + 54726219540 c[9] + 3228504075 + 54712913832 c[10] + 83568034368 c[11] + 80583461712 c[12]) A[6] + ( 2026915974 c[5] + 2026915974 c[6] + 2026915948 c[7] + 2026915272 c[8] + 2026897020 c[9] + 2026404216 c[10] + 3095112384 c[11] + 2984572656 c[12] + 119574225) A[7] + (75070962 c[5] + 75070962 c[6] + 75070962 c[7] + 75070936 c[8] + 75070260 c[9] + 75052008 c[10] + 114633792 c[11] + 110539728 c[12] + 4428675) A[8] + (2780406 c[5] + 2780406 c[6] + 2780406 c[7] + 2780406 c[8] + 2780380 c[9] + 2779704 c[10] + 4245696 c[11] + 4094064 c[12] + 164025) A[9] + (102978 c[5] + 102978 c[6] + 102978 c[7] + 102978 c[8] + 102978 c[9] + 6075 + 102952 c[10] + 157248 c[11] + 151632 c[12]) A[10] + (12393 c[5] + 12393 c[6] + 12393 c[7] + 12393 c[8] + 12393 c[9] + 12393 c[10] + 18928 c[11] + 18252 c[12] + 729) A[11] + (459 c[5] + 459 c[6] + 459 c[7] + 459 c[8] + 459 c[9] + 459 c[10] + 702 c[11] + 676 c[12] + 27) A[12] + ( 17 c[5] + 17 c[6] + 17 c[7] + 17 c[8] + 17 c[9] + 17 c[10] + 26 c[11] + 26 c[12] + 1) A[13], (1477621745020 c[5] + 1477621744344 c[6] + 1477621726092 c[7] + 1477621233288 c[8] + 1477607927580 c[9] + 1477248673464 c[10] + 2256336927936 c[11] + 2175753466224 c[12] + 87169610025) A[5] + (54726731298 c[5] + 54726731272 c[6] + 54726730596 c[7] + 54726712344 c[8] + 54726219540 c[9] + 3228504075 + 54712913832 c[10] + 83568034368 c[11] + 80583461712 c[12]) A[6] + ( 2026915974 c[5] + 2026915974 c[6] + 2026915948 c[7] + 2026915272 c[8] + 2026897020 c[9] + 2026404216 c[10] + 3095112384 c[11] + 2984572656 c[12] + 119574225) A[7] + (75070962 c[5] + 75070962 c[6] + 75070962 c[7] + 75070936 c[8] + 75070260 c[9] + 75052008 c[10] + 114633792 c[11] + 110539728 c[12] + 4428675) A[8] + (2780406 c[5] + 2780406 c[6] + 2780406 c[7] + 2780406 c[8] + 2780380 c[9] + 2779704 c[10] + 4245696 c[11] + 4094064 c[12] + 164025) A[9] + (102978 c[5] + 102978 c[6] + 102978 c[7] + 102978 c[8] + 102978 c[9] + 6075 + 102952 c[10] + 157248 c[11] + 151632 c[12]) A[10] + (12393 c[5] + 12393 c[6] + 12393 c[7] + 12393 c[8] + 12393 c[9] + 12393 c[10] + 18928 c[11] + 18252 c[12] + 729) A[11] + (459 c[5] + 459 c[6] + 459 c[7] + 459 c[8] + 459 c[9] + 459 c[10] + 702 c[11] + 676 c[12] + 27) A[12] + ( 17 c[5] + 17 c[6] + 17 c[7] + 17 c[8] + 17 c[9] + 17 c[10] + 26 c[11] + 26 c[12] + 1) A[13]]], [1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 19, 19, 19]] For example, C(100000), mudolo , 27, equals , 1 The congruence classes mod, 27, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 37, : Let C(n) be the constant term, in x, of n (3/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 13, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], 26 A[5], A[5]], [ (190 c[5] + 27) A[5] + (10 c[5] + 1) A[6], (190 c[5] + 513) A[5] + (10 c[5] + 26) A[6], (190 c[5] + 27) A[5] + (10 c[5] + 1) A[6]], [ (3214 c[5] + 3780 c[6] + 540) A[5] + (162 c[5] + 190 c[6] + 27) A[6] + (8 c[5] + 10 c[6] + 1) A[7], (3214 c[5] + 3780 c[6] + 10206) A[5] + (162 c[5] + 190 c[6] + 513) A[6] + (8 c[5] + 10 c[6] + 26) A[7], (3214 c[5] + 3780 c[6] + 540) A[5] + (162 c[5] + 190 c[6] + 27) A[6] + (8 c[5] + 10 c[6] + 1) A[7]], [ (10746 + 87103 c[5] + 152199 c[6] + 141966 c[7]) A[5] + (4374 c[5] + 7642 c[6] + 7128 c[7] + 540) A[6] + (216 c[5] + 378 c[6] + 352 c[7] + 27) A[7] + (13 c[5] + 23 c[6] + 22 c[7] + 1) A[8], (174231 + 87103 c[5] + 152199 c[6] + 141966 c[7]) A[5] + (4374 c[5] + 7642 c[6] + 7128 c[7] + 8748) A[6] + (216 c[5] + 378 c[6] + 352 c[7] + 432) A[7] + (13 c[5] + 23 c[6] + 22 c[7] + 26) A[8], (10746 + 87103 c[5] + 152199 c[6] + 141966 c[7]) A[5] + (4374 c[5] + 7642 c[6] + 7128 c[7] + 540) A[6] + (216 c[5] + 378 c[6] + 352 c[7] + 27) A[7] + (13 c[5] + 23 c[6] + 22 c[7] + 1) A[8]], [ (740368 c[5] + 1850688 c[6] + 3146985 c[7] + 1306530 c[8] + 184977) A[5] + (37179 c[5] + 9288 + 92935 c[6] + 158031 c[7] + 65610 c[8]) A[6] + (1836 c[5] + 4590 c[6] + 7804 c[7] + 3240 c[8] + 459) A[7] + (108 c[5] + 270 c[6] + 459 c[7] + 190 c[8] + 27) A[8] + (5 c[5] + 13 c[6] + 23 c[7] + 10 c[8] + 1) A[9], (740368 c[5] + 1850688 c[6] + 3146985 c[7] + 1306530 c[8] + 3527631) A[5] + (37179 c[5] + 177147 + 92935 c[6] + 158031 c[7] + 65610 c[8]) A[6] + (1836 c[5] + 4590 c[6] + 7804 c[7] + 3240 c[8] + 8748) A[7] + (108 c[5] + 270 c[6] + 459 c[7] + 190 c[8] + 513) A[8] + (5 c[5] + 13 c[6] + 23 c[7] + 10 c[8] + 26) A[9], (740368 c[5] + 1850688 c[6] + 3146985 c[7] + 1306530 c[8] + 184977) A[5] + (37179 c[5] + 9288 + 92935 c[6] + 158031 c[7] + 65610 c[8]) A[6] + (1836 c[5] + 4590 c[6] + 7804 c[7] + 3240 c[8] + 459) A[7] + (108 c[5] + 270 c[6] + 459 c[7] + 190 c[8] + 27) A[8] + (5 c[5] + 13 c[6] + 23 c[7] + 10 c[8] + 1) A[9]], [(11147464 c[5] + 14859558 c[6] + 36970047 c[7] + 22120722 c[8] + 26017740 c[9] + 3712608) A[5] + (559791 c[5] + 746200 c[6] + 1856520 c[7] + 1110834 c[8] + 1306530 c[9] + 186435) A[6] + (27648 c[5] + 36855 c[6] + 9207 + 91693 c[7] + 54864 c[8] + 64530 c[9]) A[7] + (1620 c[5] + 2160 c[6] + 5373 c[7] + 3214 c[8] + 3780 c[9] + 540) A[8] + (81 c[5] + 108 c[6] + 270 c[7] + 162 c[8] + 190 c[9] + 27) A[9] + (4 c[5] + 5 c[6] + 13 c[7] + 8 c[8] + 10 c[9] + 1) A[10], (11147464 c[5] + 14859558 c[6] + 36970047 c[7] + 22120722 c[8] + 26017740 c[9] + 70247898 ) A[5] + (559791 c[5] + 746200 c[6] + 1856520 c[7] + 1110834 c[8] + 1306530 c[9] + 3527631) A[6] + (27648 c[5] + 36855 c[6] + 174231 + 91693 c[7] + 54864 c[8] + 64530 c[9]) A[7] + (1620 c[5] + 2160 c[6] + 5373 c[7] + 3214 c[8] + 3780 c[9] + 10206) A[8] + (81 c[5] + 108 c[6] + 270 c[7] + 162 c[8] + 190 c[9] + 513) A[9] + (4 c[5] + 5 c[6] + 13 c[7] + 8 c[8] + 10 c[9] + 26) A[10], ( 11147464 c[5] + 14859558 c[6] + 36970047 c[7] + 22120722 c[8] + 26017740 c[9] + 3712608) A[5] + (559791 c[5] + 746200 c[6] + 1856520 c[7] + 1110834 c[8] + 1306530 c[9] + 186435) A[6] + (27648 c[5] + 36855 c[6] + 9207 + 91693 c[7] + 54864 c[8] + 64530 c[9]) A[7] + (1620 c[5] + 2160 c[6] + 5373 c[7] + 3214 c[8] + 3780 c[9] + 540) A[8] + (81 c[5] + 108 c[6] + 270 c[7] + 162 c[8] + 190 c[9] + 27) A[9] + (4 c[5] + 5 c[6] + 13 c[7] + 8 c[8] + 10 c[9] + 1) A[10]], [( 899246044 c[5] + 1124982783 c[6] + 525424266 c[7] + 599384799 c[8] + 1047331107 c[9] + 976908438 c[10] + 73960506) A[5] + (45157311 c[5] + 56493100 c[6] + 26385156 c[7] + 30099222 c[8] + 52593678 c[9] + 49057272 c[10] + 3714066) A[6] + (2230308 c[5] + 2790180 c[6] + 1303156 c[7] + 1486593 c[8] + 2597589 c[9] + 2422926 c[10] + 183438) A[7] + (130680 c[5] + 163485 c[6] + 76356 c[7] + 10746 + 87103 c[8] + 152199 c[9] + 141966 c[10]) A[8] + (6561 c[5] + 8208 c[6] + 3834 c[7] + 4374 c[8] + 7642 c[9] + 7128 c[10] + 540) A[9] + (324 c[5] + 405 c[6] + 189 c[7] + 216 c[8] + 378 c[9] + 352 c[10] + 27) A[10] + (20 c[5] + 25 c[6] + 11 c[7] + 13 c[8] + 23 c[9] + 22 c[10] + 1) A[11], (899246044 c[5] + 1124982783 c[6] + 525424266 c[7] + 599384799 c[8] + 1047331107 c[9] + 976908438 c[10] + 1198933083) A[5] + (45157311 c[5] + 56493100 c[6] + 26385156 c[7] + 30099222 c[8] + 52593678 c[9] + 49057272 c[10] + 60206652) A[6] + (2230308 c[5] + 2790180 c[6] + 1303156 c[7] + 1486593 c[8] + 2597589 c[9] + 2422926 c[10] + 2973591) A[7] + (130680 c[5] + 163485 c[6] + 76356 c[7] + 174231 + 87103 c[8] + 152199 c[9] + 141966 c[10]) A[8] + (6561 c[5] + 8208 c[6] + 3834 c[7] + 4374 c[8] + 7642 c[9] + 7128 c[10] + 8748) A[9] + (324 c[5] + 405 c[6] + 189 c[7] + 216 c[8] + 378 c[9] + 352 c[10] + 432) A[10] + (20 c[5] + 25 c[6] + 11 c[7] + 13 c[8] + 23 c[9] + 22 c[10] + 26) A[11], (899246044 c[5] + 1124982783 c[6] + 525424266 c[7] + 599384799 c[8] + 1047331107 c[9] + 976908438 c[10] + 73960506) A[5] + (45157311 c[5] + 56493100 c[6] + 26385156 c[7] + 30099222 c[8] + 52593678 c[9] + 49057272 c[10] + 3714066) A[6] + (2230308 c[5] + 2790180 c[6] + 1303156 c[7] + 1486593 c[8] + 2597589 c[9] + 2422926 c[10] + 183438) A[7] + (130680 c[5] + 163485 c[6] + 76356 c[7] + 10746 + 87103 c[8] + 152199 c[9] + 141966 c[10]) A[8] + (6561 c[5] + 8208 c[6] + 3834 c[7] + 4374 c[8] + 7642 c[9] + 7128 c[10] + 540) A[9] + (324 c[5] + 405 c[6] + 189 c[7] + 216 c[8] + 378 c[9] + 352 c[10] + 27) A[10] + (20 c[5] + 25 c[6] + 11 c[7] + 13 c[8] + 23 c[9] + 22 c[10] + 1) A[11]], [(15284395036 c[5] + 19106419023 c[6] + 23002219566 c[7] + 5094743859 c[8] + 1272893589 + 12735264663 c[9] + 21655560114 c[10] + 8990718120 c[11]) A[5] + (767534319 c[5] + 959464360 c[6] + 1155099231 c[7] + 255842037 c[8] + 639524997 c[9] + 1087474221 c[10] + 451485630 c[11] + 63920718) A[6] + ( 37908324 c[5] + 47387700 c[6] + 57050056 c[7] + 12635973 c[8] + 31585977 c[9] + 53710074 c[10] + 22298760 c[11] + 3157029) A[7] + ( 2221128 c[5] + 2776545 c[6] + 3342681 c[7] + 740368 c[8] + 1850688 c[9] + 3146985 c[10] + 1306530 c[11] + 184977) A[8] + (111537 c[5] + 139428 c[6] + 167859 c[7] + 37179 c[8] + 9288 + 92935 c[9] + 158031 c[10] + 65610 c[11]) A[9] + (5508 c[5] + 6885 c[6] + 8289 c[7] + 1836 c[8] + 4590 c[9] + 7804 c[10] + 3240 c[11] + 459) A[10] + (324 c[5] + 405 c[6] + 486 c[7] + 108 c[8] + 270 c[9] + 459 c[10] + 190 c[11] + 27) A[11] + (16 c[5] + 20 c[6] + 25 c[7] + 5 c[8] + 13 c[9] + 23 c[10] + 10 c[11] + 1) A[12], (15284395036 c[5] + 19106419023 c[6] + 23002219566 c[7] + 5094743859 c[8] + 24274938924 + 12735264663 c[9] + 21655560114 c[10] + 8990718120 c[11]) A[5] + (767534319 c[5] + 959464360 c[6] + 1155099231 c[7] + 255842037 c[8] + 639524997 c[9] + 1087474221 c[10] + 451485630 c[11] + 1219011201) A[6] + (37908324 c[5] + 47387700 c[6] + 57050056 c[7] + 12635973 c[8] + 31585977 c[9] + 53710074 c[10] + 22298760 c[11] + 60206652) A[7] + (2221128 c[5] + 2776545 c[6] + 3342681 c[7] + 740368 c[8] + 1850688 c[9] + 3146985 c[10] + 1306530 c[11] + 3527631) A[8] + (111537 c[5] + 139428 c[6] + 167859 c[7] + 37179 c[8] + 177147 + 92935 c[9] + 158031 c[10] + 65610 c[11]) A[9] + ( 5508 c[5] + 6885 c[6] + 8289 c[7] + 1836 c[8] + 4590 c[9] + 7804 c[10] + 3240 c[11] + 8748) A[10] + (324 c[5] + 405 c[6] + 486 c[7] + 108 c[8] + 270 c[9] + 459 c[10] + 190 c[11] + 513) A[11] + (16 c[5] + 20 c[6] + 25 c[7] + 5 c[8] + 13 c[9] + 23 c[10] + 10 c[11] + 26) A[12], (15284395036 c[5] + 19106419023 c[6] + 23002219566 c[7] + 5094743859 c[8] + 1272893589 + 12735264663 c[9] + 21655560114 c[10] + 8990718120 c[11]) A[5] + (767534319 c[5] + 959464360 c[6] + 1155099231 c[7] + 255842037 c[8] + 639524997 c[9] + 1087474221 c[10] + 451485630 c[11] + 63920718) A[6] + (37908324 c[5] + 47387700 c[6] + 57050056 c[7] + 12635973 c[8] + 31585977 c[9] + 53710074 c[10] + 22298760 c[11] + 3157029) A[7] + (2221128 c[5] + 2776545 c[6] + 3342681 c[7] + 740368 c[8] + 1850688 c[9] + 3146985 c[10] + 1306530 c[11] + 184977) A[8] + (111537 c[5] + 139428 c[6] + 167859 c[7] + 37179 c[8] + 9288 + 92935 c[9] + 158031 c[10] + 65610 c[11]) A[9] + ( 5508 c[5] + 6885 c[6] + 8289 c[7] + 1836 c[8] + 4590 c[9] + 7804 c[10] + 3240 c[11] + 459) A[10] + (324 c[5] + 405 c[6] + 486 c[7] + 108 c[8] + 270 c[9] + 459 c[10] + 190 c[11] + 27) A[11] + (16 c[5] + 20 c[6] + 25 c[7] + 5 c[8] + 13 c[9] + 23 c[10] + 10 c[11] + 1) A[12]], [(51091963138 c[5] + 305566963839 c[6] + 381077900586 c[7] + 76709880063 c[8] + 102254184918 c[9] + 254404752777 c[10] + 152220990528 c[11] + 179037840510 c[12] + 25547832513) A[5] + ( 2565677943 c[5] + 15344613352 c[6] + 19136535471 c[7] + 3852129285 c[8] + 5134884057 c[9] + 1282931919 + 12775407777 c[10] + 7644060126 c[11] + 8990718120 c[12]) A[6] + (126718182 c[5] + 757866564 c[6] + 945148636 c[7] + 190255689 c[8] + 253610622 c[9] + 630974151 c[10] + 377538192 c[11] + 444049290 c[12] + 63363681) A[7] + (7424676 c[5] + 44404929 c[6] + 55378161 c[7] + 11147464 c[8] + 14859558 c[9] + 36970047 c[10] + 22120722 c[11] + 26017740 c[12] + 3712608) A[8] + ( 372843 c[5] + 2229876 c[6] + 2780919 c[7] + 559791 c[8] + 746200 c[9] + 1856520 c[10] + 1110834 c[11] + 1306530 c[12] + 186435) A[9] + ( 18414 c[5] + 110133 c[6] + 137349 c[7] + 27648 c[8] + 36855 c[9] + 9207 + 91693 c[10] + 54864 c[11] + 64530 c[12]) A[10] + (1080 c[5] + 6453 c[6] + 8046 c[7] + 1620 c[8] + 2160 c[9] + 5373 c[10] + 3214 c[11] + 3780 c[12] + 540) A[11] + (54 c[5] + 324 c[6] + 405 c[7] + 81 c[8] + 108 c[9] + 270 c[10] + 162 c[11] + 190 c[12] + 27) A[12] + (2 c[5] + 16 c[6] + 20 c[7] + 4 c[8] + 5 c[9] + 13 c[10] + 8 c[11] + 10 c[12] + 1) A[13], ( 51091963138 c[5] + 305566963839 c[6] + 381077900586 c[7] + 76709880063 c[8] + 102254184918 c[9] + 254404752777 c[10] + 152220990528 c[11] + 179037840510 c[12] + 483402169377) A[5] + (2565677943 c[5] + 15344613352 c[6] + 19136535471 c[7] + 3852129285 c[8] + 5134884057 c[9] + 24274938924 + 12775407777 c[10] + 7644060126 c[11] + 8990718120 c[12]) A[6] + (126718182 c[5] + 757866564 c[6] + 945148636 c[7] + 190255689 c[8] + 253610622 c[9] + 630974151 c[10] + 377538192 c[11] + 444049290 c[12] + 1198933083) A[7] + (7424676 c[5] + 44404929 c[6] + 55378161 c[7] + 11147464 c[8] + 14859558 c[9] + 36970047 c[10] + 22120722 c[11] + 26017740 c[12] + 70247898) A[8] + (372843 c[5] + 2229876 c[6] + 2780919 c[7] + 559791 c[8] + 746200 c[9] + 1856520 c[10] + 1110834 c[11] + 1306530 c[12] + 3527631) A[9] + (18414 c[5] + 110133 c[6] + 137349 c[7] + 27648 c[8] + 36855 c[9] + 174231 + 91693 c[10] + 54864 c[11] + 64530 c[12]) A[10] + (1080 c[5] + 6453 c[6] + 8046 c[7] + 1620 c[8] + 2160 c[9] + 5373 c[10] + 3214 c[11] + 3780 c[12] + 10206) A[11] + ( 54 c[5] + 324 c[6] + 405 c[7] + 81 c[8] + 108 c[9] + 270 c[10] + 162 c[11] + 190 c[12] + 513) A[12] + (2 c[5] + 16 c[6] + 20 c[7] + 4 c[8] + 5 c[9] + 13 c[10] + 8 c[11] + 10 c[12] + 26) A[13], (51091963138 c[5] + 305566963839 c[6] + 381077900586 c[7] + 76709880063 c[8] + 102254184918 c[9] + 254404752777 c[10] + 152220990528 c[11] + 179037840510 c[12] + 25547832513) A[5] + (2565677943 c[5] + 15344613352 c[6] + 19136535471 c[7] + 3852129285 c[8] + 5134884057 c[9] + 1282931919 + 12775407777 c[10] + 7644060126 c[11] + 8990718120 c[12]) A[6] + (126718182 c[5] + 757866564 c[6] + 945148636 c[7] + 190255689 c[8] + 253610622 c[9] + 630974151 c[10] + 377538192 c[11] + 444049290 c[12] + 63363681) A[7] + (7424676 c[5] + 44404929 c[6] + 55378161 c[7] + 11147464 c[8] + 14859558 c[9] + 36970047 c[10] + 22120722 c[11] + 26017740 c[12] + 3712608) A[8] + (372843 c[5] + 2229876 c[6] + 2780919 c[7] + 559791 c[8] + 746200 c[9] + 1856520 c[10] + 1110834 c[11] + 1306530 c[12] + 186435) A[9] + (18414 c[5] + 110133 c[6] + 137349 c[7] + 27648 c[8] + 36855 c[9] + 9207 + 91693 c[10] + 54864 c[11] + 64530 c[12] ) A[10] + (1080 c[5] + 6453 c[6] + 8046 c[7] + 1620 c[8] + 2160 c[9] + 5373 c[10] + 3214 c[11] + 3780 c[12] + 540) A[11] + (54 c[5] + 324 c[6] + 405 c[7] + 81 c[8] + 108 c[9] + 270 c[10] + 162 c[11] + 190 c[12] + 27) A[12] + (2 c[5] + 16 c[6] + 20 c[7] + 4 c[8] + 5 c[9] + 13 c[10] + 8 c[11] + 10 c[12] + 1) A[13]]], [1, 1, 2, 4, 1, 8, 10, 2, 16, 20, 4, 5, 13]] For example, C(100000), mudolo , 27, equals , 25 The congruence classes mod, 27, in the following set , {0, 3, 6, 7, 9, 11, 12, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 38, : Let C(n) be the constant term, in x, of n (3/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 46, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [2 A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [10 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[32], A[33], A[34]], [A[41], A[42], A[43]], [ 17 A[5], 9 A[5] + 2 A[6] + 3 A[8] + 6 A[14] + 6 A[16], 18 A[5] + 5 A[6] + 20 A[7] + 15 A[8] + 18 A[9] + 4 A[14] + 6 A[15] + 3 A[16]], [ 15 A[5] + 9 A[8] + 5 A[16], 9 A[5] + 13 A[6] + 3 A[8] + 24 A[9] + 2 A[14] + 6 A[16] + 8 A[17], 9 A[5] + 13 A[6] + 9 A[7] + 3 A[8] + 5 A[9] + 23 A[10] + 3 A[12] + 2 A[14] + 6 A[16] + A[17]], [A[44], A[45], A[46]], [7 A[5] + 4 A[6] + 6 A[7] + 17 A[8] + 4 A[9] + 23 A[10] + 9 A[11] + 3 A[12] + 5 A[14] + 6 A[15] + 7 A[16] + 2 A[17] + 4 A[18] + 3 A[19], 9 A[6] + 10 A[14], 9 A[7] + 10 A[15]], [18 A[5] + 20 A[6] + 18 A[8] + 15 A[9] + 16 A[14] + 10 A[16] + 12 A[17] + 18 A[19], 24 A[6] + 18 A[8] + 21 A[9] + 12 A[11] + 18 A[12] + 12 A[14] + 7 A[17] + 6 A[19] + 9 A[20], 24 A[5] + A[6] + 8 A[7] + 11 A[8] + 3 A[9] + A[10] + 15 A[11] + 12 A[12] + 14 A[14] + 7 A[15] + 4 A[16] + 18 A[17] + 24 A[19] + 12 A[20]], [12 A[5] + 14 A[6] + 9 A[7] + 4 A[8] + 3 A[9] + 23 A[10] + 8 A[11] + 2 A[12] + 4 A[14] + 3 A[15] + 2 A[16] + 4 A[18] + 2 A[19] + A[20], 24 A[6] + 18 A[8] + 21 A[9] + 12 A[11] + 9 A[12] + 12 A[14] + 15 A[17] + 6 A[19] + 10 A[20], 18 A[5] + 5 A[6] + 20 A[7] + 15 A[8] + 5 A[9] + 20 A[10] + 11 A[11] + 26 A[12] + 15 A[13] + 16 A[14] + 10 A[15] + 6 A[16] + 16 A[17] + 7 A[18] + 19 A[19] + 13 A[20] + 4 A[21]], [18 A[5] + 5 A[6] + 9 A[7] + 21 A[8] + 4 A[9] + 4 A[10] + 18 A[11] + 14 A[12] + 26 A[13] + 12 A[14] + 3 A[15] + 8 A[16] + 2 A[17] + 2 A[18] + 9 A[19] + 4 A[20] + A[21] + 2 A[22], 20 A[6] + 4 A[7] + 10 A[9] + 25 A[10] + 18 A[12] + 8 A[14] + 2 A[15] + 4 A[17] + 2 A[18] + 6 A[20] + A[22], 6 A[5] + A[6] + A[7] + 23 A[8] + 14 A[9] + 24 A[10] + 17 A[11] + 24 A[12] + 25 A[13] + 13 A[14] + 11 A[15] + 4 A[16] + 7 A[17] + 8 A[18] + 7 A[19] + 9 A[20] + 2 A[21] + 2 A[22]], [15 A[5] + 4 A[6] + 13 A[7] + 8 A[8] + 2 A[9] + 16 A[11] + 7 A[12] + 2 A[14] + 2 A[15] + 4 A[16] + A[17] + 4 A[19] + 2 A[20] + 6 A[23], 24 A[6] + 2 A[7] + 8 A[9] + 8 A[10] + 16 A[12] + 25 A[13] + 9 A[14] + 4 A[17] + 4 A[18] + 4 A[20] + 2 A[21] + 2 A[23], 17 A[6] + 2 A[7] + 9 A[8] + A[9] + A[10] + 24 A[11] + 3 A[12] + 9 A[13] + 20 A[14] + 2 A[15] + 20 A[17] + 17 A[18] + 12 A[19] + 12 A[20] + 2 A[21] + A[22] + 19 A[23]], [7 A[5] + 6 A[6] + 3 A[7] + 9 A[8] + 6 A[9] + 24 A[10] + 6 A[11] + 6 A[12] + 3 A[14] + 3 A[16] + 3 A[17] + 3 A[18] + 3 A[19] + 3 A[20] + 3 A[23] + 3 A[24], 6 A[14] + 15 A[6] + 2 A[22], 15 A[6] + 18 A[7] + 10 A[9] + 25 A[10] + 18 A[12] + 6 A[14] + 2 A[15] + 2 A[17] + 2 A[18] + 6 A[20] + 10 A[23]], [6 A[5] + 5 A[7] + 12 A[8] + 24 A[9] + 6 A[11] + 10 A[14] + 2 A[16] + 4 A[17] + 3 A[19] + A[22] + 2 A[23] + 4 A[24] + 10 A[25], 15 A[6] + 10 A[7] + 21 A[9] + 6 A[14] + 4 A[17] + 4 A[23] + 6 A[25], 12 A[6] + 9 A[7] + 18 A[9] + 12 A[10] + 12 A[12] + 6 A[14] + 6 A[17] + 7 A[18] + 6 A[20] + 6 A[23] + 6 A[25]], [6 A[5] + 18 A[6] + 9 A[7] + 12 A[8] + A[9] + 4 A[10] + 13 A[11] + 26 A[13] + 8 A[14] + 2 A[15] + 2 A[16] + 4 A[17] + 2 A[18] + 6 A[19] + A[21] + 2 A[22] + 5 A[23] + 2 A[24] + 14 A[25], 20 A[6] + 15 A[9] + 3 A[10] + 3 A[12] + 8 A[14] + 4 A[17] + A[20] + A[22] + 4 A[25] + 3 A[26], 11 A[6] + 19 A[7] + 12 A[9] + 3 A[10] + 2 A[12] + 17 A[13] + 5 A[14] + A[15] + 3 A[17] + 8 A[18] + A[20] + 5 A[21] + A[22] + 14 A[23] + 3 A[25] + 8 A[26]], [2 A[5], 7 A[6] + 5 A[7] + 18 A[9] + 12 A[14] + 4 A[17] + 2 A[22] + 2 A[23] + 4 A[25], 15 A[7] + 2 A[15] + 6 A[23]], [ 3 A[5] + 9 A[8] + 3 A[9] + 3 A[11] + 3 A[16] + A[24] + 3 A[25] + 3 A[27], 18 A[6] + 4 A[7] + 6 A[9] + 3 A[10] + 9 A[12] + 6 A[14] + A[15] + A[17] + 2 A[20] + 2 A[23] + 2 A[25] + 3 A[26] + 5 A[28], 3 A[7] + 9 A[10] + 5 A[18] + 3 A[23] + 3 A[26]], [9 A[9] + 2 A[15] + 7 A[14] + 3 A[28] + 19 A[6] + 4 A[26] + 10 A[7] + 2 A[22] + 5 A[12] + 11 A[11] + 7 A[25] + 4 A[24] + 3 A[16] + 7 A[27] + A[17] + 11 A[10] + 6 A[23] + A[19] + 13 A[8] + 6 A[5] + 2 A[18] + A[20], 11 A[9] + 2 A[15] + 10 A[14] + 7 A[28] + 7 A[26] + 10 A[7] + A[22] + 13 A[12] + 4 A[11] + 7 A[25] + 2 A[24] + 2 A[16] + 4 A[27] + 2 A[17] + 14 A[10] + 6 A[23] + 12 A[8] + 6 A[5] + 2 A[18] + 2 A[20], 8 A[9] + 2 A[15] + 4 A[14] + 4 A[28] + 11 A[6] + 4 A[26] + 11 A[7] + 6 A[12] + 2 A[11] + 4 A[25] + A[24] + A[16] + 2 A[27] + 2 A[17] + 8 A[10] + 7 A[23] + 4 A[13] + 5 A[29] + 6 A[8] + 3 A[5] + 2 A[18] + A[20]], [3 A[9] + 6 A[14] + 18 A[6] + A[22] + 9 A[11] + 3 A[25] + A[30] + A[16] + 5 A[27] + 2 A[19] + 3 A[8], 6 A[9] + A[15] + 7 A[14] + 5 A[28] + 18 A[6] + 3 A[26] + 4 A[7] + 9 A[12] + 3 A[25] + 2 A[30] + A[17] + 3 A[10] + 2 A[23] + 2 A[20], 2 A[9] + 10 A[14] + 4 A[28] + 3 A[6] + A[21] + 6 A[26] + 4 A[7] + 2 A[22] + 8 A[12] + 2 A[11] + 2 A[25] + A[24] + A[30] + A[16] + 2 A[27] + 11 A[10] + 4 A[23] + 3 A[13] + A[29] + 6 A[8] + 3 A[5] + 2 A[18] + 2 A[20]], [ 2 A[9] + 6 A[14] + 17 A[6] + A[22] + 3 A[11] + 2 A[25] + 2 A[24] + 2 A[30] + 3 A[16] + 2 A[27] + 8 A[8] + 3 A[5], 2 A[9] + 5 A[14] + 2 A[28] + 14 A[6] + A[21] + 4 A[26] + 6 A[7] + A[31] + A[22] + 3 A[12] + 2 A[25] + 8 A[10] + 4 A[23] + 3 A[13] + A[29] + 2 A[18], 3 A[9] + A[15] + 7 A[14] + 5 A[28] + 19 A[6] + 4 A[26] + 12 A[7] + A[31] + 2 A[22] + 9 A[12] + A[25] + A[17] + 6 A[10] + 8 A[23] + 3 A[13] + 5 A[29] + A[18] + 2 A[20]] , [5 A[9] + A[15] + 7 A[14] + 3 A[28] + 18 A[6] + A[21] + 5 A[32] + 4 A[26] + 10 A[7] + A[31] + 2 A[22] + 5 A[12] + 5 A[11] + 3 A[25] + A[24] + A[30] + 2 A[16] + 3 A[27] + A[17] + 8 A[10] + 6 A[23] + A[19] + 3 A[13] + A[29] + 15 A[8] + 2 A[5] + 2 A[18] + A[20], 6 A[14] + 18 A[6] + A[22], 3 A[9] + 7 A[14] + 5 A[28] + 19 A[6] + 3 A[26] + 8 A[7] + A[31] + 2 A[22] + 9 A[12] + A[25] + A[17] + 3 A[10] + 4 A[23] + 2 A[20]], [10 A[9] + 2 A[15] + 10 A[14] + 6 A[28] + A[6] + 2 A[21] + 2 A[32] + 8 A[26] + 20 A[7] + 2 A[31] + 2 A[22] + 10 A[12] + 10 A[11] + 6 A[25] + A[24] + A[16] + 6 A[27] + 16 A[10] + 12 A[23] + 2 A[19] + 6 A[13] + 2 A[29] + 9 A[8] + 2 A[33] + 4 A[18] + 2 A[20], 3 A[9] + 2 A[15] + 2 A[14] + 4 A[28] + 6 A[6] + 3 A[26] + 8 A[7] + 6 A[12] + A[25] + A[30] + 3 A[10] + 4 A[23] + A[20], A[21] + 4 A[26] + 7 A[7] + A[31] + 12 A[10] + 5 A[23] + 3 A[13] + A[29] + 3 A[18]], [8 A[9] + 10 A[14] + 6 A[28] + 26 A[6] + 2 A[21] + 3 A[32] + 8 A[26] + 8 A[7] + A[31] + 2 A[22] + 10 A[12] + 7 A[11] + 4 A[25] + 2 A[24] + 4 A[34] + 2 A[30] + A[16] + 3 A[27] + 16 A[10] + 6 A[23] + A[19] + 6 A[13] + 2 A[29] + 10 A[8] + 2 A[33] + 2 A[20], 4 A[9] + 5 A[14] + A[28] + 11 A[6] + 2 A[21] + 5 A[26] + 8 A[7] + A[31] + A[22] + 3 A[12] + 2 A[25] + 4 A[34] + 13 A[10] + 6 A[23] + 6 A[13] + 2 A[29] + A[33], 2 A[15] + 2 A[14] + 4 A[28] + 6 A[6] + 2 A[21] + 7 A[26] + 18 A[7] + A[31] + 8 A[12] + 4 A[34] + A[30] + 15 A[10] + 12 A[23] + 8 A[13] + 2 A[29] + 2 A[20]], [5 A[9] + 2 A[35] + 5 A[14] + 3 A[28] + 14 A[6] + A[21] + 2 A[32] + 4 A[26] + 10 A[7] + 2 A[31] + 5 A[12] + 8 A[11] + 3 A[25] + 2 A[34] + 2 A[30] + A[16] + 4 A[27] + 8 A[10] + 6 A[23] + 3 A[13] + A[29] + 6 A[8] + A[5] + A[33] + A[20], 3 A[9] + 7 A[14] + 5 A[28] + 19 A[6] + 2 A[21] + 7 A[26] + 4 A[7] + 2 A[22] + 9 A[12] + A[25] + 8 A[34] + A[30] + 4 A[23] + 6 A[13] + 2 A[29] + A[33] + 2 A[18] + 2 A[20], A[15] + 2 A[21] + 4 A[26] + 17 A[7] + 2 A[31] + 8 A[34] + 24 A[10] + 10 A[23] + 6 A[13] + 2 A[29] + 2 A[18]], [2 A[36] + 8 A[9] + A[35] + 2 A[15] + 4 A[14] + 8 A[28] + 10 A[6] + 2 A[21] + A[32] + 9 A[26] + 20 A[7] + 2 A[31] + 14 A[12] + 5 A[11] + 4 A[25] + 6 A[34] + A[30] + 3 A[27] + 23 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + 3 A[8] + 2 A[33] + A[18] + A[20], A[36] + 6 A[9] + A[15] + 5 A[14] + 5 A[28] + 12 A[6] + 3 A[26] + 4 A[7] + 9 A[12] + 3 A[25] + A[30] + 3 A[10] + 2 A[23] + A[33] + A[20], 2 A[15] + 2 A[21] + 6 A[26] + 18 A[7] + A[31] + 6 A[34] + 21 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + A[18]], [ A[37] + 4 A[9] + A[15] + 3 A[14] + 3 A[28] + 8 A[6] + 6 A[32] + 4 A[26] + 14 A[7] + 2 A[31] + 5 A[12] + 2 A[11] + 2 A[25] + A[24] + 4 A[34] + A[30] + 2 A[16] + A[27] + 14 A[10] + 8 A[23] + 3 A[13] + A[29] + 17 A[8] + A[33] + A[18] + A[20], 2 A[37] + 8 A[9] + 2 A[15] + 5 A[14] + 6 A[28] + 12 A[6] + 9 A[26] + 20 A[7] + 2 A[31] + 9 A[12] + 4 A[25] + 4 A[34] + A[30] + 17 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + 2 A[33] + A[20], 2 A[14] + 4 A[28] + 6 A[6] + 4 A[26] + 10 A[7] + A[31] + 8 A[12] + A[34] + A[30] + 6 A[10] + 8 A[23] + 5 A[13] + 10 A[29] + 2 A[20]], [2 A[36] + A[37] + A[38] + 2 A[9] + A[35] + 2 A[14] + 4 A[28] + 6 A[6] + 3 A[32] + 4 A[26] + 10 A[7] + 2 A[31] + 8 A[12] + 5 A[11] + 2 A[25] + A[24] + 4 A[34] + A[30] + 3 A[27] + 14 A[10] + 6 A[23] + 3 A[13] + A[29] + 9 A[8] + 2 A[5] + A[18], 2 A[36] + A[37] + A[38] + 2 A[9] + 2 A[35] + 2 A[14] + 4 A[28] + 6 A[6] + 2 A[32] + 4 A[26] + 10 A[7] + 2 A[31] + 8 A[12] + 8 A[11] + 2 A[25] + 4 A[34] + 4 A[27] + 14 A[10] + 6 A[23] + 3 A[13] + A[29] + 6 A[8] + A[18], A[37] + A[15] + 2 A[26] + 14 A[7] + A[31] + 6 A[34] + 18 A[10] + 8 A[23] + 3 A[13] + A[29] + 2 A[18]], [ A[36] + 2 A[37] + 9 A[9] + A[35] + 4 A[14] + 4 A[28] + 10 A[6] + 7 A[26] + 4 A[7] + 6 A[12] + 6 A[11] + 5 A[25] + A[24] + 6 A[34] + A[30] + 4 A[27] + 21 A[10] + 4 A[23] + 6 A[13] + 2 A[29] + 3 A[8] + 2 A[33] + A[18], 2 A[37] + 2 A[38] + 10 A[9] + A[35] + A[15] + 6 A[14] + 4 A[28] + 16 A[6] + 4 A[32] + 8 A[26] + 12 A[7] + A[31] + 4 A[12] + 4 A[11] + 4 A[25] + 8 A[34] + 2 A[30] + 2 A[27] + A[10] + 8 A[23] + 6 A[13] + 2 A[29] + 12 A[8] + 2 A[33] + 2 A[18], A[37] + A[15] + 4 A[26] + 11 A[7] + A[31] + 5 A[34] + 18 A[10] + 7 A[23] + 3 A[13] + A[29] + A[18]], [2 A[36] + A[37] + 3 A[9] + 2 A[35] + A[15] + 4 A[14] + 4 A[28] + 12 A[6] + 5 A[32] + 4 A[26] + 10 A[7] + A[31] + 8 A[12] + 11 A[11] + 3 A[25] + 2 A[24] + 4 A[34] + 2 A[30] + 2 A[16] + 5 A[27] + 14 A[10] + 6 A[23] + 3 A[13] + A[29] + 16 A[8] + A[40], A[36] + A[37] + A[38] + 9 A[9] + 2 A[35] + A[15] + 4 A[14] + 4 A[28] + 8 A[6] + 2 A[32] + 5 A[26] + 10 A[7] + A[31] + 8 A[12] + 8 A[11] + 5 A[25] + 6 A[34] + 4 A[27] + 21 A[10] + 6 A[23] + 3 A[13] + A[29] + 6 A[8] + 2 A[33] + 2 A[40], A[38] + 5 A[9] + 2 A[35] + 2 A[15] + A[14] + A[28] + 2 A[6] + 2 A[32] + 2 A[26] + 10 A[7] + A[12] + 8 A[11] + 3 A[25] + 4 A[27] + 2 A[10] + 6 A[23] + 2 A[13] + 6 A[8] + A[33]], [A[5], 2 A[36] + A[37] + 3 A[9] + 2 A[15] + 5 A[14] + 5 A[28] + 15 A[6] + 5 A[26] + 14 A[7] + A[31] + 9 A[12] + A[25] + 6 A[34] + 2 A[30] + 21 A[10] + 8 A[23] + 3 A[13] + A[29] + A[33] + 2 A[40], 2 A[37] + 4 A[26] + 13 A[7] + 2 A[31] + 6 A[34] + 18 A[10] + 8 A[23] + 6 A[13] + 2 A[29] + A[40]], [2 A[36] + A[37] + 10 A[9] + 2 A[35] + 2 A[14] + 6 A[28] + 4 A[6] + A[32] + 6 A[26] + 6 A[7] + A[31] + 10 A[12] + 13 A[11] + 6 A[25] + 2 A[34] + 7 A[27] + 10 A[10] + 4 A[23] + A[41] + 3 A[13] + A[29] + 3 A[8] + 2 A[33], A[37] + A[38] + 11 A[9] + 2 A[35] + 6 A[14] + 2 A[28] + 14 A[6] + 2 A[32] + 4 A[26] + 6 A[7] + A[31] + 2 A[12] + 8 A[11] + 6 A[25] + 4 A[34] + A[30] + 4 A[27] + 14 A[10] + 4 A[23] + 3 A[13] + A[29] + 6 A[8] + 2 A[33] + A[40], 2 A[37] + A[15] + 6 A[26] + 18 A[7] + 2 A[31] + 6 A[34] + 21 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + A[40]], [A[36] + 2 A[37] + 3 A[9] + A[35] + 2 A[15] + 2 A[14] + 2 A[28] + 6 A[6] + 2 A[32] + 5 A[26] + 20 A[7] + 2 A[31] + 4 A[12] + 10 A[11] + 3 A[25] + A[24] + 8 A[34] + A[30] + 5 A[27] + 25 A[10] + 12 A[23] + A[41] + 6 A[13] + 2 A[29] + 5 A[8] + 2 A[40], 2 A[37] + 2 A[38] + 6 A[9] + A[35] + 2 A[15] + 6 A[14] + 3 A[28] + 16 A[6] + 4 A[32] + 7 A[26] + 20 A[7] + 2 A[31] + 4 A[12] + 4 A[11] + 4 A[25] + 6 A[34] + 2 A[30] + 2 A[27] + 21 A[10] + 12 A[23] + 6 A[13] + 2 A[29] + 12 A[8] + A[33] + A[40], A[37] + 2 A[38] + 7 A[9] + A[35] + A[15] + 2 A[14] + 2 A[28] + 4 A[6] + 4 A[32] + 4 A[26] + 16 A[7] + 2 A[31] + 2 A[12] + 4 A[11] + 3 A[25] + 5 A[34] + 2 A[27] + 16 A[10] + 10 A[23] + 4 A[13] + A[29] + 12 A[8] + 2 A[33] + A[40]], [2 A[36] + A[37] + A[38] + 5 A[9] + 2 A[35] + A[15] + 2 A[14] + 4 A[28] + 6 A[6] + 3 A[32] + 4 A[26] + 10 A[7] + A[31] + 8 A[12] + 17 A[11] + 5 A[25] + A[24] + 4 A[34] + A[30] + 9 A[27] + 14 A[10] + 6 A[23] + 2 A[41] + 3 A[13] + A[29] + 9 A[8] + 2 A[5] + A[40], 6 A[14] + 18 A[6] + 2 A[30], 12 A[7] + 2 A[31] + 6 A[23]], [2 A[38] + 3 A[9] + 2 A[35] + 6 A[32] + 12 A[11] + 3 A[25] + 6 A[27] + A[41] + 18 A[8], 9 A[9] + 7 A[14] + 3 A[28] + 18 A[6] + 3 A[26] + 3 A[12] + 3 A[25] + 2 A[30] + 3 A[10] + 2 A[33], 2 A[15] + 12 A[7] + A[31] + 6 A[34] + 18 A[10] + 6 A[23] + 2 A[40]], [ 2 A[38] + 11 A[9] + 2 A[35] + 2 A[15] + 6 A[14] + 2 A[28] + 16 A[6] + 4 A[32] + 2 A[26] + 8 A[7] + 2 A[12] + 20 A[11] + 7 A[25] + 4 A[34] + 2 A[30] + 10 A[27] + 14 A[10] + 4 A[23] + 2 A[41] + 12 A[8] + 2 A[33] + 2 A[40], A[37] + A[38] + 5 A[9] + 2 A[35] + 2 A[15] + 3 A[14] + 3 A[28] + 8 A[6] + 2 A[32] + 4 A[26] + 14 A[7] + A[31] + 5 A[12] + 8 A[11] + 3 A[25] + 4 A[34] + A[30] + 4 A[27] + 14 A[10] + 8 A[23] + 3 A[13] + A[29] + 6 A[8] + A[33] + A[40], 2 A[37] + 6 A[26] + 14 A[7] + 2 A[31] + 10 A[34] + 3 A[10] + 10 A[23] + 9 A[13] + 3 A[29] + 2 A[40]]], [1, 1, 2, 10, 1, 17, 19, 2, 25, 2, 10, 26, 19, 26, 1, 17, 10, 17, 19, 8, 19, 25, 2, 25, 2, 25, 2, 25, 2, 17, 10, 26, 1, 26, 19, 8, 19, 26, 1, 26, 1, 26, 1, 17, 10, 17]] For example, C(100000), mudolo , 27, equals , 25 The congruence classes mod, 27, in the following set , {0, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 23, 24}, never show up! Theorem Number, 39, : Let C(n) be the constant term, in x, of n (3/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 46, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [2 A[5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [16 A[5], A[30], A[31]], [A[32], A[33], A[34]], [A[35], A[36], A[37]], [A[38], A[39], A[40]], [A[41], A[42], A[43]], [ 26 A[5], 13 A[6] + 8 A[7] + 9 A[9] + 4 A[14] + A[15], 6 A[7] + 9 A[12] + 2 A[15]], [9 A[5] + 16 A[7] + 19 A[8] + 22 A[9] + 3 A[11] + 2 A[15] + 4 A[16] + 8 A[17], 18 A[6] + 8 A[7] + 24 A[9] + 6 A[14] + A[15] + 8 A[17], 12 A[5] + 22 A[6] + 11 A[7] + 7 A[8] + 18 A[9] + 20 A[10] + 3 A[11] + 24 A[12] + 2 A[14] + 4 A[15] + 2 A[16] + 3 A[17]], [A[44], A[45], A[46]], [A[5], 18 A[6] + 10 A[14], 3 A[5] + 17 A[6] + 5 A[7] + 13 A[8] + 13 A[9] + 18 A[11] + 10 A[14] + 5 A[15] + 5 A[16] + 8 A[17] + 12 A[19]], [15 A[5] + 17 A[6] + 4 A[7] + 23 A[8] + 18 A[9] + 13 A[10] + 26 A[11] + 11 A[12] + 25 A[14] + 5 A[15] + 8 A[16] + 6 A[17] + 8 A[18] + 16 A[19] + 22 A[20], 24 A[5] + 16 A[6] + 14 A[8] + 5 A[9] + 15 A[11] + 12 A[12] + 11 A[14] + 4 A[16] + 2 A[17] + 9 A[19] + 6 A[20], 15 A[5] + 20 A[6] + 20 A[8] + 4 A[9] + 14 A[10] + A[11] + 22 A[12] + 10 A[14] + 13 A[16] + 2 A[17] + 11 A[18] + 14 A[19] + 8 A[20]], [21 A[5] + 8 A[6] + 14 A[7] + 22 A[8] + 9 A[9] + 23 A[10] + 6 A[11] + 3 A[12] + 16 A[14] + 4 A[15] + 14 A[16] + 3 A[17] + 10 A[18] + 16 A[19] + 12 A[20], 3 A[5] + 5 A[6] + A[7] + 13 A[8] + 6 A[9] + 14 A[10] + 18 A[11] + 4 A[12] + 10 A[13] + 19 A[14] + 14 A[15] + 5 A[16] + 3 A[17] + 10 A[18] + 12 A[19] + 12 A[20] + 14 A[21] , 24 A[5] + 24 A[6] + 14 A[8] + 14 A[9] + 9 A[10] + 14 A[11] + 22 A[12] + 14 A[13] + 15 A[14] + 12 A[15] + 7 A[16] + 4 A[17] + 9 A[18] + 10 A[19] + 8 A[20] + 17 A[21]], [15 A[5] + 8 A[6] + 7 A[7] + 8 A[8] + 14 A[9] + 12 A[10] + 10 A[11] + 18 A[12] + 11 A[13] + 14 A[14] + 14 A[15] + 9 A[16] + 4 A[17] + 6 A[18] + 5 A[19] + 6 A[20] + 10 A[21] + A[22], 11 A[6] + 9 A[9] + 6 A[12] + 5 A[14] + 2 A[17] + 3 A[20] + A[22], 18 A[5] + 26 A[6] + 24 A[7] + 6 A[8] + 5 A[9] + 8 A[10] + 14 A[11] + 12 A[12] + 19 A[13] + 10 A[14] + 9 A[15] + 6 A[16] + A[17] + 9 A[18] + 4 A[19] + 6 A[20] + 5 A[21]], [6 A[5] + 8 A[6] + 2 A[7] + 17 A[8] + 10 A[9] + 11 A[10] + 26 A[11] + 18 A[12] + 13 A[13] + 15 A[14] + 13 A[16] + 2 A[17] + 4 A[18] + 9 A[19] + 9 A[20] + 2 A[21] + 2 A[22] + 2 A[23], 18 A[5] + A[6] + 4 A[7] + 6 A[8] + 14 A[9] + 5 A[10] + 13 A[11] + 16 A[12] + 11 A[14] + A[15] + 9 A[16] + 4 A[17] + A[18] + 5 A[19] + 7 A[20] + 2 A[23], 18 A[5] + 26 A[6] + 17 A[7] + 6 A[8] + 9 A[9] + 13 A[11] + 18 A[12] + 17 A[13] + 12 A[14] + 2 A[15] + 9 A[16] + 3 A[17] + 5 A[19] + 6 A[20] + 6 A[21] + 2 A[22] + 10 A[23]], [7 A[5], 12 A[5] + 22 A[6] + 2 A[8] + 2 A[9] + 6 A[11] + 12 A[12] + 10 A[14] + 4 A[17] + 6 A[20] + A[22] + 2 A[24], 3 A[7] + 2 A[23]], [ 12 A[5] + 14 A[6] + 21 A[8] + 4 A[14] + 4 A[16] + 6 A[24], 20 A[6] + A[7] + 25 A[9] + 19 A[10] + 2 A[12] + 20 A[13] + 8 A[14] + 5 A[17] + 2 A[18] + A[20] + A[21] + A[22] + A[23] + 8 A[25], 6 A[5] + 17 A[6] + 14 A[7] + A[8] + 3 A[9] + 5 A[10] + 3 A[11] + 4 A[12] + 13 A[13] + 8 A[14] + A[15] + 7 A[17] + 2 A[18] + 2 A[20] + 2 A[21] + A[22] + 6 A[23] + A[24] + 10 A[25]], [6 A[5] + 8 A[6] + 10 A[8] + 22 A[9] + 5 A[11] + 2 A[12] + 4 A[14] + 3 A[16] + 5 A[17] + 2 A[19] + A[20] + 4 A[24] + 6 A[25], 6 A[5] + 13 A[6] + A[8] + A[9] + 3 A[11] + 3 A[12] + 16 A[14] + 7 A[17] + A[20] + 2 A[22] + A[24] + 8 A[25], 12 A[5] + 22 A[6] + 2 A[7] + 2 A[8] + 21 A[9] + 10 A[10] + 6 A[11] + 2 A[12] + 5 A[13] + 10 A[14] + 5 A[17] + 3 A[18] + A[20] + 2 A[21] + 2 A[22] + 2 A[23] + 2 A[24] + 8 A[25] + 4 A[26]], [ 2 A[5] + 14 A[6] + 15 A[8] + 6 A[11] + 4 A[14] + 3 A[16] + 2 A[19] + 6 A[24] + 2 A[27], 9 A[6] + 2 A[14], 3 A[5] + 17 A[6] + 16 A[7] + 11 A[8] + 8 A[9] + 4 A[11] + 4 A[12] + 6 A[14] + A[15] + 3 A[16] + 2 A[17] + 2 A[20] + 2 A[22] + 6 A[23] + 2 A[24] + 4 A[25] + 4 A[27]], [ 3 A[5] + 7 A[6] + 24 A[8] + 9 A[11] + 2 A[14] + 6 A[16] + 2 A[19] + 7 A[24] + 5 A[27], 6 A[5] + 3 A[6] + 9 A[7] + 22 A[8] + 7 A[9] + 8 A[11] + 8 A[12] + 11 A[14] + A[15] + 6 A[16] + 2 A[20] + 2 A[22] + 4 A[23] + 4 A[24] + 5 A[25] + 8 A[27] + 4 A[28], 14 A[7] + 3 A[10] + 2 A[15] + 7 A[23] + A[26]], [5 A[9] + 2 A[15] + 8 A[14] + 5 A[28] + 21 A[6] + 18 A[7] + 2 A[22] + 9 A[12] + 9 A[11] + 5 A[25] + 7 A[24] + 5 A[16] + 3 A[27] + 8 A[23] + 2 A[19] + 20 A[8] + 2 A[20], 7 A[9] + A[15] + 10 A[14] + 7 A[28] + 24 A[6] + A[26] + 13 A[7] + A[22] + 13 A[12] + 4 A[11] + 5 A[25] + 2 A[24] + 3 A[16] + 4 A[27] + A[17] + 5 A[10] + 8 A[23] + 3 A[13] + 3 A[29] + 11 A[8] + 3 A[5] + 2 A[18] + 2 A[20], A[21] + 4 A[26] + 9 A[7] + 6 A[10] + 6 A[23] + 12 A[13] + 5 A[29] + A[18]], [3 A[11] + 3 A[24] + A[16] + A[27] + A[19] + 6 A[8], 6 A[9] + 4 A[14] + 3 A[28] + 12 A[6] + 3 A[12] + A[25] + 2 A[30] + 2 A[17], 2 A[9] + 2 A[15] + 4 A[14] + A[28] + 11 A[6] + A[26] + 18 A[7] + A[22] + A[12] + A[30] + A[17] + 2 A[10] + 8 A[23] + A[13] + A[29]], [5 A[9] + A[15] + 5 A[14] + 4 A[28] + 13 A[6] + 9 A[7] + A[22] + 6 A[12] + 6 A[11] + 3 A[25] + 2 A[24] + A[30] + 3 A[16] + 3 A[27] + A[17] + 4 A[23] + A[19] + 8 A[8] + 3 A[5] + A[20], 8 A[9] + 10 A[14] + 7 A[28] + 22 A[6] + A[26] + 4 A[7] + 2 A[22] + 12 A[12] + 4 A[25] + 2 A[17] + 5 A[10] + 4 A[23] + 3 A[13] + 3 A[29] + 2 A[18] + 2 A[20], 4 A[9] + 2 A[15] + 8 A[14] + 2 A[28] + 22 A[6] + 2 A[21] + 5 A[26] + 19 A[7] + 2 A[22] + 2 A[12] + 2 A[30] + 2 A[17] + 7 A[10] + 12 A[23] + 14 A[13] + 6 A[29] + A[18]], [ 2 A[6] + A[32] + A[24] + A[30] + 3 A[8] + 5 A[5], 4 A[9] + 2 A[15] + 6 A[14] + 4 A[28] + 20 A[6] + 4 A[32] + 10 A[7] + A[31] + 2 A[22] + 8 A[12] + 5 A[11] + 4 A[25] + 3 A[24] + 2 A[30] + A[16] + 3 A[27] + 4 A[23] + A[19] + 13 A[8] + 3 A[5] + 2 A[20], 4 A[9] + 4 A[14] + 2 A[28] + 12 A[6] + 4 A[32] + 4 A[7] + A[31] + 2 A[12] + 5 A[11] + 3 A[24] + 2 A[30] + A[16] + 3 A[27] + 2 A[17] + A[19] + 13 A[8] + 3 A[5]], [ 6 A[32] + 9 A[11] + 4 A[24] + 5 A[27] + 2 A[19] + 18 A[8], 12 A[9] + A[15] + 7 A[14] + 5 A[28] + 17 A[6] + 8 A[7] + 2 A[31] + A[22] + 9 A[12] + 6 A[25] + A[30] + 2 A[23] + 2 A[33] + 2 A[20], 2 A[15] + 15 A[7] + A[31] + 6 A[10] + 9 A[23] + 3 A[13] + 3 A[29] + 2 A[18] ], [5 A[9] + A[15] + 5 A[14] + 2 A[28] + 14 A[6] + 3 A[32] + 2 A[26] + 10 A[7] + 2 A[31] + 2 A[22] + 2 A[12] + 5 A[11] + 3 A[25] + 4 A[24] + 2 A[34] + A[30] + 2 A[16] + 3 A[27] + 6 A[10] + 4 A[23] + 14 A[8] + A[33], A[9] + 2 A[15] + 2 A[14] + A[28] + 6 A[6] + 4 A[26] + 14 A[7] + A[31] + 3 A[12] + A[25] + 2 A[34] + A[30] + 10 A[10] + 8 A[23] + 3 A[13] + 3 A[29] + A[18], 4 A[9] + 2 A[15] + 8 A[14] + 2 A[28] + 22 A[6] + 2 A[21] + 4 A[26] + 10 A[7] + 2 A[22] + 2 A[12] + 2 A[30] + 2 A[17] + 4 A[10] + 6 A[23] + 17 A[13] + 5 A[29]], [4 A[9] + A[15] + 4 A[14] + 2 A[28] + 12 A[6] + 4 A[32] + A[26] + 9 A[7] + 2 A[31] + 2 A[12] + 5 A[11] + 3 A[24] + A[34] + 2 A[30] + A[16] + 3 A[27] + 2 A[17] + 3 A[10] + 3 A[23] + A[19] + 13 A[8] + A[5], 9 A[9] + 2 A[35] + 2 A[15] + 7 A[14] + 4 A[28] + 17 A[6] + 4 A[32] + 10 A[7] + A[31] + 4 A[12] + 10 A[11] + 3 A[25] + 6 A[24] + A[30] + 2 A[16] + 6 A[27] + 2 A[17] + 4 A[23] + 18 A[8] + A[33], 2 A[15] + A[26] + 14 A[7] + A[31] + A[34] + 3 A[10] + 7 A[23]], [ 3 A[32] + 6 A[11] + 3 A[24] + A[16] + 4 A[27] + A[19] + 12 A[8], 3 A[9] + A[35] + A[15] + 2 A[28] + 4 A[6] + 2 A[32] + 2 A[26] + 10 A[7] + 2 A[31] + 2 A[12] + 5 A[11] + 2 A[25] + 3 A[24] + 2 A[34] + 2 A[30] + A[16] + 3 A[27] + 6 A[10] + 4 A[23] + 9 A[8], 2 A[36] + 7 A[9] + A[35] + 8 A[14] + 7 A[28] + 16 A[6] + 2 A[32] + 4 A[26] + 3 A[7] + A[31] + 11 A[12] + 5 A[11] + 3 A[25] + 3 A[24] + A[34] + A[16] + 3 A[27] + 2 A[17] + 7 A[10] + A[23] + 3 A[13] + 3 A[29] + 9 A[8]], [3 A[9] + 2 A[35] + 3 A[28] + 2 A[6] + 6 A[32] + 3 A[12] + 18 A[11] + 3 A[25] + 6 A[24] + A[30] + 9 A[27] + 2 A[19] + 18 A[8], 2 A[36] + 9 A[9] + 2 A[15] + 9 A[14] + 7 A[28] + 18 A[6] + 5 A[26] + 12 A[7] + A[31] + 12 A[12] + 3 A[25] + 2 A[34] + 2 A[17] + 9 A[10] + 6 A[23] + 3 A[13] + 3 A[29] + A[33], 2 A[36] + 2 A[37] + 7 A[9] + A[35] + 2 A[15] + 8 A[14] + 7 A[28] + 16 A[6] + 2 A[32] + 6 A[26] + 15 A[7] + 2 A[31] + 11 A[12] + 5 A[11] + 3 A[25] + 3 A[24] + 2 A[34] + A[16] + 3 A[27] + 2 A[17] + 10 A[10] + 7 A[23] + 9 A[13] + 4 A[29] + 9 A[8]], [2 A[38] + 4 A[6] + 5 A[32] + 6 A[11] + 4 A[24] + 2 A[30] + 4 A[27] + A[19] + 18 A[8] + 2 A[5], A[38] + 4 A[9] + A[35] + 2 A[15] + 3 A[14] + A[28] + 10 A[6] + 4 A[32] + 10 A[7] + A[31] + A[12] + 7 A[11] + 2 A[25] + 5 A[24] + A[30] + 2 A[16] + 5 A[27] + 4 A[23] + 19 A[8] + A[33], A[15] + 12 A[7] + 2 A[31] + 4 A[23]], [A[39] + A[38] + 5 A[9] + 2 A[15] + 3 A[14] + A[28] + 8 A[6] + 4 A[32] + A[26] + 11 A[7] + A[31] + A[12] + 7 A[11] + A[25] + 2 A[24] + A[34] + A[30] + 5 A[27] + 3 A[10] + 5 A[23] + A[19] + 14 A[8] + A[33], 2 A[39] + A[38] + 16 A[9] + 2 A[35] + 8 A[14] + 6 A[28] + 16 A[6] + 3 A[32] + 6 A[12] + 9 A[11] + 6 A[25] + 5 A[24] + 5 A[27] + 13 A[8] + 2 A[33], A[38] + 8 A[9] + 2 A[35] + A[15] + 2 A[14] + 2 A[28] + 4 A[6] + 3 A[32] + 3 A[26] + 9 A[7] + A[31] + 2 A[12] + 9 A[11] + 4 A[25] + 5 A[24] + A[34] + 5 A[27] + 7 A[10] + 5 A[23] + 2 A[13] + 2 A[29] + 13 A[8] + 2 A[33]], [ 2 A[39] + 2 A[38] + 13 A[9] + 2 A[35] + A[15] + 6 A[14] + 5 A[28] + 14 A[6] + 8 A[32] + 2 A[26] + 10 A[7] + 2 A[31] + 5 A[12] + 17 A[11] + 5 A[25] + 8 A[24] + 2 A[34] + A[30] + 9 A[27] + 6 A[10] + 4 A[23] + A[19] + A[8] + 2 A[33], 2 A[36] + A[39] + 12 A[9] + A[15] + 9 A[14] + 6 A[28] + 18 A[6] + 5 A[26] + 10 A[7] + 2 A[31] + 12 A[12] + 6 A[25] + 2 A[34] + 9 A[10] + 4 A[23] + 3 A[13] + 3 A[29] + 2 A[33], A[36] + 2 A[37] + 2 A[38] + 8 A[9] + A[35] + A[15] + 4 A[14] + 3 A[28] + 8 A[6] + 2 A[21] + 3 A[32] + 6 A[26] + 10 A[7] + A[31] + 5 A[12] + 6 A[11] + 4 A[25] + 4 A[24] + 4 A[27] + 6 A[10] + 6 A[23] + 17 A[13] + 7 A[29] + 14 A[8] + 2 A[33]], [ A[5], A[39] + A[38] + 10 A[9] + 2 A[35] + 4 A[14] + 3 A[28] + 13 A[6] + 3 A[32] + 3 A[12] + 9 A[11] + 4 A[25] + 5 A[24] + 2 A[30] + 5 A[27] + 13 A[8] + 2 A[33], 2 A[26] + 7 A[7] + 2 A[31] + 2 A[34] + 6 A[10] + 2 A[23]], [A[39] + 6 A[9] + A[35] + A[15] + 3 A[14] + 2 A[28] + 10 A[6] + 4 A[32] + 8 A[7] + 2 A[31] + 2 A[12] + 8 A[11] + 2 A[25] + 3 A[24] + 2 A[30] + 4 A[27] + 2 A[23] + A[41] + 12 A[8] + A[33], 6 A[9] + 3 A[14] + 6 A[6] + A[26] + A[7] + 3 A[25] + A[34] + 3 A[10] + A[23] + A[33], 2 A[36] + 8 A[9] + 2 A[15] + 6 A[14] + 4 A[28] + 12 A[6] + 6 A[26] + 16 A[7] + 2 A[31] + 8 A[12] + 4 A[25] + 2 A[34] + 11 A[10] + 8 A[23] + 2 A[13] + 2 A[29] + 2 A[33]], [ 2 A[39] + A[38] + 14 A[9] + 2 A[35] + 6 A[14] + 6 A[28] + 12 A[6] + 7 A[32] + A[26] + A[7] + 6 A[12] + 15 A[11] + 6 A[25] + 7 A[24] + A[34] + 8 A[27] + 3 A[10] + A[23] + A[41] + 23 A[8] + 2 A[33], A[36] + 2 A[39] + 2 A[38] + 14 A[9] + A[35] + 9 A[14] + 6 A[28] + 20 A[6] + 3 A[32] + 4 A[26] + A[7] + 9 A[12] + 6 A[11] + 6 A[25] + 4 A[24] + A[34] + A[30] + 4 A[27] + 6 A[10] + A[23] + 3 A[13] + 3 A[29] + 14 A[8] + 2 A[33], A[36] + A[37] + 2 A[38] + 8 A[9] + A[35] + 2 A[15] + 4 A[14] + 3 A[28] + 8 A[6] + A[21] + 3 A[32] + 3 A[26] + 11 A[7] + A[31] + 5 A[12] + 6 A[11] + 4 A[25] + 4 A[24] + 4 A[27] + 3 A[10] + 5 A[23] + 8 A[13] + 3 A[29] + 14 A[8] + 2 A[33]], [2 A[39] + 12 A[9] + 2 A[35] + 2 A[15] + 6 A[14] + 4 A[28] + 12 A[6] + 7 A[32] + 10 A[7] + A[31] + 4 A[12] + 16 A[11] + 4 A[25] + 7 A[24] + 8 A[27] + 4 A[23] + 2 A[41] + 21 A[8] + 2 A[5] + 2 A[33], 2 A[39] + 2 A[38] + 8 A[9] + A[35] + 7 A[14] + 3 A[28] + 20 A[6] + 3 A[32] + 3 A[12] + 6 A[11] + 2 A[25] + 4 A[24] + 2 A[30] + 4 A[27] + 14 A[8] + A[33], 2 A[15] + 12 A[7] + 6 A[23]], [2 A[39] + A[38] + 10 A[9] + A[15] + 6 A[14] + 2 A[28] + 14 A[6] + 2 A[32] + 2 A[26] + 10 A[7] + 2 A[31] + 2 A[12] + 2 A[11] + 2 A[25] + 2 A[24] + 2 A[34] + A[30] + 2 A[27] + 6 A[10] + 4 A[23] + 10 A[8] + 2 A[33], 2 A[39] + 12 A[9] + 8 A[14] + 3 A[28] + 18 A[6] + A[26] + A[7] + 3 A[12] + 2 A[25] + A[34] + A[30] + 3 A[10] + A[23] + 2 A[33], 2 A[36] + 2 A[38] + 3 A[9] + A[35] + A[15] + 4 A[14] + 5 A[28] + 8 A[6] + 3 A[32] + 3 A[26] + 11 A[7] + 2 A[31] + 9 A[12] + 6 A[11] + 3 A[25] + 4 A[24] + A[34] + 4 A[27] + 7 A[10] + 5 A[23] + 3 A[13] + 3 A[29] + 14 A[8]], [A[39] + 7 A[9] + 2 A[35] + 3 A[14] + 3 A[28] + 10 A[6] + 4 A[32] + 2 A[26] + 2 A[7] + 3 A[12] + 12 A[11] + 3 A[25] + 5 A[24] + 2 A[34] + 2 A[30] + 6 A[27] + 6 A[10] + 2 A[23] + 13 A[8] + A[33], A[38] + 10 A[9] + 2 A[35] + 3 A[14] + 4 A[28] + 10 A[6] + 3 A[32] + 2 A[26] + 2 A[7] + 6 A[12] + 9 A[11] + 6 A[25] + 5 A[24] + 2 A[34] + 2 A[30] + 5 A[27] + 6 A[10] + 2 A[23] + 13 A[8] + 2 A[33], A[36] + A[38] + 3 A[9] + 2 A[35] + A[15] + 2 A[14] + 4 A[28] + 4 A[6] + A[21] + 3 A[32] + 5 A[26] + 10 A[7] + 2 A[31] + 6 A[12] + 9 A[11] + 3 A[25] + 5 A[24] + 5 A[27] + 5 A[10] + 4 A[23] + 9 A[13] + 5 A[29] + 13 A[8]]], [1, 1, 2, 16, 1, 26, 1, 2, 7, 11, 16, 20, 25, 26, 1, 26, 1, 26, 1, 26, 1, 25, 2, 7, 20, 7, 11, 16, 11, 11, 16, 20, 7, 20, 25, 2, 25, 26, 1, 26, 1, 26, 1, 26, 1, 26]] For example, C(100000), mudolo , 27, equals , 25 The congruence classes mod, 27, in the following set , {0, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24}, never show up! Theorem Number, 40, : Let C(n) be the constant term, in x, of n (3/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 22, 116, 934, 6332, 48124, 352424, 2669062, 20107628, 153277972, 1170192344, 8981891164, 69111416792, 533463087928, 4126851588176 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 13, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], 26 A[5], A[5]], [ (190 c[5] + 27) A[5] + (10 c[5] + 1) A[6], (190 c[5] + 513) A[5] + (10 c[5] + 26) A[6], (190 c[5] + 27) A[5] + (10 c[5] + 1) A[6]], [ (3214 c[5] + 3780 c[6] + 540) A[5] + (162 c[5] + 190 c[6] + 27) A[6] + (8 c[5] + 10 c[6] + 1) A[7], (3214 c[5] + 3780 c[6] + 10206) A[5] + (162 c[5] + 190 c[6] + 513) A[6] + (8 c[5] + 10 c[6] + 26) A[7], (3214 c[5] + 3780 c[6] + 540) A[5] + (162 c[5] + 190 c[6] + 27) A[6] + (8 c[5] + 10 c[6] + 1) A[7]], [ (10746 + 87103 c[5] + 152199 c[6] + 141966 c[7]) A[5] + (4374 c[5] + 7642 c[6] + 7128 c[7] + 540) A[6] + (216 c[5] + 378 c[6] + 352 c[7] + 27) A[7] + (13 c[5] + 23 c[6] + 22 c[7] + 1) A[8], (174231 + 87103 c[5] + 152199 c[6] + 141966 c[7]) A[5] + (4374 c[5] + 7642 c[6] + 7128 c[7] + 8748) A[6] + (216 c[5] + 378 c[6] + 352 c[7] + 432) A[7] + (13 c[5] + 23 c[6] + 22 c[7] + 26) A[8], (10746 + 87103 c[5] + 152199 c[6] + 141966 c[7]) A[5] + (4374 c[5] + 7642 c[6] + 7128 c[7] + 540) A[6] + (216 c[5] + 378 c[6] + 352 c[7] + 27) A[7] + (13 c[5] + 23 c[6] + 22 c[7] + 1) A[8]], [ (740368 c[5] + 1850688 c[6] + 3146985 c[7] + 1306530 c[8] + 184977) A[5] + (37179 c[5] + 9288 + 92935 c[6] + 158031 c[7] + 65610 c[8]) A[6] + (1836 c[5] + 4590 c[6] + 7804 c[7] + 3240 c[8] + 459) A[7] + (108 c[5] + 270 c[6] + 459 c[7] + 190 c[8] + 27) A[8] + (5 c[5] + 13 c[6] + 23 c[7] + 10 c[8] + 1) A[9], (740368 c[5] + 1850688 c[6] + 3146985 c[7] + 1306530 c[8] + 3527631) A[5] + (37179 c[5] + 177147 + 92935 c[6] + 158031 c[7] + 65610 c[8]) A[6] + (1836 c[5] + 4590 c[6] + 7804 c[7] + 3240 c[8] + 8748) A[7] + (108 c[5] + 270 c[6] + 459 c[7] + 190 c[8] + 513) A[8] + (5 c[5] + 13 c[6] + 23 c[7] + 10 c[8] + 26) A[9], (740368 c[5] + 1850688 c[6] + 3146985 c[7] + 1306530 c[8] + 184977) A[5] + (37179 c[5] + 9288 + 92935 c[6] + 158031 c[7] + 65610 c[8]) A[6] + (1836 c[5] + 4590 c[6] + 7804 c[7] + 3240 c[8] + 459) A[7] + (108 c[5] + 270 c[6] + 459 c[7] + 190 c[8] + 27) A[8] + (5 c[5] + 13 c[6] + 23 c[7] + 10 c[8] + 1) A[9]], [(11147464 c[5] + 14859558 c[6] + 36970047 c[7] + 22120722 c[8] + 26017740 c[9] + 3712608) A[5] + (559791 c[5] + 746200 c[6] + 1856520 c[7] + 1110834 c[8] + 1306530 c[9] + 186435) A[6] + (27648 c[5] + 36855 c[6] + 9207 + 91693 c[7] + 54864 c[8] + 64530 c[9]) A[7] + (1620 c[5] + 2160 c[6] + 5373 c[7] + 3214 c[8] + 3780 c[9] + 540) A[8] + (81 c[5] + 108 c[6] + 270 c[7] + 162 c[8] + 190 c[9] + 27) A[9] + (4 c[5] + 5 c[6] + 13 c[7] + 8 c[8] + 10 c[9] + 1) A[10], (11147464 c[5] + 14859558 c[6] + 36970047 c[7] + 22120722 c[8] + 26017740 c[9] + 70247898 ) A[5] + (559791 c[5] + 746200 c[6] + 1856520 c[7] + 1110834 c[8] + 1306530 c[9] + 3527631) A[6] + (27648 c[5] + 36855 c[6] + 174231 + 91693 c[7] + 54864 c[8] + 64530 c[9]) A[7] + (1620 c[5] + 2160 c[6] + 5373 c[7] + 3214 c[8] + 3780 c[9] + 10206) A[8] + (81 c[5] + 108 c[6] + 270 c[7] + 162 c[8] + 190 c[9] + 513) A[9] + (4 c[5] + 5 c[6] + 13 c[7] + 8 c[8] + 10 c[9] + 26) A[10], ( 11147464 c[5] + 14859558 c[6] + 36970047 c[7] + 22120722 c[8] + 26017740 c[9] + 3712608) A[5] + (559791 c[5] + 746200 c[6] + 1856520 c[7] + 1110834 c[8] + 1306530 c[9] + 186435) A[6] + (27648 c[5] + 36855 c[6] + 9207 + 91693 c[7] + 54864 c[8] + 64530 c[9]) A[7] + (1620 c[5] + 2160 c[6] + 5373 c[7] + 3214 c[8] + 3780 c[9] + 540) A[8] + (81 c[5] + 108 c[6] + 270 c[7] + 162 c[8] + 190 c[9] + 27) A[9] + (4 c[5] + 5 c[6] + 13 c[7] + 8 c[8] + 10 c[9] + 1) A[10]], [( 215553259 c[5] + 144935973 c[6] + 74134737 c[7] + 430932312 c[8] + 286170570 c[9] + 74329380 c[10] + 73960506) A[5] + (10824408 c[5] + 7278229 c[6] + 3722814 c[7] + 21640068 c[8] + 14370588 c[9] + 3732588 c[10] + 3714066) A[6] + (534627 c[5] + 359478 c[6] + 183871 c[7] + 1068822 c[8] + 709776 c[9] + 184356 c[10] + 183438) A[7] + (31320 c[5] + 21060 c[6] + 10773 c[7] + 10746 + 62614 c[8] + 41580 c[9] + 10800 c[10]) A[8] + ( 1566 c[5] + 1053 c[6] + 540 c[7] + 3132 c[8] + 2080 c[9] + 540 c[10] + 540) A[9] + (81 c[5] + 54 c[6] + 27 c[7] + 162 c[8] + 108 c[9] + 28 c[10] + 27) A[10] + (11 c[5] + 7 c[6] + 2 c[7] + 22 c[8] + 14 c[9] + 4 c[10] + 1) A[11], ( 215553259 c[5] + 144935973 c[6] + 74134737 c[7] + 430932312 c[8] + 286170570 c[9] + 74329380 c[10] + 501723315) A[5] + (10824408 c[5] + 7278229 c[6] + 3722814 c[7] + 21640068 c[8] + 14370588 c[9] + 3732588 c[10] + 25194969) A[6] + (534627 c[5] + 359478 c[6] + 183871 c[7] + 1068822 c[8] + 709776 c[9] + 184356 c[10] + 1244403) A[7] + (31320 c[5] + 21060 c[6] + 10773 c[7] + 72900 + 62614 c[8] + 41580 c[9] + 10800 c[10]) A[8] + ( 1566 c[5] + 1053 c[6] + 540 c[7] + 3132 c[8] + 2080 c[9] + 540 c[10] + 3645 ) A[9] + (81 c[5] + 54 c[6] + 27 c[7] + 162 c[8] + 108 c[9] + 28 c[10] + 189) A[10] + (11 c[5] + 7 c[6] + 2 c[7] + 22 c[8] + 14 c[9] + 4 c[10] + 26) A[11], ( 215553259 c[5] + 144935973 c[6] + 74134737 c[7] + 430932312 c[8] + 286170570 c[9] + 74329380 c[10] + 73960506) A[5] + (10824408 c[5] + 7278229 c[6] + 3722814 c[7] + 21640068 c[8] + 14370588 c[9] + 3732588 c[10] + 3714066) A[6] + (534627 c[5] + 359478 c[6] + 183871 c[7] + 1068822 c[8] + 709776 c[9] + 184356 c[10] + 183438) A[7] + (31320 c[5] + 21060 c[6] + 10773 c[7] + 10746 + 62614 c[8] + 41580 c[9] + 10800 c[10]) A[8] + ( 1566 c[5] + 1053 c[6] + 540 c[7] + 3132 c[8] + 2080 c[9] + 540 c[10] + 540) A[9] + (81 c[5] + 54 c[6] + 27 c[7] + 162 c[8] + 108 c[9] + 28 c[10] + 27) A[10] + (11 c[5] + 7 c[6] + 2 c[7] + 22 c[8] + 14 c[9] + 4 c[10] + 1) A[11]], [( 9227784034 c[5] + 4110317514 c[6] + 2597559354 c[7] + 8722184625 c[8] + 575683821 + 8216933652 c[9] + 5121178614 c[10] + 3604892310 c[11]) A[5] + (463390308 c[5] + 206407225 c[6] + 130441266 c[7] + 438000696 c[8] + 412628580 c[9] + 257169492 c[10] + 181026360 c[11] + 28909035) A[6] + ( 22887252 c[5] + 10194633 c[6] + 6442606 c[7] + 21633237 c[8] + 20380086 c[9] + 12701826 c[10] + 8941050 c[11] + 1427841) A[7] + ( 1340820 c[5] + 597240 c[6] + 377433 c[7] + 1267354 c[8] + 1193940 c[9] + 744120 c[10] + 523800 c[11] + 83646) A[8] + (67041 c[5] + 29862 c[6] + 18873 c[7] + 63369 c[8] + 4185 + 59698 c[9] + 37206 c[10] + 26190 c[11]) A[9] + (3456 c[5] + 1539 c[6] + 972 c[7] + 3267 c[8] + 3078 c[9] + 1918 c[10] + 1350 c[11] + 216) A[10] + (486 c[5] + 216 c[6] + 135 c[7] + 459 c[8] + 432 c[9] + 270 c[10] + 190 c[11] + 27) A[11] + (25 c[5] + 11 c[6] + 7 c[7] + 23 c[8] + 22 c[9] + 14 c[10] + 10 c[11] + 1) A[12], (9227784034 c[5] + 4110317514 c[6] + 2597559354 c[7] + 8722184625 c[8] + 9733209237 + 8216933652 c[9] + 5121178614 c[10] + 3604892310 c[11]) A[5] + (463390308 c[5] + 206407225 c[6] + 130441266 c[7] + 438000696 c[8] + 412628580 c[9] + 257169492 c[10] + 181026360 c[11] + 488771172) A[6] + (22887252 c[5] + 10194633 c[6] + 6442606 c[7] + 21633237 c[8] + 20380086 c[9] + 12701826 c[10] + 8941050 c[11] + 24140835) A[7] + (1340820 c[5] + 597240 c[6] + 377433 c[7] + 1267354 c[8] + 1193940 c[9] + 744120 c[10] + 523800 c[11] + 1414260) A[8] + (67041 c[5] + 29862 c[6] + 18873 c[7] + 63369 c[8] + 70713 + 59698 c[9] + 37206 c[10] + 26190 c[11]) A[9] + (3456 c[5] + 1539 c[6] + 972 c[7] + 3267 c[8] + 3078 c[9] + 1918 c[10] + 1350 c[11] + 3645) A[10] + (486 c[5] + 216 c[6] + 135 c[7] + 459 c[8] + 432 c[9] + 270 c[10] + 190 c[11] + 513) A[11] + (25 c[5] + 11 c[6] + 7 c[7] + 23 c[8] + 22 c[9] + 14 c[10] + 10 c[11] + 26) A[12], (9227784034 c[5] + 4110317514 c[6] + 2597559354 c[7] + 8722184625 c[8] + 575683821 + 8216933652 c[9] + 5121178614 c[10] + 3604892310 c[11]) A[5] + (463390308 c[5] + 206407225 c[6] + 130441266 c[7] + 438000696 c[8] + 412628580 c[9] + 257169492 c[10] + 181026360 c[11] + 28909035) A[6] + (22887252 c[5] + 10194633 c[6] + 6442606 c[7] + 21633237 c[8] + 20380086 c[9] + 12701826 c[10] + 8941050 c[11] + 1427841) A[7] + (1340820 c[5] + 597240 c[6] + 377433 c[7] + 1267354 c[8] + 1193940 c[9] + 744120 c[10] + 523800 c[11] + 83646) A[8] + (67041 c[5] + 29862 c[6] + 18873 c[7] + 63369 c[8] + 4185 + 59698 c[9] + 37206 c[10] + 26190 c[11]) A[9] + (3456 c[5] + 1539 c[6] + 972 c[7] + 3267 c[8] + 3078 c[9] + 1918 c[10] + 1350 c[11] + 216) A[10] + (486 c[5] + 216 c[6] + 135 c[7] + 459 c[8] + 432 c[9] + 270 c[10] + 190 c[11] + 27) A[11] + (25 c[5] + 11 c[6] + 7 c[7] + 23 c[8] + 22 c[9] + 14 c[10] + 10 c[11] + 1) A[12]], [(153419787694 c[5] + 184350322089 c[6] + 81903161367 c[7] + 102446987004 c[8] + 174037737861 c[9] + 163732382640 c[10] + 61281693774 c[11] + 72096001830 c[12] + 10308893058) A[5] + ( 7704259488 c[5] + 9257493700 c[6] + 4112919315 c[7] + 5144565663 c[8] + 8739628137 c[9] + 517680207 + 8222125590 c[10] + 3077374032 c[11] + 3620434590 c[12]) A[6] + (380520072 c[5] + 457235658 c[6] + 203140657 c[7] + 254094570 c[8] + 431657829 c[9] + 406097928 c[10] + 151994178 c[11] + 178816410 c[12] + 25568676) A[7] + (22292280 c[5] + 26786565 c[6] + 11900736 c[7] + 14885803 c[8] + 25288119 c[9] + 23790726 c[10] + 8904384 c[11] + 10475730 c[12] + 1497906) A[8] + ( 1114641 c[5] + 1339362 c[6] + 595053 c[7] + 744309 c[8] + 1264438 c[9] + 1189566 c[10] + 445230 c[11] + 523800 c[12] + 74898) A[9] + (57456 c[5] + 69039 c[6] + 30672 c[7] + 38367 c[8] + 65178 c[9] + 3861 + 61318 c[10] + 22950 c[11] + 27000 c[12]) A[10] + (8046 c[5] + 9666 c[6] + 4293 c[7] + 5373 c[8] + 9126 c[9] + 8586 c[10] + 3214 c[11] + 3780 c[12] + 540) A[11] + (405 c[5] + 486 c[6] + 216 c[7] + 270 c[8] + 459 c[9] + 432 c[10] + 162 c[11] + 190 c[12] + 27) A[12] + (20 c[5] + 25 c[6] + 11 c[7] + 13 c[8] + 23 c[9] + 22 c[10] + 8 c[11] + 10 c[12] + 1) A[13], ( 153419787694 c[5] + 184350322089 c[6] + 81903161367 c[7] + 102446987004 c[8] + 174037737861 c[9] + 163732382640 c[10] + 61281693774 c[11] + 72096001830 c[12] + 194659204941) A[5] + ( 7704259488 c[5] + 9257493700 c[6] + 4112919315 c[7] + 5144565663 c[8] + 8739628137 c[9] + 9775173393 + 8222125590 c[10] + 3077374032 c[11] + 3620434590 c[12]) A[6] + (380520072 c[5] + 457235658 c[6] + 203140657 c[7] + 254094570 c[8] + 431657829 c[9] + 406097928 c[10] + 151994178 c[11] + 178816410 c[12] + 482804307) A[7] + (22292280 c[5] + 26786565 c[6] + 11900736 c[7] + 14885803 c[8] + 25288119 c[9] + 23790726 c[10] + 8904384 c[11] + 10475730 c[12] + 28284471) A[8] + ( 1114641 c[5] + 1339362 c[6] + 595053 c[7] + 744309 c[8] + 1264438 c[9] + 1189566 c[10] + 445230 c[11] + 523800 c[12] + 1414260) A[9] + ( 57456 c[5] + 69039 c[6] + 30672 c[7] + 38367 c[8] + 65178 c[9] + 72900 + 61318 c[10] + 22950 c[11] + 27000 c[12]) A[10] + (8046 c[5] + 9666 c[6] + 4293 c[7] + 5373 c[8] + 9126 c[9] + 8586 c[10] + 3214 c[11] + 3780 c[12] + 10206) A[11] + (405 c[5] + 486 c[6] + 216 c[7] + 270 c[8] + 459 c[9] + 432 c[10] + 162 c[11] + 190 c[12] + 513) A[12] + (20 c[5] + 25 c[6] + 11 c[7] + 13 c[8] + 23 c[9] + 22 c[10] + 8 c[11] + 10 c[12] + 26) A[13], (153419787694 c[5] + 184350322089 c[6] + 81903161367 c[7] + 102446987004 c[8] + 174037737861 c[9] + 163732382640 c[10] + 61281693774 c[11] + 72096001830 c[12] + 10308893058) A[5] + ( 7704259488 c[5] + 9257493700 c[6] + 4112919315 c[7] + 5144565663 c[8] + 8739628137 c[9] + 517680207 + 8222125590 c[10] + 3077374032 c[11] + 3620434590 c[12]) A[6] + (380520072 c[5] + 457235658 c[6] + 203140657 c[7] + 254094570 c[8] + 431657829 c[9] + 406097928 c[10] + 151994178 c[11] + 178816410 c[12] + 25568676) A[7] + (22292280 c[5] + 26786565 c[6] + 11900736 c[7] + 14885803 c[8] + 25288119 c[9] + 23790726 c[10] + 8904384 c[11] + 10475730 c[12] + 1497906) A[8] + ( 1114641 c[5] + 1339362 c[6] + 595053 c[7] + 744309 c[8] + 1264438 c[9] + 1189566 c[10] + 445230 c[11] + 523800 c[12] + 74898) A[9] + (57456 c[5] + 69039 c[6] + 30672 c[7] + 38367 c[8] + 65178 c[9] + 3861 + 61318 c[10] + 22950 c[11] + 27000 c[12]) A[10] + (8046 c[5] + 9666 c[6] + 4293 c[7] + 5373 c[8] + 9126 c[9] + 8586 c[10] + 3214 c[11] + 3780 c[12] + 540) A[11] + (405 c[5] + 486 c[6] + 216 c[7] + 270 c[8] + 459 c[9] + 432 c[10] + 162 c[11] + 190 c[12] + 27) A[12] + (20 c[5] + 25 c[6] + 11 c[7] + 13 c[8] + 23 c[9] + 22 c[10] + 8 c[11] + 10 c[12] + 1) A[13]]], [1, 1, 2, 22, 1, 8, 10, 2, 16, 20, 22, 14, 4]] For example, C(100000), mudolo , 27, equals , 25 The congruence classes mod, 27, in the following set , {0, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 26}, never show up! Theorem Number, 41, : Let C(n) be the constant term, in x, of n (3/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[12], A[13]], [A[5], A[6], A[7]], 0, 0, [3 A[5], 3 A[6], 3 A[7]], 0, 0, [15 A[5], 15 A[6], 15 A[7]], 0, 0, 0, 0], [1, 1, 3, 15, 1, 0, 0, 3, 0, 0, 15, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, never show up! Theorem Number, 42, : Let C(n) be the constant term, in x, of n (3/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 27 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3], A[4]], [A[5], A[6], A[7]], [A[8], A[9], A[10]], [A[11], A[9], A[13]], [A[5], A[14], A[15]], 0, 0, [3 A[5], 3 A[6], 3 A[7]], 0, 0, [21 A[5], 21 A[6], 21 A[7]], 0, 0, 0, 0], [1, 1, 3, 21, 1, 0, 0, 3, 0, 0, 21, 0, 0, 0, 0]] For example, C(100000), mudolo , 27, equals , 0 The congruence classes mod, 27, in the following set , {2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26}, never show up! ------------------------------------------ This ends this fascinating book that took, 17.683, to generate. ----------------------------------------- ----------------------------------------------------- On computing the Mod, 5, of Many Interesting sequences by Shalosh B. Ekhad Theorem Number, 1, : Let C(n) be the constant term, in x, of n (1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 2, : Let C(n) be the constant term, in x, of n (1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 3, : Let C(n) be the constant term, in x, of n (2 + x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 4 A[1], 3 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 4, : Let C(n) be the constant term, in x, of n (2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 4 A[1], 3 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 5, : Let C(n) be the constant term, in x, of n (3 + x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], 4 A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 6, : Let C(n) be the constant term, in x, of n (3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], 4 A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 7, : Let C(n) be the constant term, in x, of n (1/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 4 A[1], 0, 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 8, : Let C(n) be the constant term, in x, of n (1/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, A[1], 0, 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 9, : Let C(n) be the constant term, in x, of n (1/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 0, 3 A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 10, : Let C(n) be the constant term, in x, of n (1/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 2 A[1], 4 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 2 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 11, : Let C(n) be the constant term, in x, of n (1/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 4 A[1], 3 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 12, : Let C(n) be the constant term, in x, of n (1/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1], 0, 0]], [1]] For example, C(100000), mudolo , 5, equals , 4 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 13, : Let C(n) be the constant term, in x, of n (1/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 3 A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 2 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 14, : Let C(n) be the constant term, in x, of n (1/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 0, 4 A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 15, : Let C(n) be the constant term, in x, of n (1/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], 4 A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 16, : Let C(n) be the constant term, in x, of n (1/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], A[1], 0, 0]], [1]] For example, C(100000), mudolo , 5, equals , 4 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 17, : Let C(n) be the constant term, in x, of n (1/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], 3 A[1], 3 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 2 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 18, : Let C(n) be the constant term, in x, of n (1/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], 0, A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 19, : Let C(n) be the constant term, in x, of n (2/x + x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 4 A[1], 0, 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 20, : Let C(n) be the constant term, in x, of n (2/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 2 A[1], 0, A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 21, : Let C(n) be the constant term, in x, of n (2/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 22, : Let C(n) be the constant term, in x, of n (2/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 0, 3 A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 23, : Let C(n) be the constant term, in x, of n (2/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 4 A[1], 0, 0]], [1]] For example, C(100000), mudolo , 5, equals , 4 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 24, : Let C(n) be the constant term, in x, of n (2/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 3 A[1], 2 A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 3 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 25, : Let C(n) be the constant term, in x, of n (2/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 3 A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 2 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 26, : Let C(n) be the constant term, in x, of n (2/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1], 0, 0]], [1]] For example, C(100000), mudolo , 5, equals , 4 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 27, : Let C(n) be the constant term, in x, of n (2/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], 4 A[1], 2 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 28, : Let C(n) be the constant term, in x, of n (2/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], 3 A[1], 3 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 2 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 29, : Let C(n) be the constant term, in x, of n (2/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], 2 A[1], 4 A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 3 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 30, : Let C(n) be the constant term, in x, of n (2/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], A[1], 0, 0]], [1]] For example, C(100000), mudolo , 5, equals , 4 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 31, : Let C(n) be the constant term, in x, of n (3/x + x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, A[1], 0, 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 32, : Let C(n) be the constant term, in x, of n (3/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 0, 2 A[1], 0, A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 33, : Let C(n) be the constant term, in x, of n (3/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], A[1], A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 34, : Let C(n) be the constant term, in x, of n (3/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 2 A[1], 4 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 2 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 35, : Let C(n) be the constant term, in x, of n (3/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 3 A[1], 2 A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 3 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 36, : Let C(n) be the constant term, in x, of n (3/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 19, 55, 595, 2611, 22141, 119449, 902035, 5420035, 38712169, 246360709, 1714206781, 11255897485, 77419522675, 517370395015 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], A[1], 4 A[1], 0, 0]], [1]] For example, C(100000), mudolo , 5, equals , 4 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 37, : Let C(n) be the constant term, in x, of n (3/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 4 A[1], 3 A[1], A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 1 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 38, : Let C(n) be the constant term, in x, of n (3/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 0, 4 A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 39, : Let C(n) be the constant term, in x, of n (3/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], A[1], 0, 0]], [1]] For example, C(100000), mudolo , 5, equals , 4 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 40, : Let C(n) be the constant term, in x, of n (3/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 22, 116, 934, 6332, 48124, 352424, 2669062, 20107628, 153277972, 1170192344, 8981891164, 69111416792, 533463087928, 4126851588176 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 2 A[1], 2 A[1], A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 3 The congruence classes mod, 5, in the following set , {0, 2, 3, 4}, never show up! Theorem Number, 41, : Let C(n) be the constant term, in x, of n (3/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], 0, A[1], 4 A[1]]], [1]] For example, C(100000), mudolo , 5, equals , 0 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! Theorem Number, 42, : Let C(n) be the constant term, in x, of n (3/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 5 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 1, states . Here it is: [[[A[1], 3 A[1], A[1], 0, 0]], [1]] For example, C(100000), mudolo , 5, equals , 4 The congruence classes mod, 5, in the following set , {2, 3, 4}, never show up! ------------------------------------------ This ends this fascinating book that took, 0.404, to generate. ----------------------------------------- ----------------------------------------------------- On computing the Mod, 25, of Many Interesting sequences by Shalosh B. Ekhad Theorem Number, 1, : Let C(n) be the constant term, in x, of n (1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], %4, A[11], A[12], A[13]], [A[2], 15 A[2] + 11 A[3], 5 A[3] + 21 A[4], A[14], A[15]], [A[2], %4, %3, %2, A[13]], %1, %5, [A[2], 15 A[2] + 11 A[3], 5 A[3] + 21 A[4], 20 A[4] + 6 A[5], 10 A[5] + 16 A[6]], %5, %1, [A[2], 15 A[2] + 11 A[3], 5 A[3] + 21 A[4], 20 A[4] + 6 A[5], 10 A[5] + 16 A[6]], %5, %1, %5, %1, %1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] %1 := [A[2], 10 A[2] + 16 A[3], 20 A[3] + 6 A[4], 5 A[4] + 21 A[5], 15 A[5] + 11 A[6]] %2 := 15 A[4] + 11 A[5] %3 := 10 A[3] + 16 A[4] %4 := 5 A[2] + 21 A[3] %5 := [A[2], %4, %3, %2, 20 A[5] + 6 A[6]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, never show up! Theorem Number, 2, : Let C(n) be the constant term, in x, of n (1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[%6, [A[2], A[7], A[8], A[9], A[10]], [A[2], %4, A[11], A[12], A[13]], %6, [A[2], %4, %3, %2, A[13]], %1, %5, %6, %5, %1, %6, %5, %1, %5, %1, %1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] %1 := [A[2], 20 A[2] + 6 A[3], 15 A[3] + 11 A[4], 10 A[4] + 16 A[5], 5 A[5] + 21 A[6]] %2 := 20 A[4] + 6 A[5] %3 := 5 A[3] + 21 A[4] %4 := 15 A[2] + 11 A[3] %5 := [A[2], %4, %3, %2, 10 A[5] + 16 A[6]] %6 := [A[2], A[3], A[4], A[5], A[6]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, never show up! Theorem Number, 3, : Let C(n) be the constant term, in x, of n (2 + x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], 15 A[2] + 12 A[3], A[11], A[12], A[13]], [4 A[2], 10 A[2] + 9 A[3], 20 A[3] + 14 A[4], A[14], A[15]], [8 A[2], 10 A[2] + 23 A[3], 20 A[3] + 13 A[4], 5 A[4] + 3 A[5], A[16]], [ 16 A[2], 5 A[2] + 16 A[3], 10 A[3] + 16 A[4], 15 A[4] + 16 A[5], 20 A[5] + 16 A[6]], [7 A[2], 15 A[2] + 17 A[3], 5 A[3] + 2 A[4], 20 A[4] + 12 A[5], 10 A[5] + 22 A[6]], [24 A[2], 10 A[2] + 4 A[3], 20 A[3] + 9 A[4], 5 A[4] + 14 A[5], 15 A[5] + 19 A[6]], [18 A[2], 10 A[2] + 8 A[3], 20 A[3] + 23 A[4], 5 A[4] + 13 A[5], 15 A[5] + 3 A[6]], [A[2], 5 A[2] + A[3], 10 A[3] + A[4], 15 A[4] + A[5], 20 A[5] + A[6]], [ 23 A[2], 20 A[2] + 8 A[3], 15 A[3] + 18 A[4], 10 A[4] + 3 A[5], 5 A[5] + 13 A[6]], [11 A[2], 20 A[2] + 16 A[3], 15 A[3] + 21 A[4], 10 A[4] + A[5], 5 A[5] + 6 A[6]], [2 A[2], 10 A[2] + 2 A[3], 20 A[3] + 2 A[4], 5 A[4] + 2 A[5], 15 A[5] + 2 A[6]], [22 A[2], 15 A[2] + 7 A[3], 5 A[3] + 17 A[4], 20 A[4] + 2 A[5], 10 A[5] + 12 A[6]], [ 4 A[2], 20 A[2] + 4 A[3], 15 A[3] + 4 A[4], 10 A[4] + 4 A[5], 5 A[5] + 4 A[6]], [8 A[2], 15 A[2] + 8 A[3], 5 A[3] + 8 A[4], 20 A[4] + 8 A[5], 10 A[5] + 8 A[6]]], [1, 1, 2, 4, 8, 16, 7, 24, 18, 1, 23, 11, 2, 22, 4, 8]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 3, 5, 6, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21}, never show up! Theorem Number, 4, : Let C(n) be the constant term, in x, of n (2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], 20 A[2] + 22 A[3], A[11], A[12], A[13]], [4 A[2], 20 A[2] + 4 A[3], 15 A[3] + 4 A[4], A[14], A[15]], [8 A[2], 5 A[2] + 13 A[3], 10 A[3] + 18 A[4], 15 A[4] + 23 A[5], A[16]], [ 16 A[2], 20 A[2] + 21 A[3], 15 A[3] + A[4], 10 A[4] + 6 A[5], 5 A[5] + 11 A[6]], [7 A[2], 20 A[2] + 2 A[3], 15 A[3] + 22 A[4], 10 A[4] + 17 A[5], 5 A[5] + 12 A[6]], [24 A[2], 20 A[2] + 24 A[3], 15 A[3] + 24 A[4], 10 A[4] + 24 A[5], 5 A[5] + 24 A[6]], [18 A[2], 5 A[2] + 23 A[3], 10 A[3] + 3 A[4], 15 A[4] + 8 A[5], 20 A[5] + 13 A[6]], [ A[2], 20 A[2] + 6 A[3], 15 A[3] + 11 A[4], 10 A[4] + 16 A[5], 5 A[5] + 21 A[6]], [23 A[2], 15 A[2] + 23 A[3], 5 A[3] + 23 A[4], 20 A[4] + 23 A[5], 10 A[5] + 23 A[6]], [11 A[2], 10 A[2] + 21 A[3], 20 A[3] + 6 A[4], 5 A[4] + 16 A[5], 15 A[5] + A[6]], [2 A[2], 15 A[2] + 12 A[3], 5 A[3] + 22 A[4], 20 A[4] + 7 A[5], 10 A[5] + 17 A[6]], [22 A[2], 20 A[2] + 17 A[3], 15 A[3] + 12 A[4], 10 A[4] + 7 A[5], 5 A[5] + 2 A[6]], [4 A[2], 5 A[2] + 24 A[3], 10 A[3] + 19 A[4], 15 A[4] + 14 A[5], 20 A[5] + 9 A[6]], [8 A[2], 10 A[2] + 23 A[3], 20 A[3] + 13 A[4], 5 A[4] + 3 A[5], 15 A[5] + 18 A[6]]], [1, 1, 2, 4, 8, 16, 7, 24, 18, 1, 23, 11, 2, 22, 4, 8]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 3, 5, 6, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21}, never show up! Theorem Number, 5, : Let C(n) be the constant term, in x, of n (3 + x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], 18 A[3], A[11], A[12], A[13]], [9 A[2], 20 A[2] + 14 A[3], 15 A[3] + 19 A[4], A[14], A[15]], [2 A[2], 12 A[3], 22 A[4], 7 A[5], A[16]], [6 A[2], 15 A[2] + 6 A[3], 5 A[3] + 6 A[4], 20 A[4] + 6 A[5], 10 A[5] + 6 A[6]], [18 A[2], 8 A[3], 23 A[4], 13 A[5], 3 A[6]], [24 A[2], 20 A[2] + 4 A[3], 15 A[3] + 9 A[4], 10 A[4] + 14 A[5], 5 A[5] + 19 A[6]], [7 A[2], 17 A[3], 2 A[4], 12 A[5], 22 A[6]], [A[2], 15 A[2] + A[3], 5 A[3] + A[4], 20 A[4] + A[5], 10 A[5] + A[6]], [ 22 A[2], 10 A[2] + 12 A[3], 20 A[3] + 2 A[4], 5 A[4] + 17 A[5], 15 A[5] + 7 A[6]], [21 A[2], A[3], 6 A[4], 11 A[5], 16 A[6]], [3 A[2], 20 A[2] + 3 A[3], 15 A[3] + 3 A[4], 10 A[4] + 3 A[5], 5 A[5] + 3 A[6]], [13 A[2], 3 A[3], 18 A[4], 8 A[5], 23 A[6]], [9 A[2], 10 A[2] + 9 A[3], 20 A[3] + 9 A[4], 5 A[4] + 9 A[5], 15 A[5] + 9 A[6]], [2 A[2], 5 A[2] + 2 A[3], 10 A[3] + 2 A[4], 15 A[4] + 2 A[5], 20 A[5] + 2 A[6]]], [1, 1, 3, 9, 2, 6, 18, 24, 7, 1, 22, 21, 3, 13, 9, 2]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 4, 5, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 23}, never show up! Theorem Number, 6, : Let C(n) be the constant term, in x, of n (3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], 15 A[2] + 13 A[3], A[11], A[12], A[13]], [9 A[2], 15 A[2] + 24 A[3], 5 A[3] + 14 A[4], A[14], A[15]], [2 A[2], 10 A[2] + 17 A[3], 20 A[3] + 7 A[4], 5 A[4] + 22 A[5], A[16]], [ 6 A[2], 20 A[2] + 21 A[3], 15 A[3] + 11 A[4], 10 A[4] + A[5], 5 A[5] + 16 A[6]], [18 A[2], 15 A[2] + 3 A[3], 5 A[3] + 13 A[4], 20 A[4] + 23 A[5], 10 A[5] + 8 A[6]], [24 A[2], 15 A[2] + 14 A[3], 5 A[3] + 4 A[4], 20 A[4] + 19 A[5], 10 A[5] + 9 A[6]], [7 A[2], 10 A[2] + 22 A[3], 20 A[3] + 12 A[4], 5 A[4] + 2 A[5], 15 A[5] + 17 A[6]], [A[2], 20 A[2] + 16 A[3], 15 A[3] + 6 A[4], 10 A[4] + 21 A[5], 5 A[5] + 11 A[6]], [22 A[2], 20 A[2] + 17 A[3], 15 A[3] + 12 A[4], 10 A[4] + 7 A[5], 5 A[5] + 2 A[6]], [21 A[2], 5 A[2] + 16 A[3], 10 A[3] + 11 A[4], 15 A[4] + 6 A[5], 20 A[5] + A[6]], [3 A[2], 10 A[2] + 23 A[3], 20 A[3] + 18 A[4], 5 A[4] + 13 A[5], 15 A[5] + 8 A[6]], [13 A[2], 15 A[2] + 23 A[3], 5 A[3] + 8 A[4], 20 A[4] + 18 A[5], 10 A[5] + 3 A[6]], [9 A[2], 5 A[2] + 19 A[3], 10 A[3] + 4 A[4], 15 A[4] + 14 A[5], 20 A[5] + 24 A[6]], [2 A[2], 15 A[2] + 7 A[3], 5 A[3] + 12 A[4], 20 A[4] + 17 A[5], 10 A[5] + 22 A[6]]], [1, 1, 3, 9, 2, 6, 18, 24, 7, 1, 22, 21, 3, 13, 9, 2]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 4, 5, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 23}, never show up! Theorem Number, 7, : Let C(n) be the constant term, in x, of n (1/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 22, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [0, 10 A[2], 20 A[3], A[11], A[12]], [4 A[2], A[13], A[14], A[15], A[16]], [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]], [24 A[2], A[17], A[18], A[15], A[20]], [0, 10 A[2], 20 A[3], 5 A[4], 15 A[5]], [14 A[2], 20 A[3] + 4 A[7], 5 A[4] + 4 A[8], A[21], A[20]], [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]], [19 A[2], 10 A[3] + 4 A[7], 5 A[4] + 4 A[8], 4 A[9], 20 A[6] + 4 A[10]], [20 A[2], 20 A[3], 4 A[11], 20 A[5], 20 A[6]], 0, [0, 15 A[2], 20 A[3] + 4 A[7] + 4 A[13], 4 A[11], 4 A[12]], [11 A[2], 10 A[3] + 2 A[7] + A[13], 5 A[4] + A[8] + 2 A[11], 5 A[5] + A[9], 20 A[6] + A[10]], [0, 5 A[2], 15 A[3] + 3 A[7] + 3 A[13], 3 A[11], 3 A[12]], [11 A[2], 20 A[3] + 4 A[7] + 3 A[13], 5 A[4] + A[8] + 3 A[11], 5 A[5] + A[9] + 3 A[12], 5 A[6] + 4 A[16]], [0, 15 A[2], 20 A[3] + 4 A[7] + 4 A[13], 4 A[11], 4 A[12]], [21 A[2], 5 A[3] + 4 A[13], 10 A[4] + 2 A[8] + A[18], 5 A[5] + A[9] + 4 A[12], 5 A[6] + 4 A[16]], [0, 5 A[2], 15 A[3] + 3 A[7] + 3 A[13], 10 A[4] + 2 A[8] + 2 A[18], 15 A[5] + 3 A[9] + 3 A[19]], [ A[2], 5 A[3] + A[7], 5 A[4] + 4 A[18], 5 A[5] + 4 A[19], 20 A[6] + 4 A[16]] , [0, 5 A[2], 15 A[3] + 3 A[7] + 3 A[13], 10 A[4] + 2 A[8] + 2 A[18], 5 A[5] + 3 A[19] + 2 A[21]], [ A[2], 5 A[3] + A[7], 5 A[4] + 4 A[18], 5 A[5] + 4 A[19], 20 A[6] + 4 A[16]] ], [1, 1, 0, 4, 0, 24, 0, 14, 0, 19, 20, 0, 0, 11, 0, 11, 0, 21, 0, 1, 0, 1]] For example, C(100000), mudolo , 25, equals , 15 The congruence classes mod, 25, in the following set , {2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 22, 23}, never show up! Theorem Number, 8, : Let C(n) be the constant term, in x, of n (1/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 22, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [0, 15 A[2], 5 A[3], A[11], A[12]], [6 A[2], A[13], A[14], A[9], A[16]], [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]], [4 A[2], A[17], A[18], A[19], A[20]], [0, 15 A[2], 5 A[3], 20 A[4], 10 A[5]], [11 A[2], A[7], 15 A[4] + A[8], A[21], A[22]], [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]], [19 A[2], 10 A[3] + 4 A[7], 5 A[4] + 4 A[8], 4 A[9], 20 A[6] + 4 A[10]], [20 A[2], 20 A[3], A[11], 20 A[5], 20 A[6]], 0, [0, 15 A[2], 20 A[3] + A[7] + 4 A[13], A[11], A[12]], [21 A[2], 2 A[7] + 4 A[13], 5 A[4] + A[8] + A[11], 5 A[5] + A[9] + A[12], 5 A[6] + A[10]], [0, 20 A[2], 20 A[3] + 3 A[7] + 2 A[13], 3 A[11], 3 A[12]], [4 A[2], 20 A[3] + 4 A[7], 20 A[4] + 4 A[8] + A[11], 20 A[5] + 4 A[9] + 4 A[12], 20 A[6] + 4 A[16]], [0, 10 A[2], 20 A[3] + 4 A[7] + 4 A[17], 4 A[11], 4 A[12]], [9 A[2], 20 A[3] + 3 A[7] + 4 A[17], 5 A[4] + 4 A[11] + A[18], 20 A[5] + 4 A[9] + 4 A[12], 10 A[6] + 4 A[16]], [0, 5 A[2], 10 A[3] + 2 A[7] + 2 A[17], 2 A[11], 5 A[5] + A[9] + A[19]], [21 A[2], 5 A[3] + 4 A[17], 5 A[4] + 4 A[18], 5 A[5] + 4 A[19], A[16]], [ 0, 20 A[2], 15 A[3] + 3 A[7] + 3 A[17], 3 A[11], 20 A[5] + 4 A[9] + 4 A[19] ], [24 A[2], 20 A[3] + 3 A[7] + 4 A[17], 5 A[4] + 2 A[11] + A[18], 15 A[5] + 2 A[9] + 3 A[19], 15 A[6] + 4 A[16]]], [1, 1, 0, 6, 0, 4, 0, 11, 0, 19, 20, 0, 0, 21, 0, 4, 0, 9, 0, 21, 0, 24]] For example, C(100000), mudolo , 25, equals , 15 The congruence classes mod, 25, in the following set , {2, 3, 5, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 22, 23}, never show up! Theorem Number, 9, : Let C(n) be the constant term, in x, of n (1/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 28, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], A[11], A[12], A[13], A[14]], [5 A[2], 10 A[2] + 10 A[3], 20 A[3] + 15 A[4], A[15], A[16]], [13 A[2], A[17], A[18], A[19], A[20]], [24 A[2], A[21], A[22], A[23], A[24]], [11 A[2], 15 A[2] + 15 A[3] + A[7], A[25], A[26], A[27]], [10 A[2], 10 A[2] + 15 A[3], 20 A[3] + 20 A[4], 5 A[4], 15 A[5] + 5 A[6]], [18 A[2], 20 A[2] + 20 A[3] + 3 A[7], 15 A[3] + 3 A[8], 10 A[4] + 5 A[5] + 3 A[9], A[28]], [24 A[2], 20 A[3] + 4 A[7], 20 A[4] + 4 A[8], 20 A[5] + 4 A[9], 20 A[6] + 4 A[10]], [6 A[2], 5 A[3] + A[11], 10 A[2] + 20 A[3] + 15 A[4] + 4 A[7] + A[8] + A[11], 20 A[4] + 20 A[5] + A[9], 10 A[5] + A[10]], [15 A[2], 20 A[3] + A[7] + 4 A[11], 15 A[2] + 20 A[3] + A[7] + 4 A[11], 20 A[2] + 20 A[3] + 20 A[4] + 5 A[5] + 3 A[7] + 2 A[8] + 2 A[11] + 3 A[12], 15 A[5] + 10 A[6]], [13 A[2], 20 A[2] + 15 A[3] + 3 A[7], 15 A[4] + 3 A[12], 20 A[2] + 20 A[3] + 20 A[4] + 10 A[5] + 3 A[7] + 2 A[8] + 4 A[9] + 2 A[11] + 3 A[12] + 4 A[13], 15 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 5 A[6] + A[7] + 4 A[8] + 3 A[9] + 3 A[10] + 4 A[11] + A[12] + 2 A[13]], [4 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 3 A[7] + 2 A[8] + 4 A[9] + 4 A[10] + 2 A[11] + 3 A[12] + A[13] + A[14], 20 A[3] + 20 A[4] + 20 A[5] + 15 A[6] + 4 A[7] + 3 A[8] + A[9] + A[10] + 2 A[12] + 4 A[13] + 4 A[14], 20 A[4] + 20 A[5] + 10 A[6] + 4 A[8] + 2 A[9] + 2 A[10] + 3 A[13] + 3 A[14] , 20 A[5] + 4 A[9] + 4 A[10] + A[14], 5 A[6] + 4 A[10]], [20 A[3] + 20 A[4] + 20 A[5] + 5 A[6] + A[7] + 4 A[8] + 3 A[9] + 3 A[10] + 4 A[11] + A[12] + 2 A[13] + 2 A[14], 20 A[4] + 20 A[5] + 15 A[6] + 3 A[8] + A[9] + A[10] + 2 A[12] + 4 A[13] + 4 A[14], 20 A[5] + 10 A[6] + 2 A[9] + 2 A[10] + 3 A[13] + 3 A[14], 4 A[10] + A[14], 10 A[6]], [20 A[3] + 20 A[4] + 20 A[5] + 2 A[7] + 3 A[8] + A[9] + 3 A[11] + 2 A[12] + 4 A[13] + A[16], 20 A[4] + 20 A[5] + A[8] + 2 A[9] + 4 A[12] + 3 A[13] + 2 A[16], 20 A[5] + 4 A[9] + A[13] + 4 A[16], 3 A[16], 20 A[6]], [ 3 A[2] + 5 A[3] + 4 A[7] + 2 A[11] + 3 A[17], 15 A[3] + 3 A[11], 15 A[4] + 3 A[12], 10 A[3] + 20 A[4] + 15 A[5] + 4 A[8] + 3 A[9] + A[11] + A[12] + 3 A[16] + 3 A[17], 3 A[10] + 2 A[16]], [ 20 A[3] + 2 A[7] + A[11] + 4 A[17], 20 A[3] + 3 A[7] + 2 A[11], 20 A[3] + 3 A[11] + 4 A[17], 10 A[3] + 20 A[4] + 3 A[8] + 2 A[11] + A[16] + A[17] + 4 A[18], 5 A[6] + 3 A[16]], [4 A[2], 10 A[3] + 20 A[4] + 10 A[5] + 4 A[8] + A[9] + 2 A[16] + 3 A[17] + 2 A[18] + 3 A[19], 10 A[4] + 10 A[5] + A[9] + 2 A[16] + 3 A[18] + 3 A[19], 10 A[5] + 2 A[16] + 3 A[19], 10 A[6] + 4 A[10] + A[16]], [2 A[2] + 20 A[3] + 20 A[4] + 10 A[5] + 3 A[7] + 4 A[8] + A[9] + 2 A[16] + 4 A[17] + 2 A[18] + 3 A[19], 10 A[3] + 10 A[4] + 20 A[5] + 2 A[7] + A[8] + 4 A[9] + 3 A[16] + 3 A[18] + 2 A[19], 10 A[4] + 20 A[5] + 2 A[8] + 4 A[9] + 3 A[16] + 2 A[19], 10 A[5] + 2 A[9] + 3 A[16], 2 A[16] + 4 A[20]], [ 4 A[2] + 20 A[3] + 3 A[7] + 3 A[17] + 2 A[21], 5 A[3] + 4 A[7] + 3 A[17] + 4 A[21], 10 A[4] + 3 A[18], 10 A[5] + 2 A[16] + 3 A[19], 20 A[6] + 3 A[20]], [ 15 A[3] + 2 A[7] + 2 A[17] + 3 A[21], 20 A[3] + 4 A[7] + 2 A[17], 10 A[3] + 10 A[4] + 2 A[7] + A[8] + 3 A[18] + 2 A[21], 20 A[3] + 20 A[4] + 4 A[7] + 4 A[8] + 2 A[18] + 4 A[21], 20 A[6] + 4 A[16]] , [2 A[2] + 4 A[7] + 4 A[17] + A[21], 5 A[3] + 4 A[7] + 4 A[17] + 4 A[21], 10 A[4] + 5 A[5] + A[16] + 4 A[18] + 2 A[19] + A[23], 10 A[5] + 3 A[16] + 4 A[19], 5 A[6] + 4 A[20]], [A[2], 5 A[3] + 4 A[21], 20 A[3] + 5 A[4] + 4 A[7] + 2 A[18] + 4 A[21], 5 A[5] + 4 A[23], 10 A[6] + A[16] + 2 A[20]], [15 A[3] + 2 A[7] + 2 A[17] + 3 A[21], 15 A[3] + 2 A[7] + 3 A[17] + A[21], 20 A[3] + 4 A[7] + 4 A[21], 10 A[5] + A[16] + 3 A[19] + 4 A[23], 10 A[6] + A[16]], [ 15 A[3] + 3 A[2] + 3 A[21] + 2 A[7] + 2 A[17], 5 A[3] + A[17], 5 A[4] + A[18], A[19] + 5 A[5], 5 A[6] + A[20]], [ 10 A[3] + 4 A[2] + 4 A[21] + A[7] + A[17], 20 A[3] + 4 A[7], 10 A[4] + 5 A[3] + A[21] + A[7] + A[16] + A[23] + 2 A[19] + 5 A[5] + 3 A[18], 3 A[16] + A[23] + 5 A[5], 4 A[16] + 3 A[20]], [2 A[2], 5 A[3] + 3 A[21], 10 A[4] + 15 A[3] + 3 A[21] + 3 A[7] + 4 A[18], 3 A[23] + 5 A[5], 20 A[6] + 2 A[16] + 4 A[20]]], [1, 1, 1, 5, 13, 24, 11, 10, 18, 24, 6, 15, 13, 9, 10, 20, 18, 20, 4, 7, 14, 15, 7, 1, 15, 18, 24, 2]] For example, C(100000), mudolo , 25, equals , 0 The congruence classes mod, 25, in the following set , {0, 3, 8, 12, 16, 17, 19, 21, 22, 23}, never show up! Theorem Number, 10, : Let C(n) be the constant term, in x, of n (1/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 32, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], A[11], A[12], A[13], A[14]], [7 A[2], A[15], A[16], A[17], A[18]], [19 A[2], A[19], A[20], A[21], A[22]], [16 A[2], A[23], A[24], A[25], A[10]], [6 A[2], 5 A[2] + 10 A[3] + A[7], A[27], A[28], A[29]], [22 A[2], 20 A[2] + 5 A[3] + 2 A[7], 15 A[3] + 15 A[4] + 2 A[8], A[30], A[31]], [ 14 A[2], 20 A[2] + 15 A[3] + 4 A[7], 15 A[3] + 20 A[4] + 4 A[8], 10 A[4] + 4 A[9], A[32]], [16 A[2], 5 A[3] + A[7], 20 A[4] + A[8], 10 A[5] + A[9], A[10]], [16 A[2], 15 A[2] + 3 A[7] + 3 A[11], 15 A[2] + 20 A[3] + 3 A[7] + A[8] + 2 A[11], 15 A[4] + 5 A[5] + A[9], 20 A[5] + 10 A[6] + A[10]], [2 A[2], 20 A[2] + 10 A[3] + 2 A[7], 20 A[2] + 20 A[3] + 10 A[4] + 4 A[7] + A[8] + A[11] + A[12], 10 A[2] + 20 A[3] + 20 A[4] + 5 A[5] + 2 A[7] + 4 A[8] + 2 A[9] + 3 A[11] + A[12], 5 A[5] + 15 A[6] + 2 A[10]], [4 A[2], 10 A[2] + 20 A[3] + 2 A[7] + 2 A[11], 20 A[4] + 4 A[12], 15 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 3 A[7] + A[8] + A[9] + 2 A[11] + 4 A[12] + 3 A[13], 20 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 20 A[6] + 4 A[7] + 3 A[8] + 3 A[9] + 4 A[10] + A[11] + 2 A[12] + 2 A[13]], [A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 10 A[6] + 2 A[7] + 4 A[8] + 4 A[9] + 2 A[10] + 3 A[11] + A[12] + A[13] + 3 A[14], 5 A[3] + A[7], 5 A[4] + 20 A[5] + 10 A[6] + A[8] + 4 A[9] + 2 A[10] + A[13] + 3 A[14], 5 A[5] + 15 A[6] + A[9] + A[10] + 4 A[14], A[10]], [2 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 15 A[6] + A[7] + 2 A[8] + 2 A[9] + A[10] + 4 A[11] + 3 A[12] + 3 A[13] + 4 A[14], 10 A[3] + 20 A[4] + 20 A[5] + 10 A[6] + 4 A[8] + 4 A[9] + 2 A[10] + 2 A[11] + A[12] + A[13] + 3 A[14], 10 A[4] + 20 A[5] + 5 A[6] + A[9] + 3 A[10] + 2 A[12] + 4 A[13] + 2 A[14], 10 A[5] + 10 A[6] + 2 A[10] + 2 A[13] + 3 A[14], 20 A[6] + 2 A[14]], [ 4 A[2] + 20 A[3] + 15 A[4] + 4 A[7] + A[11] + 3 A[12] + A[16], 20 A[3] + 10 A[4] + A[7] + 3 A[11] + A[12] + 2 A[16], 15 A[4] + 4 A[8] + 2 A[12] + 4 A[16], 20 A[5] + A[9] + 3 A[13], 20 A[5] + 10 A[6] + A[8] + A[9] + 4 A[10] + 3 A[12] + 4 A[13] + 3 A[16]], [ 3 A[2] + 20 A[3] + 10 A[4] + 3 A[7] + 2 A[11] + A[12] + 2 A[16], 15 A[3] + 10 A[4] + 3 A[11] + A[12] + 2 A[16], 4 A[8] + 2 A[16], 15 A[4] + 15 A[5] + 3 A[8] + 3 A[9] + 4 A[12] + 4 A[16], 15 A[4] + 10 A[5] + 2 A[8] + A[9] + 3 A[10] + A[12] + A[16] + 2 A[17]], [ 2 A[2] + 20 A[3] + 10 A[4] + 15 A[5] + 10 A[6] + A[7] + A[8] + 3 A[9] + 2 A[10] + 4 A[11] + 2 A[16] + A[17] + 4 A[18], 10 A[3] + 15 A[4] + 5 A[6] + 2 A[7] + 3 A[8] + 4 A[9] + A[10] + A[16] + 3 A[17] + 2 A[18], 10 A[4] + 10 A[5] + 5 A[6] + 2 A[8] + A[9] + 4 A[10] + 2 A[17] + 3 A[18], 10 A[5] + 2 A[9], 15 A[6] + 2 A[10]], [4 A[2] + 15 A[3] + 20 A[4] + 10 A[5] + 5 A[6] + 3 A[7] + 2 A[8] + A[9] + 4 A[10] + 4 A[16] + 2 A[17] + 3 A[18] + 3 A[19], 5 A[3] + 10 A[4] + 15 A[5] + 10 A[6] + A[8] + 3 A[9] + 2 A[10] + 2 A[16] + A[17] + 4 A[18] + A[19], 15 A[4] + 15 A[5] + 10 A[6] + 2 A[8] + 3 A[9] + 2 A[10] + A[16] + A[17] + 4 A[18], 15 A[5] + 10 A[6] + 2 A[9] + 2 A[10] + A[17] + 4 A[18], 20 A[6] + 2 A[10] + A[18]], [3 A[2], 15 A[3] + 2 A[7] + 3 A[8] + 3 A[16] + 4 A[19] + 4 A[20], 15 A[4] + 2 A[8] + 4 A[20], 15 A[4] + 10 A[5] + A[8] + A[9] + 3 A[16] + A[17] + 2 A[20], 4 A[18]], [ A[2] + 5 A[3] + 15 A[4] + A[7] + 2 A[8] + 2 A[16] + A[19] + A[20], 5 A[3] + 5 A[4] + 4 A[8] + 4 A[16] + 4 A[19] + 2 A[20], 3 A[8] + 4 A[16], 15 A[4] + 15 A[5] + A[8] + 3 A[16] + 4 A[17] + 2 A[20] + 2 A[21], 15 A[4] + 10 A[5] + 10 A[6] + 2 A[8] + A[16] + 3 A[17] + 3 A[18] + 4 A[20] + A[21] ], [4 A[2] + 10 A[3] + 10 A[4] + 15 A[5] + 20 A[6] + 2 A[7] + 2 A[16] + 4 A[17] + 4 A[18] + 2 A[19] + 4 A[20] + 3 A[21] + 3 A[22], 20 A[3] + 15 A[4] + 10 A[5] + 20 A[6] + 4 A[7] + 4 A[16] + 3 A[17] + 3 A[18] + 3 A[20] + A[21] + A[22], 15 A[4] + 10 A[5] + 3 A[16] + 2 A[17] + 2 A[18] + 2 A[20] + 4 A[21] + 4 A[22], 15 A[5] + 3 A[17] + 2 A[18] + 2 A[21] + 4 A[22], 15 A[6] + 3 A[18] + 2 A[22]], [A[2] + 15 A[3] + 10 A[4] + 5 A[5] + 3 A[7] + 3 A[16] + A[17] + A[18] + 3 A[19] + A[20] + 2 A[21] + 2 A[22], 5 A[3] + 10 A[4] + 15 A[5] + 20 A[6] + 2 A[16] + 4 A[17] + 4 A[18] + 4 A[19] + 4 A[20] + 3 A[21] + 3 A[22], 5 A[4] + 4 A[20], 5 A[5] + 20 A[6] + 4 A[18] + 4 A[21] + 3 A[22], 4 A[22]], [%1, 15 A[3] + 15 A[4] + 10 A[5] + 20 A[6] + 3 A[7] + 4 A[16] + 3 A[17] + 3 A[18] + A[19] + 3 A[20] + A[21] + A[22], 5 A[4] + A[16], 5 A[5] + A[17], 5 A[6] + A[18]], [4 A[2], 5 A[3] + 15 A[4] + 10 A[5] + 20 A[6] + 4 A[16] + 3 A[17] + 3 A[18] + A[19] + 3 A[20] + A[21] + A[22], 5 A[4] + 5 A[5] + A[17] + A[18] + A[20] + 2 A[21] + 2 A[22], 5 A[5] + 2 A[18] + A[21] + 4 A[22], 10 A[6] + A[22]], [10 A[4] + 10 A[3] + A[2] + 20 A[6] + 3 A[21] + 2 A[7] + 3 A[22] + 2 A[16] + 4 A[17] + 2 A[19] + 15 A[5] + 4 A[18] + 4 A[20], 5 A[3] + A[7], 10 A[4] + 20 A[6] + 3 A[21] + 3 A[22] + 2 A[16] + 4 A[17] + 15 A[5] + 4 A[18] + 3 A[20], 3 A[21] + 2 A[17] + 10 A[5], 15 A[6] + 3 A[22] + 2 A[18]], [%1, 15 A[3] + 15 A[4] + 10 A[5] + 20 A[6] + 3 A[7] + 4 A[16] + 3 A[17] + 3 A[18] + A[19] + 3 A[20] + A[21] + A[22], 5 A[4] + A[16], 5 A[5] + A[17], 5 A[6] + A[18]], [5 A[4] + 5 A[3] + 4 A[2] + 4 A[21] + A[7] + 4 A[22] + A[16] + 2 A[17] + A[19] + 10 A[5] + 2 A[18] + 2 A[20], 5 A[3] + A[19], 5 A[4] + A[20], A[21] + 5 A[5], 5 A[6] + A[22]], [5 A[4] + 5 A[3] + A[2] + 4 A[21] + A[7] + 4 A[22] + A[16] + 2 A[17] + A[19] + 10 A[5] + 2 A[18] + 2 A[20], 15 A[4] + 5 A[3] + 20 A[6] + A[21] + A[7] + A[22] + 4 A[16] + 3 A[17] + 10 A[5] + 3 A[18] + 3 A[20], 10 A[4] + 2 A[16] + 3 A[20], 3 A[21] + 4 A[22] + 2 A[17] + 10 A[5] + 2 A[18], 20 A[6] + 3 A[22] + 2 A[18]], [5 A[4] + 5 A[3] + 3 A[2] + 4 A[21] + A[7] + 4 A[22] + A[16] + 2 A[17] + A[19] + 10 A[5] + 2 A[18] + 2 A[20], 15 A[4] + 5 A[3] + 20 A[6] + A[21] + A[22] + 4 A[16] + 3 A[17] + 2 A[19] + 10 A[5] + 3 A[18] + 3 A[20], 5 A[4] + 2 A[21] + 2 A[22] + A[17] + 5 A[5] + A[18] + 2 A[20], 2 A[21] + 4 A[22] + 5 A[5] + 2 A[18], 10 A[6] + 2 A[22]], [%1, 15 A[4] + 10 A[3] + 20 A[6] + A[21] + 2 A[7] + A[22] + 4 A[16] + 3 A[17] + 10 A[5] + 3 A[18] + 3 A[20], 15 A[4] + 20 A[6] + 3 A[21] + 3 A[22] + 4 A[16] + 4 A[17] + 15 A[5] + 4 A[18] + A[20], A[21] + 4 A[22] + 4 A[17] + 15 A[5] + 2 A[18], 5 A[6] + A[22] + 4 A[18]], [ 10 A[4] + 15 A[3] + 4 A[2] + 2 A[21] + 3 A[7] + 2 A[22] + 3 A[16] + A[17] + 3 A[19] + 5 A[5] + A[18] + A[20], 20 A[3] + 4 A[7], 15 A[4] + 2 A[21] + 2 A[22] + 3 A[16] + A[17] + 5 A[5] + A[18] + 2 A[20], 2 A[21] + 3 A[17] + 15 A[5], 10 A[6] + 2 A[22] + 3 A[18]]], [1, 1, 1, 7, 19, 16, 6, 22, 14, 16, 16, 2, 4, 16, 22, 9, 13, 22, 14, 3, 21, 19, 11, 12, 4, 16, 12, 24, 21, 23, 12, 14]] %1 := 2 A[2] + 15 A[3] + 10 A[4] + 5 A[5] + 3 A[7] + 3 A[16] + A[17] + A[18] + 3 A[19] + A[20] + 2 A[21] + 2 A[22] For example, C(100000), mudolo , 25, equals , 17 The congruence classes mod, 25, in the following set , {0, 5, 8, 10, 15, 17, 18, 20}, never show up! Theorem Number, 11, : Let C(n) be the constant term, in x, of n (1/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], 15 A[2] + 12 A[3], A[11], A[12], A[13]], [4 A[2], 10 A[2] + 9 A[3], 20 A[3] + 14 A[4], A[14], A[15]], [8 A[2], 10 A[2] + 23 A[3], 20 A[3] + 13 A[4], 5 A[4] + 3 A[5], A[16]], [ 16 A[2], 5 A[2] + 16 A[3], 10 A[3] + 16 A[4], 15 A[4] + 16 A[5], 20 A[5] + 16 A[6]], [7 A[2], 15 A[2] + 17 A[3], 5 A[3] + 2 A[4], 20 A[4] + 12 A[5], 10 A[5] + 22 A[6]], [24 A[2], 10 A[2] + 4 A[3], 20 A[3] + 9 A[4], 5 A[4] + 14 A[5], 15 A[5] + 19 A[6]], [18 A[2], 10 A[2] + 8 A[3], 20 A[3] + 23 A[4], 5 A[4] + 13 A[5], 15 A[5] + 3 A[6]], [A[2], 5 A[2] + A[3], 10 A[3] + A[4], 15 A[4] + A[5], 20 A[5] + A[6]], [ 23 A[2], 20 A[2] + 8 A[3], 15 A[3] + 18 A[4], 10 A[4] + 3 A[5], 5 A[5] + 13 A[6]], [11 A[2], 20 A[2] + 16 A[3], 15 A[3] + 21 A[4], 10 A[4] + A[5], 5 A[5] + 6 A[6]], [2 A[2], 10 A[2] + 2 A[3], 20 A[3] + 2 A[4], 5 A[4] + 2 A[5], 15 A[5] + 2 A[6]], [22 A[2], 15 A[2] + 7 A[3], 5 A[3] + 17 A[4], 20 A[4] + 2 A[5], 10 A[5] + 12 A[6]], [ 4 A[2], 20 A[2] + 4 A[3], 15 A[3] + 4 A[4], 10 A[4] + 4 A[5], 5 A[5] + 4 A[6]], [8 A[2], 15 A[2] + 8 A[3], 5 A[3] + 8 A[4], 20 A[4] + 8 A[5], 10 A[5] + 8 A[6]]], [1, 1, 2, 4, 8, 16, 7, 24, 18, 1, 23, 11, 2, 22, 4, 8]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 3, 5, 6, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21}, never show up! Theorem Number, 12, : Let C(n) be the constant term, in x, of n (1/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[3], A[4], A[5], A[6]], [2 A[2], 12 A[3], A[11], A[12], A[13]], [6 A[2], 16 A[3], A[4], A[14], A[15]], [20 A[2], 10 A[3], 0, 15 A[5], A[16]], %1, [2 A[2], 12 A[3], 22 A[4], 7 A[5], 17 A[6]], [6 A[2], 16 A[3], A[4], 11 A[5], 21 A[6]], %1, %1, [7 A[2], 2 A[3], 22 A[4], 17 A[5], 12 A[6]], [15 A[2], 20 A[3], 0, 5 A[5], 10 A[6]], [15 A[2], 20 A[3], 0, 5 A[5], 10 A[6]], %1, %1, 0], [1, 1, 2, 6, 20, 20, 2, 6, 20, 20, 7, 15, 15, 20, 20, 0]] %1 := [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]] For example, C(100000), mudolo , 25, equals , 9 The congruence classes mod, 25, in the following set , {3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24}, never show up! Theorem Number, 13, : Let C(n) be the constant term, in x, of n (1/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 32, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], A[11], A[12], A[13], A[14]], [8 A[2], A[15], A[16], A[17], A[18]], [7 A[2], A[19], A[20], A[21], A[14]], [11 A[2], A[7], A[24], A[25], A[26]], [17 A[2], 20 A[2] + 2 A[7], A[20], A[28], A[29]], [ 3 A[2], 10 A[2] + 15 A[3] + 3 A[7], 20 A[3] + 5 A[4] + 3 A[8], A[30], A[31] ], [2 A[2], 20 A[2] + 15 A[3] + 2 A[7], 15 A[3] + 5 A[4] + 2 A[8], 10 A[4] + 20 A[5] + 2 A[9], A[32]], [16 A[2], 5 A[3] + A[7], 20 A[4] + A[8], 10 A[5] + A[9], A[10]], [24 A[2], 10 A[2] + 20 A[3] + A[7] + 4 A[11], 5 A[2] + 15 A[3] + 10 A[4] + 3 A[7] + 4 A[8] + A[11], 20 A[4] + 5 A[5] + 4 A[9], 10 A[5] + 4 A[10]], [21 A[2], 15 A[2] + 3 A[7] + 4 A[11], 15 A[2] + 5 A[4] + 4 A[7] + A[8] + 3 A[11], 15 A[2] + 20 A[4] + 10 A[5] + 4 A[7] + 2 A[8] + A[9] + 3 A[11] + 4 A[12], 5 A[5] + 15 A[6] + A[10]], [4 A[2] + 10 A[3] + 15 A[4] + 15 A[5] + A[7] + 3 A[8] + 3 A[9] + 2 A[11] + A[12] + A[13], 10 A[3] + 15 A[4] + 15 A[5] + 3 A[8] + 3 A[9] + 2 A[11] + A[12] + A[13], 10 A[4] + 2 A[12], 20 A[5] + 2 A[13], 10 A[5] + 10 A[6] + 4 A[10]], [2 A[2] + 20 A[3] + 10 A[4] + 10 A[5] + 2 A[7] + A[8] + A[9] + 4 A[11] + 2 A[12] + 2 A[13], 10 A[3] + 2 A[7], 10 A[4] + 2 A[8] + 4 A[9] + 3 A[13], 10 A[5] + 15 A[6] + 2 A[9] + A[10] + 2 A[14], 2 A[10]], [A[2], 10 A[3] + 20 A[4] + 20 A[5] + 2 A[8] + 2 A[9] + 3 A[11] + 4 A[12] + 4 A[13] , 10 A[4] + 15 A[5] + 3 A[9] + 3 A[12] + A[13], 10 A[5] + 15 A[6] + A[10] + 3 A[13] + 2 A[14], 20 A[6] + 3 A[14]], [4 A[2], 20 A[3] + 20 A[4] + A[7] + 3 A[8] + 4 A[11] + 4 A[16], 2 A[8] + 3 A[12] + 2 A[16], 20 A[5] + 20 A[6] + 4 A[8] + 4 A[9] + 4 A[10] + A[12] + 3 A[14] + 3 A[16], 15 A[6] + A[10] + 4 A[14]], [ A[2] + 15 A[3] + 10 A[4] + 3 A[7] + 2 A[8] + A[11] + A[16], 10 A[3] + 10 A[4] + 2 A[8] + 3 A[11] + A[16], 15 A[4] + A[8] + 2 A[12] + 2 A[16], 20 A[4] + 5 A[5] + 3 A[8] + A[9] + 2 A[12] + A[16], 10 A[5] + 2 A[8] + 2 A[9] + A[10] + 3 A[12] + 4 A[16] + A[17]], [ 3 A[2] + 10 A[3] + 20 A[4] + A[7] + 4 A[8] + 2 A[11] + 2 A[16], 15 A[3] + 10 A[4] + 3 A[7] + 2 A[8] + A[16], 15 A[4] + 10 A[5] + 10 A[6] + 3 A[8] + A[9] + 4 A[10] + 3 A[17] + 2 A[18], 15 A[5] + 3 A[9] + 3 A[10] + 4 A[18], 20 A[6] + 3 A[10]], [ 4 A[2] + 10 A[3] + 10 A[4] + A[7] + 2 A[8] + A[16] + 2 A[19], 10 A[3] + 10 A[4] + A[8] + 3 A[16] + 2 A[19], 20 A[4] + 20 A[5] + 20 A[6] + 2 A[8] + 4 A[9] + A[10] + 4 A[16] + 2 A[17] + 3 A[18], 20 A[5] + 20 A[6] + 2 A[9] + A[10] + 4 A[17] + 3 A[18], 2 A[10] + 4 A[18]], [A[2] + 10 A[3] + 10 A[4] + A[7] + 2 A[8] + A[16] + 2 A[19], 20 A[3] + 4 A[7] + A[19], A[8] + 4 A[16] + 4 A[20], 20 A[4] + 5 A[5] + 3 A[8] + A[9] + A[16] + 2 A[20], 2 A[18]], [ 4 A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 4 A[8] + 2 A[16] + 4 A[19], 10 A[3] + 10 A[4] + A[8] + 3 A[16] + 2 A[19], 5 A[4] + 3 A[8] + A[16] + 4 A[20], 20 A[4] + 15 A[5] + 3 A[8] + A[16] + A[17] + 2 A[20] + 3 A[21], 20 A[5] + 2 A[8] + 4 A[16] + 4 A[17] + 3 A[18] + 3 A[20] + 4 A[21]], [ 2 A[2] + 20 A[3] + 20 A[4] + 10 A[5] + 10 A[6] + 2 A[7] + 4 A[16] + 2 A[17] + 4 A[18] + 4 A[19] + 4 A[20] + 2 A[21] + 4 A[22], 10 A[3] + 2 A[7], 10 A[4] + 15 A[5] + 15 A[6] + A[16] + 3 A[17] + A[18] + 2 A[20] + 3 A[21] + A[22], 10 A[5] + 10 A[6] + A[17] + 4 A[18] + 2 A[21] + 4 A[22], A[18] + 2 A[22]], [2 A[2] + 10 A[3] + 10 A[4] + 5 A[5] + 5 A[6] + A[7] + 2 A[16] + A[17] + 2 A[18] + 2 A[19] + 2 A[20] + A[21] + 2 A[22], 5 A[3] + A[19], 5 A[4] + 15 A[5] + 15 A[6] + 3 A[17] + A[18] + A[20] + 3 A[21] + A[22], 5 A[5] + 5 A[6] + 2 A[18] + A[21] + 2 A[22], 15 A[6] + A[22]], [3 A[2] + 15 A[3] + 5 A[4] + 15 A[5] + 15 A[6] + 3 A[7] + A[16] + 3 A[17] + A[18] + A[19] + A[20] + 3 A[21] + A[22], 20 A[3] + 2 A[7] + 3 A[19], 5 A[4] + 20 A[5] + 20 A[6] + A[16] + 4 A[17] + 3 A[18] + 4 A[21] + 3 A[22], 5 A[5] + 20 A[6] + A[17] + 3 A[18] + 3 A[22], 15 A[6] + A[18]], [2 A[2] + 15 A[4] + 20 A[5] + 20 A[6] + 4 A[7] + 3 A[16] + 4 A[17] + 3 A[18] + 3 A[19] + 3 A[20] + 4 A[21] + 3 A[22], 5 A[3] + A[19], 5 A[4] + A[20], A[21] + 5 A[5], 5 A[6] + A[22]], [15 A[4] + A[2] + 20 A[6] + 4 A[21] + 4 A[7] + 3 A[22] + 3 A[16] + 4 A[17] + 3 A[19] + 20 A[5] + 3 A[18] + 3 A[20], 10 A[4] + 5 A[3] + 5 A[6] + A[21] + A[7] + 2 A[22] + 2 A[16] + A[17] + 5 A[5] + 2 A[18] + 2 A[20], 10 A[4] + 10 A[6] + 2 A[21] + 4 A[22] + 3 A[16] + 2 A[17] + 10 A[5] + 4 A[18] + A[20], 20 A[6] + A[21] + 3 A[22] + 3 A[17] + 10 A[5] + 3 A[18], A[22] + 3 A[18]], [10 A[4] + 10 A[3] + A[2] + 5 A[6] + A[21] + A[7] + 2 A[22] + 2 A[16] + A[17] + 2 A[19] + 5 A[5] + 2 A[18] + 2 A[20], 20 A[3] + 4 A[7] + A[19], 5 A[4] + 15 A[6] + 3 A[21] + A[22] + 2 A[16] + 3 A[17] + 15 A[5] + A[18], 15 A[6] + A[22] + 2 A[17] + 5 A[5] + A[18], 2 A[18]], [5 A[4] + 15 A[3] + 4 A[2] + 15 A[6] + 3 A[21] + 3 A[7] + A[22] + A[16] + 3 A[17] + A[19] + 15 A[5] + A[18] + A[20], 10 A[3] + 2 A[19], 10 A[4] + 2 A[20], 2 A[21] + 10 A[5], 10 A[6] + 2 A[22]], [5 A[4] + 15 A[3] + 2 A[2] + 15 A[6] + 3 A[21] + 3 A[7] + A[22] + A[16] + 3 A[17] + A[19] + 15 A[5] + A[18] + A[20], 20 A[4] + 10 A[3] + 10 A[6] + 2 A[21] + 2 A[7] + 4 A[22] + 4 A[16] + 2 A[17] + 10 A[5] + 4 A[18] + 4 A[20], 10 A[4] + 20 A[6] + 4 A[21] + 3 A[22] + A[16] + 4 A[17] + 20 A[5] + 3 A[18] + 2 A[20], 15 A[6] + 2 A[21] + A[22] + A[17] + 10 A[5] + A[18], 15 A[6] + 2 A[22] + A[18]], [15 A[4] + A[2] + 20 A[6] + 4 A[21] + 4 A[7] + 3 A[22] + 3 A[16] + 4 A[17] + 3 A[19] + 20 A[5] + 3 A[18] + 3 A[20], 15 A[4] + 10 A[3] + 20 A[6] + 4 A[21] + 3 A[22] + 3 A[16] + 4 A[17] + 3 A[19] + 20 A[5] + 3 A[18] + 3 A[20], 10 A[4] + 10 A[6] + 2 A[21] + 4 A[22] + 2 A[17] + 10 A[5] + 4 A[18] + 3 A[20], 10 A[6] + 3 A[21] + 4 A[22] + 10 A[5] + 4 A[18], 15 A[6] + 3 A[22]], [ 15 A[4] + 3 A[2] + 20 A[6] + 4 A[21] + 4 A[7] + 3 A[22] + 3 A[16] + 4 A[17] + 3 A[19] + 20 A[5] + 3 A[18] + 3 A[20], 20 A[4] + 15 A[3] + 10 A[6] + 2 A[21] + 3 A[7] + 4 A[22] + 4 A[16] + 2 A[17] + 10 A[5] + 4 A[18] + 4 A[20], 20 A[4] + 15 A[6] + 3 A[21] + A[22] + 4 A[16] + 3 A[17] + 15 A[5] + A[18] + 3 A[20], 20 A[6] + 3 A[21] + 3 A[22] + 4 A[17] + 20 A[5] + 3 A[18], 20 A[6] + 3 A[22] + 4 A[18]], [2 A[2], 10 A[4] + 10 A[3] + 5 A[6] + A[21] + 2 A[7] + 2 A[22] + 2 A[16] + A[17] + 5 A[5] + 2 A[18] + 2 A[20], 10 A[4] + 5 A[6] + A[21] + 2 A[22] + A[16] + A[17] + 5 A[5] + 2 A[18] + 2 A[20], 2 A[21] + A[17] + 10 A[5], 20 A[6] + 2 A[22] + A[18]]], [1, 1, 2, 8, 7, 11, 17, 3, 2, 16, 24, 21, 19, 7, 1, 4, 21, 18, 19, 16, 9, 7, 17, 23, 12, 11, 16, 24, 22, 11, 13, 2]] For example, C(100000), mudolo , 25, equals , 17 The congruence classes mod, 25, in the following set , {0, 5, 6, 10, 14, 15, 20}, never show up! Theorem Number, 14, : Let C(n) be the constant term, in x, of n (1/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 28, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], A[11], A[12], A[13], A[14]], [10 A[2], 5 A[2] + 5 A[3], 10 A[3], A[15], A[16]], [19 A[2], A[17], A[18], A[19], A[20]], [14 A[2], A[21], A[22], A[23], A[24]], [2 A[2], 10 A[2] + 10 A[3] + 2 A[7], A[12], A[13], A[14]], [15 A[2], 5 A[2] + 10 A[3], 10 A[3] + 5 A[4], 15 A[4], 20 A[5] + 20 A[6]], [24 A[2], 20 A[2] + 10 A[3] + 4 A[7], 15 A[3] + 4 A[8], 10 A[4] + 15 A[5] + 4 A[9], A[28]], [24 A[2], 20 A[3] + 4 A[7], 20 A[4] + 4 A[8], 20 A[5] + 4 A[9], 20 A[6] + 4 A[10]], [9 A[2], 10 A[2] + 15 A[3] + 2 A[7] + A[11], 20 A[2] + 20 A[4] + 4 A[7] + 4 A[8] + 3 A[11], 10 A[4] + 15 A[5] + 4 A[9], 5 A[5] + 10 A[6] + 4 A[10]], [20 A[2], 15 A[2] + 10 A[3] + A[7] + 2 A[11], 10 A[2] + 20 A[3] + 2 A[7] + 4 A[11], 5 A[2] + 10 A[3] + 10 A[4] + 15 A[5] + A[7] + A[8] + 2 A[11] + 2 A[12], 15 A[5] + 5 A[6]], [3 A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 2 A[8] + 4 A[9] + 4 A[11] + 4 A[12] + 3 A[13], 15 A[3] + 15 A[5] + 4 A[8] + 3 A[9] + 4 A[11] + 3 A[12] + A[13], 15 A[4] + 4 A[12], 5 A[5] + 4 A[13], 10 A[5] + 10 A[6] + 3 A[10]], [3 A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 2 A[8] + 4 A[9] + 4 A[11] + 4 A[12] + 3 A[13], 15 A[3] + 3 A[7], 15 A[4] + 3 A[8], 15 A[5] + 3 A[9], 15 A[6] + 3 A[10]], [ 15 A[5] + 4 A[7] + 4 A[8] + 3 A[9] + 3 A[11] + 3 A[12] + A[13], 20 A[4] + 2 A[8] + 4 A[9] + 4 A[12] + 3 A[13], 15 A[5] + 3 A[9] + A[13], 15 A[6] + A[10] + 2 A[14], 5 A[6]], [ 20 A[3] + 20 A[4] + 2 A[7] + 2 A[8] + 4 A[9] + 4 A[11] + 4 A[12] + 3 A[13], 10 A[4] + 20 A[5] + A[8] + 2 A[9] + 2 A[12] + 4 A[13], 4 A[9] + 3 A[13], 20 A[6] + 2 A[16], 15 A[6]], [3 A[2] + 3 A[7] + 3 A[11] + 4 A[17], 15 A[3] + 3 A[11] + 3 A[17], 10 A[3] + 15 A[4] + 3 A[8] + 3 A[11] + A[17], 15 A[3] + 15 A[5] + 20 A[6] + 4 A[8] + 3 A[9] + 4 A[11] + 3 A[12] + 2 A[16] + 3 A[17], 3 A[10] + 3 A[16]], [15 A[3] + 2 A[7] + 2 A[11] + A[17], 15 A[3] + A[7] + 3 A[11] + 2 A[17], 15 A[3] + 4 A[11] + 3 A[17], 10 A[3] + 10 A[4] + 15 A[6] + 2 A[8] + 2 A[11] + 4 A[16] + 4 A[17] + 2 A[18], 4 A[16]], [A[2] + 15 A[3] + 10 A[4] + 20 A[5] + 10 A[6] + 3 A[7] + 2 A[8] + 4 A[9] + A[16] + 3 A[17] + 2 A[18] + 4 A[19], 5 A[3] + 10 A[4] + 20 A[5] + 10 A[6] + 2 A[8] + 4 A[9] + A[16] + 4 A[17] + 2 A[18] + 4 A[19], 5 A[4] + 15 A[5] + 20 A[6] + 3 A[9] + 2 A[16] + 4 A[18] + 3 A[19], 5 A[5] + 15 A[6] + 4 A[16] + 4 A[19], 10 A[6] + A[10] + A[16]], [A[2] + 10 A[3] + 15 A[4] + 5 A[5] + 15 A[6] + 2 A[7] + 3 A[8] + A[9] + 4 A[16] + 2 A[17] + 3 A[18] + A[19], 5 A[3] + 20 A[4] + 15 A[5] + 20 A[6] + A[7] + 4 A[8] + 3 A[9] + 2 A[16] + 4 A[18] + 3 A[19], 5 A[4] + 5 A[5] + 15 A[6] + A[8] + A[9] + 4 A[16] + A[19], 5 A[5] + 5 A[6] + A[9] + 3 A[16], 15 A[6] + 4 A[16] + 4 A[20]], [3 A[2], 10 A[3] + A[7] + 2 A[17] + A[21], 15 A[3] + 10 A[4] + 3 A[7] + A[8] + 3 A[18] + 3 A[21], 5 A[5] + 20 A[6] + 2 A[16] + 2 A[19], 5 A[6] + 2 A[20]], [ 15 A[3] + 2 A[7] + 3 A[17] + 4 A[21], 5 A[3] + A[7] + A[17], 15 A[3] + 5 A[4] + 3 A[7] + 2 A[18] + 3 A[21] + 3 A[22], 15 A[3] + 5 A[4] + 3 A[7] + A[18] + 3 A[21] + 4 A[22], 20 A[6] + 4 A[16]], [%1, 4 A[7] + 4 A[17] + 4 A[21], 20 A[3] + 5 A[4] + 4 A[7] + 2 A[18] + 4 A[21] + 2 A[22], 5 A[5] + 15 A[6] + 4 A[16] + 4 A[19], 10 A[6] + 4 A[20]], [%1, 5 A[3] + 4 A[21], 5 A[4] + 4 A[22], 15 A[3] + 5 A[4] + 5 A[5] + 3 A[7] + A[18] + 4 A[19] + 3 A[21] + 4 A[22], 10 A[6] + 4 A[16] + 4 A[20]], [20 A[3] + 4 A[7] + A[17] + 3 A[21], 5 A[3] + 3 A[17] + 2 A[21], 5 A[3] + A[7] + A[21], 20 A[3] + 5 A[4] + 20 A[6] + 4 A[7] + 2 A[16] + 3 A[18] + 4 A[21] + 2 A[22] , 2 A[16]], [15 A[3] + 3 A[2] + A[21] + 3 A[7] + 2 A[17], 5 A[3] + 2 A[17], 5 A[4] + 2 A[18], 2 A[19] + 5 A[5], 5 A[6] + 2 A[20]], [ 15 A[3] + 3 A[2] + A[21] + 3 A[7] + 2 A[17], 15 A[3] + 3 A[7], 5 A[4] + 15 A[3] + 3 A[21] + 3 A[7] + A[22] + A[18], 5 A[4] + 20 A[3] + 5 A[6] + 4 A[21] + 4 A[7] + 2 A[22] + 3 A[16] + 2 A[19] + 5 A[5] + 3 A[18], 5 A[6] + 2 A[16] + 2 A[20]], [%1, 5 A[3] + 4 A[21], 5 A[4] + 4 A[22], 15 A[3] + 5 A[4] + 5 A[5] + 3 A[7] + A[18] + 4 A[19] + 3 A[21] + 4 A[22], 10 A[6] + 4 A[16] + 4 A[20]]], [1, 1, 2, 10, 19, 14, 2, 15, 24, 24, 9, 20, 18, 18, 5, 15, 13, 15, 11, 16, 3, 10, 11, 11, 20, 18, 18, 11]] %1 := A[2] + 15 A[3] + 2 A[7] + 3 A[17] + 4 A[21] For example, C(100000), mudolo , 25, equals , 0 The congruence classes mod, 25, in the following set , {0, 4, 6, 7, 8, 12, 17, 21, 22, 23}, never show up! Theorem Number, 15, : Let C(n) be the constant term, in x, of n (1/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], 18 A[3], A[11], A[12], A[13]], [9 A[2], 20 A[2] + 14 A[3], 15 A[3] + 19 A[4], A[14], A[15]], [2 A[2], 12 A[3], 22 A[4], 7 A[5], A[16]], [6 A[2], 15 A[2] + 6 A[3], 5 A[3] + 6 A[4], 20 A[4] + 6 A[5], 10 A[5] + 6 A[6]], [18 A[2], 8 A[3], 23 A[4], 13 A[5], 3 A[6]], [24 A[2], 20 A[2] + 4 A[3], 15 A[3] + 9 A[4], 10 A[4] + 14 A[5], 5 A[5] + 19 A[6]], [7 A[2], 17 A[3], 2 A[4], 12 A[5], 22 A[6]], [A[2], 15 A[2] + A[3], 5 A[3] + A[4], 20 A[4] + A[5], 10 A[5] + A[6]], [ 22 A[2], 10 A[2] + 12 A[3], 20 A[3] + 2 A[4], 5 A[4] + 17 A[5], 15 A[5] + 7 A[6]], [21 A[2], A[3], 6 A[4], 11 A[5], 16 A[6]], [3 A[2], 20 A[2] + 3 A[3], 15 A[3] + 3 A[4], 10 A[4] + 3 A[5], 5 A[5] + 3 A[6]], [13 A[2], 3 A[3], 18 A[4], 8 A[5], 23 A[6]], [9 A[2], 10 A[2] + 9 A[3], 20 A[3] + 9 A[4], 5 A[4] + 9 A[5], 15 A[5] + 9 A[6]], [2 A[2], 5 A[2] + 2 A[3], 10 A[3] + 2 A[4], 15 A[4] + 2 A[5], 20 A[5] + 2 A[6]]], [1, 1, 3, 9, 2, 6, 18, 24, 7, 1, 22, 21, 3, 13, 9, 2]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 4, 5, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 23}, never show up! Theorem Number, 16, : Let C(n) be the constant term, in x, of n (1/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], 10 A[2] + 18 A[3], A[11], A[12], A[13]], [11 A[2], 20 A[2] + 21 A[3], 15 A[3] + 6 A[4], A[14], A[10]], [20 A[2], 5 A[3], 15 A[4], 0, A[16]], %1, [23 A[2], 10 A[2] + 13 A[3], 20 A[3] + 3 A[4], 5 A[4] + 18 A[5], 15 A[5] + 8 A[6]], [6 A[2], 20 A[2] + 16 A[3], 15 A[3] + A[4], 10 A[4] + 11 A[5], 5 A[5] + 21 A[6]], [5 A[2], 15 A[3], 0, 10 A[5], 20 A[6]], %1, [23 A[2], 10 A[2] + 3 A[3], 20 A[3] + 8 A[4], 5 A[4] + 13 A[5], 15 A[5] + 18 A[6]], [15 A[2], 20 A[3], 0, 5 A[5], 10 A[6]], [10 A[2], 5 A[3], 0, 20 A[5], 15 A[6]], [5 A[2], 15 A[3], 0, 10 A[5], 20 A[6]], %1, 0], [1, 1, 3, 11, 20, 20, 23, 6, 5, 20, 23, 15, 10, 5, 20, 0]] %1 := [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]] For example, C(100000), mudolo , 25, equals , 9 The congruence classes mod, 25, in the following set , {2, 4, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 22, 24}, never show up! Theorem Number, 17, : Let C(n) be the constant term, in x, of n (1/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 32, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], A[11], A[12], A[13], A[14]], [13 A[2], A[15], A[16], A[17], A[18]], [13 A[2], A[19], A[12], A[21], A[22]], [21 A[2], A[23], A[8], A[25], A[26]], [8 A[2], 20 A[2] + 20 A[3] + 3 A[7], A[27], A[28], A[29]], [ 3 A[2], 15 A[2] + 15 A[3] + 3 A[7], 5 A[3] + 5 A[4] + 3 A[8], A[30], A[31]] , [23 A[2], 20 A[2] + 5 A[3] + 3 A[7], 15 A[3] + 15 A[4] + 3 A[8], 10 A[4] + 3 A[9], A[14]], [16 A[2], 5 A[3] + A[7], 20 A[4] + A[8], 10 A[5] + A[9], A[10]], [14 A[2], 20 A[2] + 20 A[3] + 3 A[7] + 2 A[11], 20 A[2] + 20 A[3] + 3 A[7] + 4 A[8] + 4 A[11], 5 A[4] + 20 A[5] + 4 A[9], 15 A[5] + 15 A[6] + 4 A[10]], [19 A[2], 10 A[2] + 10 A[3] + 3 A[11], 20 A[2] + 20 A[3] + 10 A[4] + 3 A[7] + 4 A[11] + 3 A[12], 5 A[2] + 10 A[3] + 10 A[4] + 2 A[7] + A[8] + 4 A[9] + A[11] + 3 A[12], 5 A[5] + 20 A[6] + 4 A[10]], [9 A[2], 10 A[3] + 3 A[11], 5 A[2] + 10 A[3] + 10 A[4] + 2 A[7] + A[8] + A[11] + A[12], 15 A[2] + 10 A[3] + 20 A[4] + 20 A[5] + A[7] + 3 A[8] + 4 A[9] + 3 A[11] + 4 A[12] , 5 A[2] + 10 A[3] + 10 A[4] + 20 A[5] + 2 A[7] + A[8] + 3 A[9] + 4 A[10] + A[11] + 3 A[12] + 4 A[13]], [3 A[2] + 10 A[3] + 10 A[4] + 20 A[5] + 20 A[6] + 2 A[7] + A[8] + 3 A[9] + 4 A[10] + A[11] + 3 A[12] + 4 A[13] + 2 A[14], 15 A[3] + 3 A[7], 15 A[4] + 20 A[5] + 10 A[6] + 3 A[8] + 4 A[9] + 2 A[10] + 2 A[13] + A[14], 15 A[5] + 10 A[6] + 3 A[9] + A[10] + 3 A[14], 3 A[10]], [4 A[2] + 10 A[3] + 20 A[4] + 20 A[5] + 10 A[6] + A[7] + 3 A[8] + 4 A[9] + 2 A[10] + 3 A[11] + 4 A[12] + 2 A[13] + A[14], 10 A[3] + 10 A[4] + 20 A[5] + 20 A[6] + A[8] + 3 A[9] + 4 A[10] + 3 A[11] + 3 A[12] + 4 A[13] + 2 A[14], 10 A[4] + 20 A[5] + 10 A[6] + 4 A[9] + 2 A[10] + 3 A[12] + 2 A[13] + A[14], 10 A[5] + 20 A[6] + 3 A[10] + 3 A[13] + 4 A[14], 5 A[6] + 3 A[14]], [ 4 A[2] + 20 A[3] + 20 A[4] + 4 A[7] + 3 A[8] + 2 A[11] + 2 A[12] + 2 A[16], 10 A[3] + 20 A[4] + A[7] + 2 A[8] + A[11] + 3 A[12] + 3 A[16], 5 A[4] + 4 A[8] + 4 A[12] + A[16], 10 A[4] + 10 A[5] + A[8] + A[12] + 3 A[13] + 2 A[16], 5 A[4] + 10 A[5] + 20 A[6] + 4 A[8] + A[9] + 4 A[10] + 4 A[12] + 3 A[13] + 3 A[16]], [ 4 A[2] + 20 A[3] + 20 A[4] + 4 A[7] + 3 A[8] + 2 A[11] + 2 A[12] + 2 A[16], 10 A[3] + 20 A[4] + 2 A[8] + 3 A[11] + 3 A[12] + 3 A[16], 5 A[4] + 4 A[8] + 2 A[12] + 3 A[16], 5 A[4] + 10 A[5] + 4 A[8] + 4 A[12] + 3 A[16] + 3 A[17], 20 A[4] + 20 A[5] + 10 A[6] + 2 A[8] + 4 A[9] + 4 A[10] + 2 A[12] + 4 A[16] + 2 A[17]], [ 3 A[2] + 10 A[3] + 20 A[4] + 20 A[5] + 2 A[7] + 3 A[8] + 3 A[9] + 4 A[10] + A[11] + 4 A[16] + 4 A[17] + 2 A[18], 15 A[3] + 3 A[7], 15 A[4] + 10 A[5] + 5 A[6] + 3 A[8] + 2 A[9] + A[10] + A[17] + 3 A[18], 15 A[5] + 5 A[6] + 3 A[9] + 3 A[10] + 4 A[18], 3 A[10]], [4 A[2], 10 A[3] + 20 A[4] + 20 A[5] + 3 A[8] + 3 A[9] + 4 A[10] + 4 A[16] + 4 A[17] + 2 A[18] + 3 A[19], 20 A[4] + 2 A[8] + 4 A[16], 20 A[5] + 2 A[9] + 2 A[10] + 4 A[17] + A[18], 15 A[6] + 2 A[10] + 4 A[18]], [4 A[2] + 20 A[3] + 20 A[4] + 4 A[7] + A[8] + 4 A[16] + 2 A[19] + 4 A[20], 10 A[3] + 20 A[4] + A[7] + A[8] + 4 A[16] + A[19] + 4 A[20], 5 A[4] + 4 A[8] + A[16] + 4 A[20], 10 A[4] + 20 A[5] + A[8] + 2 A[9] + 2 A[16] + 4 A[17] + A[20], 5 A[6] + 3 A[18]], [ 4 A[2] + 20 A[3] + 20 A[4] + 4 A[7] + A[8] + 4 A[16] + 2 A[19] + 4 A[20], 10 A[3] + 20 A[4] + 4 A[8] + A[16] + 3 A[19] + A[20], 5 A[4] + 4 A[8] + 3 A[16] + 2 A[20], 5 A[4] + 10 A[5] + 4 A[8] + 3 A[16] + 3 A[17] + 4 A[20], 20 A[4] + 10 A[5] + 20 A[6] + 2 A[8] + 4 A[16] + 4 A[17] + 3 A[18] + 2 A[20] + A[21]], [%1, 15 A[3] + 10 A[4] + 10 A[5] + 3 A[7] + A[16] + 4 A[17] + 3 A[18] + 4 A[20] + A[21] + 2 A[22], 15 A[4] + 4 A[16] + 2 A[20], 15 A[5] + 4 A[17] + 3 A[18] + 2 A[21] + 2 A[22], 20 A[6] + 4 A[18] + 2 A[22]], [%1, 5 A[3] + A[19], 5 A[4] + 10 A[5] + 15 A[6] + 3 A[17] + A[18] + A[20] + 2 A[21] + 4 A[22], 5 A[5] + 20 A[6] + 2 A[18] + A[21] + 3 A[22], 20 A[6] + A[22]], [3 A[2], 15 A[3] + 10 A[4] + 10 A[5] + 15 A[6] + 2 A[7] + 2 A[16] + 3 A[17] + A[18] + 2 A[19] + 3 A[20] + 2 A[21] + 4 A[22], 5 A[4] + 10 A[5] + 20 A[6] + A[16] + A[17] + 2 A[18] + 4 A[21] + 3 A[22], 5 A[5] + 20 A[6] + A[17] + 2 A[18] + 3 A[22], 15 A[6] + A[18]], [3 A[2] + 20 A[3] + 10 A[4] + 10 A[5] + 15 A[6] + 4 A[7] + 2 A[16] + 3 A[17] + A[18] + 2 A[19] + 3 A[20] + 2 A[21] + 4 A[22], 5 A[3] + A[19], 5 A[4] + A[20], A[21] + 5 A[5], 5 A[6] + A[22]], [10 A[4] + 10 A[3] + A[2] + A[21] + 2 A[7] + 2 A[22] + A[16] + 4 A[17] + A[19] + 10 A[5] + 3 A[18] + 4 A[20], 10 A[4] + 5 A[3] + 15 A[6] + 2 A[21] + A[7] + 4 A[22] + 2 A[16] + 3 A[17] + 10 A[5] + A[18] + 3 A[20], 15 A[4] + 3 A[16] + 4 A[20], 15 A[6] + 4 A[21] + 4 A[22] + 3 A[17] + 15 A[5] + A[18], 4 A[22] + 3 A[18]] , [10 A[4] + 20 A[3] + 4 A[2] + 20 A[6] + 4 A[21] + 3 A[7] + 3 A[22] + 4 A[16] + A[17] + 4 A[19] + 10 A[5] + 2 A[18] + A[20], 10 A[4] + 10 A[3] + 5 A[6] + 3 A[21] + A[7] + A[22] + 3 A[16] + 2 A[17] + A[19] + 10 A[5] + 4 A[18] + 2 A[20], 10 A[4] + 20 A[6] + 4 A[21] + 3 A[22] + 3 A[16] + A[17] + 10 A[5] + 2 A[18] , 20 A[6] + 3 A[22] + 3 A[17] + 10 A[5] + 2 A[18], 20 A[6] + 3 A[18]], [ 10 A[4] + 10 A[3] + 4 A[2] + A[21] + 2 A[7] + 2 A[22] + A[16] + 4 A[17] + A[19] + 10 A[5] + 3 A[18] + 4 A[20], 10 A[3] + 3 A[19], 10 A[4] + 3 A[20], 3 A[21] + 10 A[5], 10 A[6] + 3 A[22]], [3 A[2], 10 A[4] + 15 A[3] + 15 A[6] + 2 A[21] + 3 A[7] + 4 A[22] + 2 A[16] + 3 A[17] + 10 A[5] + A[18] + 3 A[20], 15 A[4] + A[21] + 2 A[22] + 4 A[16] + 4 A[17] + 10 A[5] + 3 A[18] + 2 A[20] , 15 A[6] + 2 A[21] + 4 A[22] + 4 A[17] + 15 A[5] + A[18], 10 A[6] + 2 A[22] + 4 A[18]], [10 A[4] + 10 A[3] + 4 A[2] + 5 A[6] + 3 A[21] + A[7] + A[22] + 3 A[16] + 2 A[17] + 3 A[19] + 10 A[5] + 4 A[18] + 2 A[20], 10 A[4] + 10 A[3] + A[21] + 2 A[22] + A[16] + 4 A[17] + 3 A[19] + 10 A[5] + 3 A[18] + 4 A[20], 10 A[4] + A[21] + 2 A[22] + 4 A[17] + 10 A[5] + 3 A[18] + 3 A[20], 3 A[21] + 2 A[22] + 10 A[5] + 3 A[18], 3 A[22]], [%1, 15 A[3] + 10 A[4] + 10 A[5] + 3 A[7] + A[16] + 4 A[17] + 3 A[18] + 4 A[20] + A[21] + 2 A[22] , 15 A[4] + 4 A[16] + 2 A[20], 15 A[5] + 4 A[17] + 3 A[18] + 2 A[21] + 2 A[22], 20 A[6] + 4 A[18] + 2 A[22]], [10 A[4] + 10 A[3] + 3 A[2] + A[21] + 2 A[7] + 2 A[22] + A[16] + 4 A[17] + A[19] + 10 A[5] + 3 A[18] + 4 A[20], 15 A[3] + 3 A[7], 15 A[4] + 20 A[6] + 4 A[21] + 3 A[22] + 4 A[16] + A[17] + 10 A[5] + 2 A[18] + 2 A[20], 2 A[21] + 4 A[17] + 15 A[5], 5 A[6] + 2 A[22] + 4 A[18]]], [1, 1, 3, 13, 13, 21, 8, 3, 23, 16, 14, 19, 9, 23, 14, 19, 19, 23, 4, 19, 19, 13, 13, 3, 18, 21, 9, 24, 3, 14, 13, 23]] %1 := 3 A[2] + 10 A[3] + 10 A[4] + 10 A[5] + 5 A[6] + A[7] + 3 A[16] + 2 A[17] + 4 A[18] + 3 A[19] + 2 A[20] + 3 A[21] + A[22] For example, C(100000), mudolo , 25, equals , 17 The congruence classes mod, 25, in the following set , {0, 2, 5, 6, 7, 10, 11, 12, 15, 17, 20, 22}, never show up! Theorem Number, 18, : Let C(n) be the constant term, in x, of n (1/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 27, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], A[11], A[12], A[13], A[14]], [15 A[2], 20 A[2] + 10 A[3], 15 A[3] + 5 A[4], 10 A[4], A[15]], [6 A[2], A[16], A[17], A[18], A[19]], [9 A[2], A[20], A[21], A[22], A[23]], [23 A[2], 10 A[2] + 10 A[3] + 3 A[7], A[24], A[25], A[26]], [15 A[2], 20 A[2] + 10 A[3], 15 A[3] + 5 A[4], 10 A[4], 5 A[5] + 20 A[6]], [A[2], 20 A[2] + 10 A[3] + A[7], 15 A[3] + 20 A[4] + A[8], 10 A[4] + 5 A[5] + A[9], A[27]], [24 A[2], 20 A[3] + 4 A[7], 20 A[4] + 4 A[8], 20 A[5] + 4 A[9], 20 A[6] + 4 A[10]], [24 A[2], 15 A[2] + 20 A[3] + 2 A[7] + 4 A[11], 5 A[2] + 20 A[3] + 10 A[4] + 4 A[7] + 4 A[8] + 2 A[11], 15 A[4] + 5 A[5] + 4 A[9], 20 A[5] + 4 A[10]], [5 A[2], 5 A[2] + 10 A[3] + A[7] + 3 A[11], 10 A[2] + 20 A[3] + 3 A[7] + 4 A[11], 10 A[2] + 20 A[3] + 20 A[4] + 10 A[5] + 3 A[7] + 3 A[8] + 4 A[11] + 4 A[12] , 15 A[5] + 20 A[6]], [23 A[2], 15 A[2] + 15 A[3] + 2 A[7] + 2 A[11], 15 A[2] + 10 A[3] + 15 A[4] + 2 A[7] + 2 A[8] + A[11] + 2 A[12], 15 A[2] + 10 A[3] + 10 A[4] + 15 A[5] + 2 A[7] + 2 A[8] + 2 A[9] + A[11] + A[12] + 2 A[13], 10 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 15 A[6] + 3 A[7] + 3 A[8] + 3 A[9] + 3 A[10] + 4 A[11] + 4 A[12] + 4 A[13]], [2 A[2], 10 A[3] + 20 A[4] + 20 A[5] + 20 A[6] + 2 A[7] + 4 A[8] + 4 A[9] + 4 A[10] + 2 A[12] + 2 A[13] + 2 A[14], 10 A[4] + 10 A[5] + 10 A[6] + 2 A[8] + A[9] + A[10] + 3 A[13] + 3 A[14], 10 A[5] + 10 A[6] + 2 A[9] + 2 A[10] + A[14], 2 A[10]], [10 A[3] + 10 A[4] + 10 A[5] + 10 A[6] + 2 A[7] + 2 A[8] + 2 A[9] + 2 A[10] + A[11] + A[12] + A[13] + A[14], 10 A[4] + 10 A[5] + 10 A[6] + A[8] + A[9] + A[10] + 3 A[12] + 3 A[13] + 3 A[14], 20 A[5] + 20 A[6] + 4 A[9] + 4 A[10] + 2 A[13] + 2 A[14], 20 A[6] + 3 A[10] + 4 A[14], 10 A[6]], [ 3 A[2] + 5 A[3] + 2 A[7] + 3 A[11] + 4 A[16], 15 A[3] + 4 A[11] + A[16], 20 A[3] + 15 A[4] + 3 A[8] + 2 A[11] + 4 A[16], 20 A[3] + 10 A[4] + 15 A[5] + A[8] + A[9] + 4 A[11] + 3 A[12] + 4 A[13] + 3 A[16], 20 A[3] + 20 A[4] + 20 A[5] + 20 A[6] + 3 A[8] + 3 A[9] + 3 A[10] + 2 A[11] + 4 A[12] + 4 A[13] + 4 A[16]], [ 20 A[3] + 20 A[4] + 2 A[7] + 3 A[8] + 3 A[16] + 2 A[17], 20 A[3] + 20 A[4] + 4 A[7] + 4 A[8] + A[16] + A[17], 20 A[4] + 4 A[8] + A[17], 20 A[5] + 4 A[9] + 2 A[13], 10 A[4] + 10 A[5] + 15 A[6] + A[9] + 3 A[12] + 3 A[13] + A[17]], [ A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 3 A[8] + 3 A[16] + 2 A[17], 5 A[3] + A[16], 5 A[4] + A[17], 5 A[5] + A[18], 10 A[4] + 20 A[5] + 2 A[9] + A[10] + A[12] + 2 A[17] + 3 A[18]], [ 4 A[2] + 20 A[3] + 20 A[4] + A[7] + 4 A[8] + 4 A[16] + A[17], 20 A[3] + 20 A[4] + 4 A[7] + A[8] + 4 A[17], 20 A[4] + 20 A[5] + 10 A[6] + 4 A[8] + 4 A[9] + 2 A[10] + A[18] + 3 A[19], 20 A[5] + 5 A[6] + 4 A[9] + 3 A[10] + 2 A[19], 5 A[6] + 4 A[10]], [ 2 A[2] + 2 A[7] + 4 A[16] + A[20], 10 A[3] + 2 A[16], 10 A[4] + 20 A[5] + 5 A[6] + A[9] + 3 A[10] + 2 A[17] + 4 A[18] + 2 A[19], 10 A[5] + 4 A[10] + 2 A[18] + A[19], 5 A[6] + 2 A[19]], [ 5 A[3] + 4 A[7] + 3 A[16] + 2 A[20], 5 A[3] + 3 A[7] + 4 A[16] + 2 A[20], 5 A[3] + 15 A[4] + A[7] + 3 A[17] + A[20] + 3 A[21], 20 A[3] + 5 A[4] + 4 A[7] + A[17] + 4 A[20] + A[21], 15 A[3] + 10 A[4] + 20 A[5] + 5 A[6] + 3 A[7] + 3 A[9] + 2 A[17] + 2 A[18] + 3 A[20] + 2 A[21]], [4 A[2] + 2 A[7] + 4 A[16] + A[20], A[7] + 4 A[16] + A[20], 20 A[3] + 5 A[4] + 4 A[7] + 4 A[20] + A[21], 5 A[3] + 20 A[4] + 20 A[5] + A[7] + A[9] + 4 A[17] + 3 A[18] + A[20] + 4 A[21], 10 A[6] + 4 A[19]], [ A[2] + 15 A[3] + A[7] + 2 A[16] + 3 A[20], 5 A[3] + 4 A[20], 5 A[4] + 4 A[21], 20 A[3] + 5 A[4] + 5 A[5] + 4 A[7] + A[17] + A[18] + 4 A[20] + A[21], 5 A[6] + 4 A[23]], [2 A[7] + 4 A[16] + A[20], 15 A[3] + 3 A[16] + 3 A[20], 15 A[3] + 3 A[7] + 3 A[20], 15 A[3] + 10 A[4] + 15 A[6] + 3 A[7] + 2 A[17] + 2 A[19] + 3 A[20] + 2 A[21] + 2 A[23], 5 A[6] + 3 A[19] + 3 A[23]], [ 3 A[2] + 15 A[3] + A[7] + 2 A[16] + 3 A[20], 10 A[3] + 4 A[7] + 3 A[16] + 4 A[20], 5 A[3] + 15 A[4] + A[7] + 2 A[17] + A[20] + 4 A[21], 15 A[5] + 15 A[6] + 3 A[18] + 2 A[19] + 2 A[23], 3 A[19]], [ 2 A[2] + 2 A[7] + 4 A[16] + A[20], 5 A[3] + 3 A[20], 5 A[4] + 3 A[21], 10 A[4] + 15 A[3] + 2 A[21] + 3 A[7] + 2 A[17] + 10 A[5] + 2 A[18] + 3 A[20], 5 A[6] + 3 A[23]], [ 5 A[3] + 4 A[2] + 4 A[7] + 3 A[16] + 2 A[20], 5 A[3] + A[20], 5 A[4] + A[21], 20 A[4] + 5 A[3] + 4 A[21] + A[7] + 4 A[17] + 20 A[5] + 4 A[18] + A[20], 5 A[6] + A[23]]], [1, 1, 3, 15, 6, 9, 23, 15, 1, 24, 24, 5, 23, 2, 10, 13, 10, 11, 9, 7, 10, 9, 16, 5, 18, 7, 14]] For example, C(100000), mudolo , 25, equals , 0 The congruence classes mod, 25, in the following set , {0, 4, 8, 12, 17, 19, 20, 21, 22}, never show up! Theorem Number, 19, : Let C(n) be the constant term, in x, of n (2/x + x) For the record, the first 15 terms of the sequence are: 0, 4, 0, 24, 0, 160, 0, 1120, 0, 8064, 0, 59136, 0, 439296, 0 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 22, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [0, 10 A[2], 20 A[3], A[11], A[12]], [4 A[2], A[13], A[14], A[15], A[16]], [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]], [24 A[2], A[17], A[18], A[15], A[20]], [0, 10 A[2], 20 A[3], 5 A[4], 15 A[5]], [14 A[2], 20 A[3] + 4 A[7], 5 A[4] + 4 A[8], A[21], A[20]], [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]], [19 A[2], 10 A[3] + 4 A[7], 5 A[4] + 4 A[8], 4 A[9], 20 A[6] + 4 A[10]], [20 A[2], 20 A[3], 4 A[11], 20 A[5], 20 A[6]], 0, [0, 15 A[2], 20 A[3] + 4 A[7] + 4 A[13], 4 A[11], 4 A[12]], [11 A[2], 10 A[3] + 2 A[7] + A[13], 5 A[4] + A[8] + 2 A[11], 5 A[5] + A[9], 20 A[6] + A[10]], [0, 5 A[2], 15 A[3] + 3 A[7] + 3 A[13], 3 A[11], 3 A[12]], [11 A[2], 20 A[3] + 4 A[7] + 3 A[13], 5 A[4] + A[8] + 3 A[11], 5 A[5] + A[9] + 3 A[12], 5 A[6] + 4 A[16]], [0, 15 A[2], 20 A[3] + 4 A[7] + 4 A[13], 4 A[11], 4 A[12]], [21 A[2], 5 A[3] + 4 A[13], 10 A[4] + 2 A[8] + A[18], 5 A[5] + A[9] + 4 A[12], 5 A[6] + 4 A[16]], [0, 5 A[2], 15 A[3] + 3 A[7] + 3 A[13], 10 A[4] + 2 A[8] + 2 A[18], 15 A[5] + 3 A[9] + 3 A[19]], [ A[2], 5 A[3] + A[7], 5 A[4] + 4 A[18], 5 A[5] + 4 A[19], 20 A[6] + 4 A[16]] , [0, 5 A[2], 15 A[3] + 3 A[7] + 3 A[13], 10 A[4] + 2 A[8] + 2 A[18], 5 A[5] + 3 A[19] + 2 A[21]], [ A[2], 5 A[3] + A[7], 5 A[4] + 4 A[18], 5 A[5] + 4 A[19], 20 A[6] + 4 A[16]] ], [1, 1, 0, 4, 0, 24, 0, 14, 0, 19, 20, 0, 0, 11, 0, 11, 0, 21, 0, 1, 0, 1]] For example, C(100000), mudolo , 25, equals , 15 The congruence classes mod, 25, in the following set , {2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 22, 23}, never show up! Theorem Number, 20, : Let C(n) be the constant term, in x, of n (2/x + 3 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 22, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [0, 20 A[2], 15 A[3], A[11], A[12]], [12 A[2], A[13], A[14], A[15], A[16]], %1, [16 A[2], A[17], A[8], A[19], A[20]], %1, [2 A[2], 5 A[3] + 2 A[7], 10 A[4] + 2 A[8], A[21], A[22]], %1, [6 A[2], 10 A[3] + A[7], 15 A[4] + A[8], 20 A[5] + A[9], A[10]], [20 A[2], 20 A[3], 2 A[11], 20 A[5], 20 A[6]], 0, [0, 15 A[2], 4 A[7] + 3 A[13], 2 A[11], 2 A[12]], [19 A[2], 3 A[7] + 3 A[13], 20 A[4] + 4 A[8] + 4 A[11], 20 A[5] + 4 A[9], 5 A[6] + 4 A[10]], [0, 15 A[2], 4 A[7] + 3 A[13], 2 A[11], 2 A[12]], [ 17 A[2], 5 A[3] + 4 A[7] + 4 A[13], 10 A[4] + 2 A[8], 10 A[5] + 2 A[9] + 2 A[12], A[16]], [0, 20 A[2], 20 A[3] + 3 A[7] + 2 A[17], A[11], A[12]], [2 A[2], 10 A[3] + A[7] + A[17], 10 A[4] + 2 A[18], 10 A[5] + 2 A[9] + A[12], 15 A[6] + A[16]], [0, 20 A[2], 20 A[3] + 3 A[7] + 2 A[17], A[11], 20 A[5] + 4 A[9] + A[19]], [16 A[2], 4 A[7] + 2 A[17], 5 A[4] + 2 A[11] + A[18], 5 A[5] + A[9], 15 A[6] + 3 A[16]], [0, 15 A[2], 20 A[3] + A[7] + 4 A[17], 2 A[11], 20 A[5] + 3 A[9] + 2 A[19]] , [7 A[2], 5 A[3] + 3 A[7] + 4 A[17], 10 A[4] + 4 A[11] + 2 A[18], 10 A[5] + 2 A[9], 15 A[6] + A[16]]], [1, 1, 0, 12, 0, 16, 0, 2, 0, 6, 20, 0, 0, 19, 0, 17, 0, 2, 0, 16, 0, 7]] %1 := [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]] For example, C(100000), mudolo , 25, equals , 15 The congruence classes mod, 25, in the following set , {3, 4, 5, 8, 9, 10, 11, 13, 14, 15, 18, 21, 22, 23, 24}, never show up! Theorem Number, 21, : Let C(n) be the constant term, in x, of n (2/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], %4, A[11], A[12], A[13]], [A[2], 15 A[2] + 11 A[3], 5 A[3] + 21 A[4], A[14], A[15]], [A[2], %4, %3, %2, A[13]], %1, %5, [A[2], 15 A[2] + 11 A[3], 5 A[3] + 21 A[4], 20 A[4] + 6 A[5], 10 A[5] + 16 A[6]], %5, %1, [A[2], 15 A[2] + 11 A[3], 5 A[3] + 21 A[4], 20 A[4] + 6 A[5], 10 A[5] + 16 A[6]], %5, %1, %5, %1, %1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] %1 := [A[2], 10 A[2] + 16 A[3], 20 A[3] + 6 A[4], 5 A[4] + 21 A[5], 15 A[5] + 11 A[6]] %2 := 15 A[4] + 11 A[5] %3 := 10 A[3] + 16 A[4] %4 := 5 A[2] + 21 A[3] %5 := [A[2], %4, %3, %2, 20 A[5] + 6 A[6]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, never show up! Theorem Number, 22, : Let C(n) be the constant term, in x, of n (2/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 28, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], A[11], A[12], A[13], A[14]], [5 A[2], 10 A[2] + 10 A[3], 20 A[3] + 15 A[4], A[15], A[16]], [13 A[2], A[17], A[18], A[19], A[20]], [24 A[2], A[21], A[22], A[23], A[24]], [11 A[2], 15 A[2] + 15 A[3] + A[7], A[25], A[26], A[27]], [10 A[2], 10 A[2] + 15 A[3], 20 A[3] + 20 A[4], 5 A[4], 15 A[5] + 5 A[6]], [18 A[2], 20 A[2] + 20 A[3] + 3 A[7], 15 A[3] + 3 A[8], 10 A[4] + 5 A[5] + 3 A[9], A[28]], [24 A[2], 20 A[3] + 4 A[7], 20 A[4] + 4 A[8], 20 A[5] + 4 A[9], 20 A[6] + 4 A[10]], [6 A[2], 5 A[3] + A[11], 10 A[2] + 20 A[3] + 15 A[4] + 4 A[7] + A[8] + A[11], 20 A[4] + 20 A[5] + A[9], 10 A[5] + A[10]], [15 A[2], 20 A[3] + A[7] + 4 A[11], 15 A[2] + 20 A[3] + A[7] + 4 A[11], 20 A[2] + 20 A[3] + 20 A[4] + 5 A[5] + 3 A[7] + 2 A[8] + 2 A[11] + 3 A[12], 15 A[5] + 10 A[6]], [13 A[2], 20 A[2] + 15 A[3] + 3 A[7], 15 A[4] + 3 A[12], 20 A[2] + 20 A[3] + 20 A[4] + 10 A[5] + 3 A[7] + 2 A[8] + 4 A[9] + 2 A[11] + 3 A[12] + 4 A[13], 15 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 5 A[6] + A[7] + 4 A[8] + 3 A[9] + 3 A[10] + 4 A[11] + A[12] + 2 A[13]], [4 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 3 A[7] + 2 A[8] + 4 A[9] + 4 A[10] + 2 A[11] + 3 A[12] + A[13] + A[14], 20 A[3] + 20 A[4] + 20 A[5] + 15 A[6] + 4 A[7] + 3 A[8] + A[9] + A[10] + 2 A[12] + 4 A[13] + 4 A[14], 20 A[4] + 20 A[5] + 10 A[6] + 4 A[8] + 2 A[9] + 2 A[10] + 3 A[13] + 3 A[14] , 20 A[5] + 4 A[9] + 4 A[10] + A[14], 5 A[6] + 4 A[10]], [20 A[3] + 20 A[4] + 20 A[5] + 5 A[6] + A[7] + 4 A[8] + 3 A[9] + 3 A[10] + 4 A[11] + A[12] + 2 A[13] + 2 A[14], 20 A[4] + 20 A[5] + 15 A[6] + 3 A[8] + A[9] + A[10] + 2 A[12] + 4 A[13] + 4 A[14], 20 A[5] + 10 A[6] + 2 A[9] + 2 A[10] + 3 A[13] + 3 A[14], 4 A[10] + A[14], 10 A[6]], [20 A[3] + 20 A[4] + 20 A[5] + 2 A[7] + 3 A[8] + A[9] + 3 A[11] + 2 A[12] + 4 A[13] + A[16], 20 A[4] + 20 A[5] + A[8] + 2 A[9] + 4 A[12] + 3 A[13] + 2 A[16], 20 A[5] + 4 A[9] + A[13] + 4 A[16], 3 A[16], 20 A[6]], [ 3 A[2] + 5 A[3] + 4 A[7] + 2 A[11] + 3 A[17], 15 A[3] + 3 A[11], 15 A[4] + 3 A[12], 10 A[3] + 20 A[4] + 15 A[5] + 4 A[8] + 3 A[9] + A[11] + A[12] + 3 A[16] + 3 A[17], 3 A[10] + 2 A[16]], [ 20 A[3] + 2 A[7] + A[11] + 4 A[17], 20 A[3] + 3 A[7] + 2 A[11], 20 A[3] + 3 A[11] + 4 A[17], 10 A[3] + 20 A[4] + 3 A[8] + 2 A[11] + A[16] + A[17] + 4 A[18], 5 A[6] + 3 A[16]], [4 A[2], 10 A[3] + 20 A[4] + 10 A[5] + 4 A[8] + A[9] + 2 A[16] + 3 A[17] + 2 A[18] + 3 A[19], 10 A[4] + 10 A[5] + A[9] + 2 A[16] + 3 A[18] + 3 A[19], 10 A[5] + 2 A[16] + 3 A[19], 10 A[6] + 4 A[10] + A[16]], [2 A[2] + 20 A[3] + 20 A[4] + 10 A[5] + 3 A[7] + 4 A[8] + A[9] + 2 A[16] + 4 A[17] + 2 A[18] + 3 A[19], 10 A[3] + 10 A[4] + 20 A[5] + 2 A[7] + A[8] + 4 A[9] + 3 A[16] + 3 A[18] + 2 A[19], 10 A[4] + 20 A[5] + 2 A[8] + 4 A[9] + 3 A[16] + 2 A[19], 10 A[5] + 2 A[9] + 3 A[16], 2 A[16] + 4 A[20]], [ 4 A[2] + 20 A[3] + 3 A[7] + 3 A[17] + 2 A[21], 5 A[3] + 4 A[7] + 3 A[17] + 4 A[21], 10 A[4] + 3 A[18], 10 A[5] + 2 A[16] + 3 A[19], 20 A[6] + 3 A[20]], [ 15 A[3] + 2 A[7] + 2 A[17] + 3 A[21], 20 A[3] + 4 A[7] + 2 A[17], 10 A[3] + 10 A[4] + 2 A[7] + A[8] + 3 A[18] + 2 A[21], 20 A[3] + 20 A[4] + 4 A[7] + 4 A[8] + 2 A[18] + 4 A[21], 20 A[6] + 4 A[16]] , [2 A[2] + 4 A[7] + 4 A[17] + A[21], 5 A[3] + 4 A[7] + 4 A[17] + 4 A[21], 10 A[4] + 5 A[5] + A[16] + 4 A[18] + 2 A[19] + A[23], 10 A[5] + 3 A[16] + 4 A[19], 5 A[6] + 4 A[20]], [A[2], 5 A[3] + 4 A[21], 20 A[3] + 5 A[4] + 4 A[7] + 2 A[18] + 4 A[21], 5 A[5] + 4 A[23], 10 A[6] + A[16] + 2 A[20]], [15 A[3] + 2 A[7] + 2 A[17] + 3 A[21], 15 A[3] + 2 A[7] + 3 A[17] + A[21], 20 A[3] + 4 A[7] + 4 A[21], 10 A[5] + A[16] + 3 A[19] + 4 A[23], 10 A[6] + A[16]], [ 15 A[3] + 3 A[2] + 3 A[21] + 2 A[7] + 2 A[17], 5 A[3] + A[17], 5 A[4] + A[18], A[19] + 5 A[5], 5 A[6] + A[20]], [ 10 A[3] + 4 A[2] + 4 A[21] + A[7] + A[17], 20 A[3] + 4 A[7], 10 A[4] + 5 A[3] + A[21] + A[7] + A[16] + A[23] + 2 A[19] + 5 A[5] + 3 A[18], 3 A[16] + A[23] + 5 A[5], 4 A[16] + 3 A[20]], [2 A[2], 5 A[3] + 3 A[21], 10 A[4] + 15 A[3] + 3 A[21] + 3 A[7] + 4 A[18], 3 A[23] + 5 A[5], 20 A[6] + 2 A[16] + 4 A[20]]], [1, 1, 1, 5, 13, 24, 11, 10, 18, 24, 6, 15, 13, 9, 10, 20, 18, 20, 4, 7, 14, 15, 7, 1, 15, 18, 24, 2]] For example, C(100000), mudolo , 25, equals , 0 The congruence classes mod, 25, in the following set , {0, 3, 8, 12, 16, 17, 19, 21, 22, 23}, never show up! Theorem Number, 23, : Let C(n) be the constant term, in x, of n (2/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], 20 A[2] + 21 A[3], A[11], A[12], A[13]], [9 A[2], 5 A[2] + 9 A[3], 10 A[3] + 9 A[4], A[14], A[15]], [0, 5 A[3], 10 A[4], 15 A[5], A[16]], %1, [11 A[2], 20 A[2] + 6 A[3], 15 A[3] + A[4], 10 A[4] + 21 A[5], 5 A[5] + 16 A[6]], [19 A[2], 5 A[2] + 19 A[3], 10 A[3] + 19 A[4], 15 A[4] + 19 A[5], 20 A[5] + 19 A[6]], [15 A[2], 20 A[3], 0, 5 A[5], 10 A[6]], %1, [9 A[2], 5 A[2] + 9 A[3], 10 A[3] + 9 A[4], 15 A[4] + 9 A[5], 20 A[5] + 9 A[6]], [15 A[2], 20 A[3], 0, 5 A[5], 10 A[6]], %1, [10 A[2], 5 A[3], 0, 20 A[5], 15 A[6]], [5 A[2], 15 A[3], 0, 10 A[5], 20 A[6]], 0], [1, 1, 1, 9, 0, 20, 11, 19, 15, 20, 9, 15, 20, 10, 5, 0]] %1 := [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]] For example, C(100000), mudolo , 25, equals , 9 The congruence classes mod, 25, in the following set , {2, 3, 4, 6, 7, 8, 12, 13, 14, 16, 17, 18, 21, 22, 23, 24}, never show up! Theorem Number, 24, : Let C(n) be the constant term, in x, of n (2/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 32, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], A[11], A[12], A[13], A[14]], [13 A[2], A[15], A[16], A[17], A[18]], [12 A[2], A[19], A[20], A[21], A[22]], [14 A[2], A[23], A[24], A[25], A[26]], [A[2], 10 A[2] + 5 A[3] + A[7], A[12], A[13], A[14]], [3 A[2], 15 A[2] + 3 A[7], 5 A[3] + 3 A[8], A[30], A[31]], [7 A[2], 20 A[2] + 10 A[3] + 2 A[7], 15 A[3] + 15 A[4] + 2 A[8], 10 A[4] + 20 A[5] + 2 A[9], A[32]], [19 A[2], 10 A[3] + 4 A[7], 5 A[4] + 4 A[8], 4 A[9], 20 A[6] + 4 A[10]], [ 16 A[2], 5 A[2] + 3 A[7] + 3 A[11], 10 A[2] + 20 A[3] + A[7] + A[8] + 4 A[11], 5 A[4] + 5 A[5] + A[9], 15 A[5] + 10 A[6] + A[10]], [3 A[2], 20 A[2] + 15 A[3] + A[7] + 2 A[11], 5 A[2] + 20 A[3] + 15 A[4] + 3 A[7] + 2 A[8] + 2 A[11] + A[12], 5 A[2] + 20 A[3] + 20 A[4] + 3 A[7] + 3 A[8] + 3 A[9] + 2 A[11] + 2 A[12], 10 A[5] + 3 A[10]], [2 A[2], 5 A[2] + 5 A[3] + 3 A[7] + 4 A[11], 20 A[2] + 20 A[3] + 10 A[4] + 2 A[7] + 2 A[8] + 3 A[11], 10 A[5] + 2 A[13], 10 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 20 A[6] + A[7] + A[8] + 3 A[9] + 2 A[10] + 4 A[11] + 4 A[12] + 2 A[13]], [4 A[2], 20 A[3] + 4 A[7], 20 A[4] + 20 A[5] + 4 A[8] + A[9] + 4 A[10] + 4 A[13] + A[14], 20 A[5] + 4 A[9] + 4 A[10] + A[14], 5 A[6] + 4 A[10]], [3 A[2], 15 A[3] + 20 A[4] + 20 A[5] + 2 A[8] + A[9] + 4 A[10] + 3 A[11] + 3 A[12] + 4 A[13] + A[14], 15 A[4] + 20 A[5] + 10 A[6] + 3 A[9] + 2 A[10] + 3 A[12] + 2 A[13] + 3 A[14], 15 A[5] + 15 A[6] + A[10] + 3 A[13] + 4 A[14], 3 A[14]], [ 4 A[2] + 20 A[3] + 10 A[4] + A[7] + 2 A[8] + 4 A[11] + A[16], 20 A[3] + 20 A[4] + 2 A[7] + 4 A[8] + 2 A[11] + 2 A[16], 20 A[4] + 2 A[8] + 2 A[12], 20 A[4] + 20 A[5] + 2 A[8] + A[9] + 2 A[12] + 3 A[13] + 2 A[16], 15 A[4] + 20 A[5] + 15 A[6] + 4 A[8] + 3 A[9] + 4 A[10] + 4 A[12] + 2 A[13] + 4 A[16]], [A[2], 5 A[3] + 10 A[4] + 2 A[8] + A[11] + A[16], 4 A[8] + 4 A[16], 5 A[5] + A[13], 10 A[5] + 10 A[6] + A[9] + A[10] + A[13] + A[17]], [ 2 A[2] + 20 A[3] + 10 A[4] + A[7] + 2 A[8] + 4 A[11] + A[16], 10 A[3] + 10 A[4] + 2 A[7] + A[8] + 3 A[16], 10 A[4] + 2 A[8], 10 A[5] + 10 A[6] + 2 A[9] + 2 A[10] + A[18], 5 A[6] + 2 A[10]], [ 2 A[2] + 20 A[4] + 4 A[7] + 3 A[8] + 4 A[16] + 3 A[19], 5 A[3] + A[19], 20 A[4] + 10 A[5] + 3 A[8] + 2 A[9] + 3 A[16] + A[17], 20 A[5] + 10 A[6] + 3 A[9] + A[10] + 3 A[17] + 3 A[18], 10 A[6] + 3 A[10] + 3 A[18]], [ A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 4 A[8] + 2 A[16] + 4 A[19], 20 A[4] + 3 A[7] + 4 A[8] + 2 A[16] + 4 A[19], 5 A[4] + 2 A[16], 5 A[5] + 2 A[17], 5 A[6] + 2 A[18]], [ 4 A[2] + 10 A[3] + 10 A[4] + A[7] + 2 A[8] + A[16] + 2 A[19], 10 A[3] + 20 A[4] + 4 A[8] + 2 A[16] + 2 A[19], 3 A[8] + 3 A[16] + A[20], 5 A[4] + 10 A[5] + 3 A[8] + 3 A[16] + 3 A[17] + 4 A[20], 5 A[4] + 20 A[5] + 20 A[6] + 4 A[8] + 4 A[16] + 4 A[17] + 3 A[18] + 2 A[20] + 4 A[21]], [ 3 A[2] + 10 A[3] + 20 A[4] + 15 A[5] + 5 A[6] + A[7] + 4 A[16] + 3 A[17] + A[18] + 2 A[19] + 4 A[20] + 3 A[21] + A[22], 15 A[3] + 10 A[4] + 20 A[5] + 15 A[6] + 3 A[7] + 2 A[16] + 4 A[17] + 3 A[18] + 2 A[20] + 4 A[21] + 3 A[22], 10 A[4] + 15 A[5] + 5 A[6] + 2 A[16] + 3 A[17] + A[18] + A[20] + 3 A[21] + A[22], 10 A[5] + 2 A[17] + A[21], 20 A[6] + 2 A[18] + A[22]], [4 A[2] + 20 A[3] + 15 A[4] + 5 A[5] + 10 A[6] + 2 A[7] + 3 A[16] + A[17] + 2 A[18] + 4 A[19] + 3 A[20] + A[21] + 2 A[22], 10 A[3] + 15 A[4] + 5 A[5] + 10 A[6] + 3 A[16] + A[17] + 2 A[18] + 2 A[19] + 3 A[20] + A[21] + 2 A[22], 10 A[4] + 20 A[5] + 15 A[6] + 4 A[17] + 3 A[18] + 2 A[20] + 4 A[21] + 3 A[22], 10 A[5] + 10 A[6] + 2 A[18] + 2 A[21] + 2 A[22], 2 A[22]], [2 A[2] + 5 A[4] + 10 A[5] + 20 A[6] + 4 A[7] + A[16] + 2 A[17] + 4 A[18] + 3 A[19] + A[20] + 2 A[21] + 4 A[22], 15 A[3] + 5 A[4] + 10 A[5] + 20 A[6] + A[7] + A[16] + 2 A[17] + 4 A[18] + 3 A[19] + A[20] + 2 A[21] + 4 A[22], 10 A[4] + 4 A[16], 10 A[5] + 4 A[17], 10 A[6] + 4 A[18]], [3 A[2] + 10 A[3] + 20 A[4] + 15 A[5] + 5 A[6] + A[7] + 4 A[16] + 3 A[17] + A[18] + 2 A[19] + 4 A[20] + 3 A[21] + A[22], 15 A[3] + 20 A[4] + 15 A[5] + 5 A[6] + 4 A[16] + 3 A[17] + A[18] + 4 A[19] + 4 A[20] + 3 A[21] + A[22], 15 A[4] + 5 A[5] + 10 A[6] + A[17] + 2 A[18] + 4 A[20] + A[21] + 2 A[22], 15 A[5] + 10 A[6] + 2 A[18] + 4 A[21] + 2 A[22], 20 A[6] + 4 A[22]], [ 5 A[4] + A[2] + 20 A[6] + 2 A[21] + 4 A[7] + 4 A[22] + A[16] + 2 A[17] + 3 A[19] + 10 A[5] + 4 A[18] + A[20], 15 A[4] + 5 A[3] + 10 A[6] + A[21] + A[7] + 2 A[22] + 3 A[16] + A[17] + 5 A[5] + 2 A[18] + 3 A[20], 15 A[4] + 20 A[6] + 2 A[21] + 4 A[22] + 4 A[16] + 2 A[17] + 10 A[5] + 4 A[18] + 2 A[20], 10 A[6] + 2 A[21] + 2 A[22] + 4 A[17] + 15 A[5] + 2 A[18], 15 A[6] + 2 A[22] + 4 A[18]], [3 A[2], 10 A[4] + 15 A[6] + 4 A[21] + 4 A[7] + 3 A[22] + 2 A[16] + 4 A[17] + 2 A[19] + 20 A[5] + 3 A[18] + 2 A[20], 5 A[4] + 20 A[6] + 2 A[21] + 4 A[22] + A[16] + 2 A[17] + 10 A[5] + 4 A[18], 20 A[6] + 4 A[22] + A[17] + 5 A[5] + 4 A[18], 15 A[6] + A[18]], [2 A[2], 15 A[4] + 5 A[3] + 10 A[6] + A[21] + 2 A[22] + 3 A[16] + A[17] + A[19] + 5 A[5] + 2 A[18] + 3 A[20], 5 A[4] + 20 A[6] + 2 A[21] + 4 A[22] + 2 A[17] + 10 A[5] + 4 A[18] + A[20], 20 A[6] + A[21] + 4 A[22] + 5 A[5] + 4 A[18], 15 A[6] + A[22]], [4 A[2], 20 A[3] + 4 A[7], 15 A[4] + A[16] + 3 A[20], 10 A[6] + 3 A[21] + 2 A[22] + A[17] + 15 A[5] + 2 A[18], 3 A[22] + A[18]], [15 A[4] + 20 A[3] + A[2] + 10 A[6] + A[21] + 2 A[7] + 2 A[22] + 3 A[16] + A[17] + 4 A[19] + 5 A[5] + 2 A[18] + 3 A[20], 15 A[4] + 10 A[3] + 10 A[6] + A[21] + 2 A[22] + 3 A[16] + A[17] + 3 A[19] + 5 A[5] + 2 A[18] + 3 A[20], 10 A[4] + 20 A[6] + 2 A[21] + 4 A[22] + 2 A[17] + 10 A[5] + 4 A[18] + 3 A[20], 20 A[6] + 3 A[21] + 4 A[22] + 10 A[5] + 4 A[18], 20 A[6] + 3 A[22]], [2 A[2], 10 A[3] + 2 A[7], 20 A[4] + 3 A[16] + 4 A[20], 5 A[6] + 4 A[21] + A[22] + 3 A[17] + 20 A[5] + A[18], 4 A[22] + 3 A[18]], [ 10 A[4] + 15 A[3] + 3 A[2] + 15 A[6] + 4 A[21] + 3 A[7] + 3 A[22] + 2 A[16] + 4 A[17] + A[19] + 20 A[5] + 3 A[18] + 2 A[20], 5 A[4] + 15 A[3] + 20 A[6] + 2 A[21] + 3 A[7] + 4 A[22] + A[16] + 2 A[17] + 10 A[5] + 4 A[18] + A[20], 10 A[4] + 15 A[6] + 4 A[21] + 3 A[22] + 2 A[16] + 4 A[17] + 20 A[5] + 3 A[18] + A[20], 10 A[6] + A[21] + 2 A[22] + 2 A[17] + 10 A[5] + 2 A[18], A[22] + 2 A[18]]], [1, 1, 1, 13, 12, 14, 1, 3, 7, 19, 16, 3, 2, 4, 3, 19, 1, 17, 12, 6, 19, 18, 9, 12, 18, 11, 3, 2, 4, 6, 2, 23]] For example, C(100000), mudolo , 25, equals , 13 The congruence classes mod, 25, in the following set , {0, 5, 8, 10, 15, 20, 21, 22, 24}, never show up! Theorem Number, 25, : Let C(n) be the constant term, in x, of n (2/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 32, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], A[11], A[12], A[13], A[14]], [8 A[2], A[15], A[16], A[17], A[18]], [7 A[2], A[19], A[20], A[21], A[14]], [11 A[2], A[7], A[24], A[25], A[26]], [17 A[2], 20 A[2] + 2 A[7], A[20], A[28], A[29]], [ 3 A[2], 10 A[2] + 15 A[3] + 3 A[7], 20 A[3] + 5 A[4] + 3 A[8], A[30], A[31] ], [2 A[2], 20 A[2] + 15 A[3] + 2 A[7], 15 A[3] + 5 A[4] + 2 A[8], 10 A[4] + 20 A[5] + 2 A[9], A[32]], [16 A[2], 5 A[3] + A[7], 20 A[4] + A[8], 10 A[5] + A[9], A[10]], [24 A[2], 10 A[2] + 20 A[3] + A[7] + 4 A[11], 5 A[2] + 15 A[3] + 10 A[4] + 3 A[7] + 4 A[8] + A[11], 20 A[4] + 5 A[5] + 4 A[9], 10 A[5] + 4 A[10]], [21 A[2], 15 A[2] + 3 A[7] + 4 A[11], 15 A[2] + 5 A[4] + 4 A[7] + A[8] + 3 A[11], 15 A[2] + 20 A[4] + 10 A[5] + 4 A[7] + 2 A[8] + A[9] + 3 A[11] + 4 A[12], 5 A[5] + 15 A[6] + A[10]], [4 A[2] + 10 A[3] + 15 A[4] + 15 A[5] + A[7] + 3 A[8] + 3 A[9] + 2 A[11] + A[12] + A[13], 10 A[3] + 15 A[4] + 15 A[5] + 3 A[8] + 3 A[9] + 2 A[11] + A[12] + A[13], 10 A[4] + 2 A[12], 20 A[5] + 2 A[13], 10 A[5] + 10 A[6] + 4 A[10]], [2 A[2] + 20 A[3] + 10 A[4] + 10 A[5] + 2 A[7] + A[8] + A[9] + 4 A[11] + 2 A[12] + 2 A[13], 10 A[3] + 2 A[7], 10 A[4] + 2 A[8] + 4 A[9] + 3 A[13], 10 A[5] + 15 A[6] + 2 A[9] + A[10] + 2 A[14], 2 A[10]], [A[2], 10 A[3] + 20 A[4] + 20 A[5] + 2 A[8] + 2 A[9] + 3 A[11] + 4 A[12] + 4 A[13] , 10 A[4] + 15 A[5] + 3 A[9] + 3 A[12] + A[13], 10 A[5] + 15 A[6] + A[10] + 3 A[13] + 2 A[14], 20 A[6] + 3 A[14]], [4 A[2], 20 A[3] + 20 A[4] + A[7] + 3 A[8] + 4 A[11] + 4 A[16], 2 A[8] + 3 A[12] + 2 A[16], 20 A[5] + 20 A[6] + 4 A[8] + 4 A[9] + 4 A[10] + A[12] + 3 A[14] + 3 A[16], 15 A[6] + A[10] + 4 A[14]], [ A[2] + 15 A[3] + 10 A[4] + 3 A[7] + 2 A[8] + A[11] + A[16], 10 A[3] + 10 A[4] + 2 A[8] + 3 A[11] + A[16], 15 A[4] + A[8] + 2 A[12] + 2 A[16], 20 A[4] + 5 A[5] + 3 A[8] + A[9] + 2 A[12] + A[16], 10 A[5] + 2 A[8] + 2 A[9] + A[10] + 3 A[12] + 4 A[16] + A[17]], [ 3 A[2] + 10 A[3] + 20 A[4] + A[7] + 4 A[8] + 2 A[11] + 2 A[16], 15 A[3] + 10 A[4] + 3 A[7] + 2 A[8] + A[16], 15 A[4] + 10 A[5] + 10 A[6] + 3 A[8] + A[9] + 4 A[10] + 3 A[17] + 2 A[18], 15 A[5] + 3 A[9] + 3 A[10] + 4 A[18], 20 A[6] + 3 A[10]], [ 4 A[2] + 10 A[3] + 10 A[4] + A[7] + 2 A[8] + A[16] + 2 A[19], 10 A[3] + 10 A[4] + A[8] + 3 A[16] + 2 A[19], 20 A[4] + 20 A[5] + 20 A[6] + 2 A[8] + 4 A[9] + A[10] + 4 A[16] + 2 A[17] + 3 A[18], 20 A[5] + 20 A[6] + 2 A[9] + A[10] + 4 A[17] + 3 A[18], 2 A[10] + 4 A[18]], [A[2] + 10 A[3] + 10 A[4] + A[7] + 2 A[8] + A[16] + 2 A[19], 20 A[3] + 4 A[7] + A[19], A[8] + 4 A[16] + 4 A[20], 20 A[4] + 5 A[5] + 3 A[8] + A[9] + A[16] + 2 A[20], 2 A[18]], [ 4 A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 4 A[8] + 2 A[16] + 4 A[19], 10 A[3] + 10 A[4] + A[8] + 3 A[16] + 2 A[19], 5 A[4] + 3 A[8] + A[16] + 4 A[20], 20 A[4] + 15 A[5] + 3 A[8] + A[16] + A[17] + 2 A[20] + 3 A[21], 20 A[5] + 2 A[8] + 4 A[16] + 4 A[17] + 3 A[18] + 3 A[20] + 4 A[21]], [ 2 A[2] + 20 A[3] + 20 A[4] + 10 A[5] + 10 A[6] + 2 A[7] + 4 A[16] + 2 A[17] + 4 A[18] + 4 A[19] + 4 A[20] + 2 A[21] + 4 A[22], 10 A[3] + 2 A[7], 10 A[4] + 15 A[5] + 15 A[6] + A[16] + 3 A[17] + A[18] + 2 A[20] + 3 A[21] + A[22], 10 A[5] + 10 A[6] + A[17] + 4 A[18] + 2 A[21] + 4 A[22], A[18] + 2 A[22]], [2 A[2] + 10 A[3] + 10 A[4] + 5 A[5] + 5 A[6] + A[7] + 2 A[16] + A[17] + 2 A[18] + 2 A[19] + 2 A[20] + A[21] + 2 A[22], 5 A[3] + A[19], 5 A[4] + 15 A[5] + 15 A[6] + 3 A[17] + A[18] + A[20] + 3 A[21] + A[22], 5 A[5] + 5 A[6] + 2 A[18] + A[21] + 2 A[22], 15 A[6] + A[22]], [3 A[2] + 15 A[3] + 5 A[4] + 15 A[5] + 15 A[6] + 3 A[7] + A[16] + 3 A[17] + A[18] + A[19] + A[20] + 3 A[21] + A[22], 20 A[3] + 2 A[7] + 3 A[19], 5 A[4] + 20 A[5] + 20 A[6] + A[16] + 4 A[17] + 3 A[18] + 4 A[21] + 3 A[22], 5 A[5] + 20 A[6] + A[17] + 3 A[18] + 3 A[22], 15 A[6] + A[18]], [2 A[2] + 15 A[4] + 20 A[5] + 20 A[6] + 4 A[7] + 3 A[16] + 4 A[17] + 3 A[18] + 3 A[19] + 3 A[20] + 4 A[21] + 3 A[22], 5 A[3] + A[19], 5 A[4] + A[20], A[21] + 5 A[5], 5 A[6] + A[22]], [15 A[4] + A[2] + 20 A[6] + 4 A[21] + 4 A[7] + 3 A[22] + 3 A[16] + 4 A[17] + 3 A[19] + 20 A[5] + 3 A[18] + 3 A[20], 10 A[4] + 5 A[3] + 5 A[6] + A[21] + A[7] + 2 A[22] + 2 A[16] + A[17] + 5 A[5] + 2 A[18] + 2 A[20], 10 A[4] + 10 A[6] + 2 A[21] + 4 A[22] + 3 A[16] + 2 A[17] + 10 A[5] + 4 A[18] + A[20], 20 A[6] + A[21] + 3 A[22] + 3 A[17] + 10 A[5] + 3 A[18], A[22] + 3 A[18]], [10 A[4] + 10 A[3] + A[2] + 5 A[6] + A[21] + A[7] + 2 A[22] + 2 A[16] + A[17] + 2 A[19] + 5 A[5] + 2 A[18] + 2 A[20], 20 A[3] + 4 A[7] + A[19], 5 A[4] + 15 A[6] + 3 A[21] + A[22] + 2 A[16] + 3 A[17] + 15 A[5] + A[18], 15 A[6] + A[22] + 2 A[17] + 5 A[5] + A[18], 2 A[18]], [5 A[4] + 15 A[3] + 4 A[2] + 15 A[6] + 3 A[21] + 3 A[7] + A[22] + A[16] + 3 A[17] + A[19] + 15 A[5] + A[18] + A[20], 10 A[3] + 2 A[19], 10 A[4] + 2 A[20], 2 A[21] + 10 A[5], 10 A[6] + 2 A[22]], [5 A[4] + 15 A[3] + 2 A[2] + 15 A[6] + 3 A[21] + 3 A[7] + A[22] + A[16] + 3 A[17] + A[19] + 15 A[5] + A[18] + A[20], 20 A[4] + 10 A[3] + 10 A[6] + 2 A[21] + 2 A[7] + 4 A[22] + 4 A[16] + 2 A[17] + 10 A[5] + 4 A[18] + 4 A[20], 10 A[4] + 20 A[6] + 4 A[21] + 3 A[22] + A[16] + 4 A[17] + 20 A[5] + 3 A[18] + 2 A[20], 15 A[6] + 2 A[21] + A[22] + A[17] + 10 A[5] + A[18], 15 A[6] + 2 A[22] + A[18]], [15 A[4] + A[2] + 20 A[6] + 4 A[21] + 4 A[7] + 3 A[22] + 3 A[16] + 4 A[17] + 3 A[19] + 20 A[5] + 3 A[18] + 3 A[20], 15 A[4] + 10 A[3] + 20 A[6] + 4 A[21] + 3 A[22] + 3 A[16] + 4 A[17] + 3 A[19] + 20 A[5] + 3 A[18] + 3 A[20], 10 A[4] + 10 A[6] + 2 A[21] + 4 A[22] + 2 A[17] + 10 A[5] + 4 A[18] + 3 A[20], 10 A[6] + 3 A[21] + 4 A[22] + 10 A[5] + 4 A[18], 15 A[6] + 3 A[22]], [ 15 A[4] + 3 A[2] + 20 A[6] + 4 A[21] + 4 A[7] + 3 A[22] + 3 A[16] + 4 A[17] + 3 A[19] + 20 A[5] + 3 A[18] + 3 A[20], 20 A[4] + 15 A[3] + 10 A[6] + 2 A[21] + 3 A[7] + 4 A[22] + 4 A[16] + 2 A[17] + 10 A[5] + 4 A[18] + 4 A[20], 20 A[4] + 15 A[6] + 3 A[21] + A[22] + 4 A[16] + 3 A[17] + 15 A[5] + A[18] + 3 A[20], 20 A[6] + 3 A[21] + 3 A[22] + 4 A[17] + 20 A[5] + 3 A[18], 20 A[6] + 3 A[22] + 4 A[18]], [2 A[2], 10 A[4] + 10 A[3] + 5 A[6] + A[21] + 2 A[7] + 2 A[22] + 2 A[16] + A[17] + 5 A[5] + 2 A[18] + 2 A[20], 10 A[4] + 5 A[6] + A[21] + 2 A[22] + A[16] + A[17] + 5 A[5] + 2 A[18] + 2 A[20], 2 A[21] + A[17] + 10 A[5], 20 A[6] + 2 A[22] + A[18]]], [1, 1, 2, 8, 7, 11, 17, 3, 2, 16, 24, 21, 19, 7, 1, 4, 21, 18, 19, 16, 9, 7, 17, 23, 12, 11, 16, 24, 22, 11, 13, 2]] For example, C(100000), mudolo , 25, equals , 17 The congruence classes mod, 25, in the following set , {0, 5, 6, 10, 14, 15, 20}, never show up! Theorem Number, 26, : Let C(n) be the constant term, in x, of n (2/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 24, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], A[11], A[12], A[13], A[14]], [16 A[2], A[7], A[16], A[17], A[18]], [5 A[2], 20 A[3], 10 A[4], 0, A[19]], %3, [2 A[2], 10 A[3] + 2 A[7], A[12], A[13], A[14]], [6 A[2], 15 A[3] + A[7], A[8], A[23], A[24]], %3, %3, [24 A[2], 20 A[3] + A[7] + 4 A[11], 10 A[4] + 4 A[8], 5 A[5] + 4 A[9], 4 A[10]], [ 7 A[2], 15 A[3] + A[7] + 3 A[11], 5 A[4] + 4 A[8] + 4 A[12], 15 A[5] + 2 A[9], 10 A[6] + 2 A[10]], [15 A[2], %2, 0, 5 A[5], 10 A[6]], [15 A[2], %2, 0, 5 A[5], 10 A[6]], [2 A[2], 10 A[3] + 2 A[7], 5 A[4] + 4 A[8] + 4 A[12], 5 A[5] + 2 A[9], 15 A[6] + 2 A[10]], [ 16 A[2], 3 A[7] + 4 A[11], 3 A[8] + 3 A[16], 20 A[5] + A[9], 5 A[6] + A[10] ], [20 A[2], %1, 0, 20 A[5] + 3 A[9] + 2 A[17], 5 A[6]], [20 A[2], %1, 0, 20 A[5] + 3 A[9] + 2 A[17], 5 A[6]], 0, [7 A[2], 15 A[3] + A[7] + 3 A[11], 20 A[4] + 3 A[16] + 2 A[20], 5 A[5] + 3 A[9] + 4 A[17], 2 A[18]], [15 A[2], %2, 0, 4 A[17] + 3 A[21], 10 A[6]], [15 A[2], %2, 0, 4 A[17] + 3 A[21], 10 A[6]], [20 A[2], %1, 0, 20 A[5] + 2 A[17] + 4 A[21], 5 A[6]], [20 A[2], %1, 0, 20 A[5] + 2 A[17] + 4 A[21], 5 A[6]]], [1, 1, 2, 16, 5, 20, 2, 6, 20, 20, 24, 7, 15, 15, 2, 16, 20, 20, 0, 7, 15, 15, 20, 20]] %1 := 10 A[3] + A[7] + 2 A[11] %2 := 20 A[3] + 2 A[7] + 4 A[11] %3 := [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]] For example, C(100000), mudolo , 25, equals , 9 The congruence classes mod, 25, in the following set , {3, 4, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 21, 22, 23}, never show up! Theorem Number, 27, : Let C(n) be the constant term, in x, of n (2/x + 3) For the record, the first 15 terms of the sequence are: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], 15 A[2] + 13 A[3], A[11], A[12], A[13]], [9 A[2], 15 A[2] + 24 A[3], 5 A[3] + 14 A[4], A[14], A[15]], [2 A[2], 10 A[2] + 17 A[3], 20 A[3] + 7 A[4], 5 A[4] + 22 A[5], A[16]], [ 6 A[2], 20 A[2] + 21 A[3], 15 A[3] + 11 A[4], 10 A[4] + A[5], 5 A[5] + 16 A[6]], [18 A[2], 15 A[2] + 3 A[3], 5 A[3] + 13 A[4], 20 A[4] + 23 A[5], 10 A[5] + 8 A[6]], [24 A[2], 15 A[2] + 14 A[3], 5 A[3] + 4 A[4], 20 A[4] + 19 A[5], 10 A[5] + 9 A[6]], [7 A[2], 10 A[2] + 22 A[3], 20 A[3] + 12 A[4], 5 A[4] + 2 A[5], 15 A[5] + 17 A[6]], [A[2], 20 A[2] + 16 A[3], 15 A[3] + 6 A[4], 10 A[4] + 21 A[5], 5 A[5] + 11 A[6]], [22 A[2], 20 A[2] + 17 A[3], 15 A[3] + 12 A[4], 10 A[4] + 7 A[5], 5 A[5] + 2 A[6]], [21 A[2], 5 A[2] + 16 A[3], 10 A[3] + 11 A[4], 15 A[4] + 6 A[5], 20 A[5] + A[6]], [3 A[2], 10 A[2] + 23 A[3], 20 A[3] + 18 A[4], 5 A[4] + 13 A[5], 15 A[5] + 8 A[6]], [13 A[2], 15 A[2] + 23 A[3], 5 A[3] + 8 A[4], 20 A[4] + 18 A[5], 10 A[5] + 3 A[6]], [9 A[2], 5 A[2] + 19 A[3], 10 A[3] + 4 A[4], 15 A[4] + 14 A[5], 20 A[5] + 24 A[6]], [2 A[2], 15 A[2] + 7 A[3], 5 A[3] + 12 A[4], 20 A[4] + 17 A[5], 10 A[5] + 22 A[6]]], [1, 1, 3, 9, 2, 6, 18, 24, 7, 1, 22, 21, 3, 13, 9, 2]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 4, 5, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 23}, never show up! Theorem Number, 28, : Let C(n) be the constant term, in x, of n (2/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 32, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], A[11], A[12], A[13], A[14]], [13 A[2], A[15], A[16], A[17], A[18]], [13 A[2], A[19], A[12], A[21], A[22]], [21 A[2], A[23], A[8], A[25], A[26]], [8 A[2], 20 A[2] + 20 A[3] + 3 A[7], A[27], A[28], A[29]], [ 3 A[2], 15 A[2] + 15 A[3] + 3 A[7], 5 A[3] + 5 A[4] + 3 A[8], A[30], A[31]] , [23 A[2], 20 A[2] + 5 A[3] + 3 A[7], 15 A[3] + 15 A[4] + 3 A[8], 10 A[4] + 3 A[9], A[14]], [16 A[2], 5 A[3] + A[7], 20 A[4] + A[8], 10 A[5] + A[9], A[10]], [14 A[2], 20 A[2] + 20 A[3] + 3 A[7] + 2 A[11], 20 A[2] + 20 A[3] + 3 A[7] + 4 A[8] + 4 A[11], 5 A[4] + 20 A[5] + 4 A[9], 15 A[5] + 15 A[6] + 4 A[10]], [19 A[2], 10 A[2] + 10 A[3] + 3 A[11], 20 A[2] + 20 A[3] + 10 A[4] + 3 A[7] + 4 A[11] + 3 A[12], 5 A[2] + 10 A[3] + 10 A[4] + 2 A[7] + A[8] + 4 A[9] + A[11] + 3 A[12], 5 A[5] + 20 A[6] + 4 A[10]], [9 A[2], 10 A[3] + 3 A[11], 5 A[2] + 10 A[3] + 10 A[4] + 2 A[7] + A[8] + A[11] + A[12], 15 A[2] + 10 A[3] + 20 A[4] + 20 A[5] + A[7] + 3 A[8] + 4 A[9] + 3 A[11] + 4 A[12] , 5 A[2] + 10 A[3] + 10 A[4] + 20 A[5] + 2 A[7] + A[8] + 3 A[9] + 4 A[10] + A[11] + 3 A[12] + 4 A[13]], [3 A[2] + 10 A[3] + 10 A[4] + 20 A[5] + 20 A[6] + 2 A[7] + A[8] + 3 A[9] + 4 A[10] + A[11] + 3 A[12] + 4 A[13] + 2 A[14], 15 A[3] + 3 A[7], 15 A[4] + 20 A[5] + 10 A[6] + 3 A[8] + 4 A[9] + 2 A[10] + 2 A[13] + A[14], 15 A[5] + 10 A[6] + 3 A[9] + A[10] + 3 A[14], 3 A[10]], [4 A[2] + 10 A[3] + 20 A[4] + 20 A[5] + 10 A[6] + A[7] + 3 A[8] + 4 A[9] + 2 A[10] + 3 A[11] + 4 A[12] + 2 A[13] + A[14], 10 A[3] + 10 A[4] + 20 A[5] + 20 A[6] + A[8] + 3 A[9] + 4 A[10] + 3 A[11] + 3 A[12] + 4 A[13] + 2 A[14], 10 A[4] + 20 A[5] + 10 A[6] + 4 A[9] + 2 A[10] + 3 A[12] + 2 A[13] + A[14], 10 A[5] + 20 A[6] + 3 A[10] + 3 A[13] + 4 A[14], 5 A[6] + 3 A[14]], [ 4 A[2] + 20 A[3] + 20 A[4] + 4 A[7] + 3 A[8] + 2 A[11] + 2 A[12] + 2 A[16], 10 A[3] + 20 A[4] + A[7] + 2 A[8] + A[11] + 3 A[12] + 3 A[16], 5 A[4] + 4 A[8] + 4 A[12] + A[16], 10 A[4] + 10 A[5] + A[8] + A[12] + 3 A[13] + 2 A[16], 5 A[4] + 10 A[5] + 20 A[6] + 4 A[8] + A[9] + 4 A[10] + 4 A[12] + 3 A[13] + 3 A[16]], [ 4 A[2] + 20 A[3] + 20 A[4] + 4 A[7] + 3 A[8] + 2 A[11] + 2 A[12] + 2 A[16], 10 A[3] + 20 A[4] + 2 A[8] + 3 A[11] + 3 A[12] + 3 A[16], 5 A[4] + 4 A[8] + 2 A[12] + 3 A[16], 5 A[4] + 10 A[5] + 4 A[8] + 4 A[12] + 3 A[16] + 3 A[17], 20 A[4] + 20 A[5] + 10 A[6] + 2 A[8] + 4 A[9] + 4 A[10] + 2 A[12] + 4 A[16] + 2 A[17]], [ 3 A[2] + 10 A[3] + 20 A[4] + 20 A[5] + 2 A[7] + 3 A[8] + 3 A[9] + 4 A[10] + A[11] + 4 A[16] + 4 A[17] + 2 A[18], 15 A[3] + 3 A[7], 15 A[4] + 10 A[5] + 5 A[6] + 3 A[8] + 2 A[9] + A[10] + A[17] + 3 A[18], 15 A[5] + 5 A[6] + 3 A[9] + 3 A[10] + 4 A[18], 3 A[10]], [4 A[2], 10 A[3] + 20 A[4] + 20 A[5] + 3 A[8] + 3 A[9] + 4 A[10] + 4 A[16] + 4 A[17] + 2 A[18] + 3 A[19], 20 A[4] + 2 A[8] + 4 A[16], 20 A[5] + 2 A[9] + 2 A[10] + 4 A[17] + A[18], 15 A[6] + 2 A[10] + 4 A[18]], [4 A[2] + 20 A[3] + 20 A[4] + 4 A[7] + A[8] + 4 A[16] + 2 A[19] + 4 A[20], 10 A[3] + 20 A[4] + A[7] + A[8] + 4 A[16] + A[19] + 4 A[20], 5 A[4] + 4 A[8] + A[16] + 4 A[20], 10 A[4] + 20 A[5] + A[8] + 2 A[9] + 2 A[16] + 4 A[17] + A[20], 5 A[6] + 3 A[18]], [ 4 A[2] + 20 A[3] + 20 A[4] + 4 A[7] + A[8] + 4 A[16] + 2 A[19] + 4 A[20], 10 A[3] + 20 A[4] + 4 A[8] + A[16] + 3 A[19] + A[20], 5 A[4] + 4 A[8] + 3 A[16] + 2 A[20], 5 A[4] + 10 A[5] + 4 A[8] + 3 A[16] + 3 A[17] + 4 A[20], 20 A[4] + 10 A[5] + 20 A[6] + 2 A[8] + 4 A[16] + 4 A[17] + 3 A[18] + 2 A[20] + A[21]], [%1, 15 A[3] + 10 A[4] + 10 A[5] + 3 A[7] + A[16] + 4 A[17] + 3 A[18] + 4 A[20] + A[21] + 2 A[22], 15 A[4] + 4 A[16] + 2 A[20], 15 A[5] + 4 A[17] + 3 A[18] + 2 A[21] + 2 A[22], 20 A[6] + 4 A[18] + 2 A[22]], [%1, 5 A[3] + A[19], 5 A[4] + 10 A[5] + 15 A[6] + 3 A[17] + A[18] + A[20] + 2 A[21] + 4 A[22], 5 A[5] + 20 A[6] + 2 A[18] + A[21] + 3 A[22], 20 A[6] + A[22]], [3 A[2], 15 A[3] + 10 A[4] + 10 A[5] + 15 A[6] + 2 A[7] + 2 A[16] + 3 A[17] + A[18] + 2 A[19] + 3 A[20] + 2 A[21] + 4 A[22], 5 A[4] + 10 A[5] + 20 A[6] + A[16] + A[17] + 2 A[18] + 4 A[21] + 3 A[22], 5 A[5] + 20 A[6] + A[17] + 2 A[18] + 3 A[22], 15 A[6] + A[18]], [3 A[2] + 20 A[3] + 10 A[4] + 10 A[5] + 15 A[6] + 4 A[7] + 2 A[16] + 3 A[17] + A[18] + 2 A[19] + 3 A[20] + 2 A[21] + 4 A[22], 5 A[3] + A[19], 5 A[4] + A[20], A[21] + 5 A[5], 5 A[6] + A[22]], [10 A[4] + 10 A[3] + A[2] + A[21] + 2 A[7] + 2 A[22] + A[16] + 4 A[17] + A[19] + 10 A[5] + 3 A[18] + 4 A[20], 10 A[4] + 5 A[3] + 15 A[6] + 2 A[21] + A[7] + 4 A[22] + 2 A[16] + 3 A[17] + 10 A[5] + A[18] + 3 A[20], 15 A[4] + 3 A[16] + 4 A[20], 15 A[6] + 4 A[21] + 4 A[22] + 3 A[17] + 15 A[5] + A[18], 4 A[22] + 3 A[18]] , [10 A[4] + 20 A[3] + 4 A[2] + 20 A[6] + 4 A[21] + 3 A[7] + 3 A[22] + 4 A[16] + A[17] + 4 A[19] + 10 A[5] + 2 A[18] + A[20], 10 A[4] + 10 A[3] + 5 A[6] + 3 A[21] + A[7] + A[22] + 3 A[16] + 2 A[17] + A[19] + 10 A[5] + 4 A[18] + 2 A[20], 10 A[4] + 20 A[6] + 4 A[21] + 3 A[22] + 3 A[16] + A[17] + 10 A[5] + 2 A[18] , 20 A[6] + 3 A[22] + 3 A[17] + 10 A[5] + 2 A[18], 20 A[6] + 3 A[18]], [ 10 A[4] + 10 A[3] + 4 A[2] + A[21] + 2 A[7] + 2 A[22] + A[16] + 4 A[17] + A[19] + 10 A[5] + 3 A[18] + 4 A[20], 10 A[3] + 3 A[19], 10 A[4] + 3 A[20], 3 A[21] + 10 A[5], 10 A[6] + 3 A[22]], [3 A[2], 10 A[4] + 15 A[3] + 15 A[6] + 2 A[21] + 3 A[7] + 4 A[22] + 2 A[16] + 3 A[17] + 10 A[5] + A[18] + 3 A[20], 15 A[4] + A[21] + 2 A[22] + 4 A[16] + 4 A[17] + 10 A[5] + 3 A[18] + 2 A[20] , 15 A[6] + 2 A[21] + 4 A[22] + 4 A[17] + 15 A[5] + A[18], 10 A[6] + 2 A[22] + 4 A[18]], [10 A[4] + 10 A[3] + 4 A[2] + 5 A[6] + 3 A[21] + A[7] + A[22] + 3 A[16] + 2 A[17] + 3 A[19] + 10 A[5] + 4 A[18] + 2 A[20], 10 A[4] + 10 A[3] + A[21] + 2 A[22] + A[16] + 4 A[17] + 3 A[19] + 10 A[5] + 3 A[18] + 4 A[20], 10 A[4] + A[21] + 2 A[22] + 4 A[17] + 10 A[5] + 3 A[18] + 3 A[20], 3 A[21] + 2 A[22] + 10 A[5] + 3 A[18], 3 A[22]], [%1, 15 A[3] + 10 A[4] + 10 A[5] + 3 A[7] + A[16] + 4 A[17] + 3 A[18] + 4 A[20] + A[21] + 2 A[22] , 15 A[4] + 4 A[16] + 2 A[20], 15 A[5] + 4 A[17] + 3 A[18] + 2 A[21] + 2 A[22], 20 A[6] + 4 A[18] + 2 A[22]], [10 A[4] + 10 A[3] + 3 A[2] + A[21] + 2 A[7] + 2 A[22] + A[16] + 4 A[17] + A[19] + 10 A[5] + 3 A[18] + 4 A[20], 15 A[3] + 3 A[7], 15 A[4] + 20 A[6] + 4 A[21] + 3 A[22] + 4 A[16] + A[17] + 10 A[5] + 2 A[18] + 2 A[20], 2 A[21] + 4 A[17] + 15 A[5], 5 A[6] + 2 A[22] + 4 A[18]]], [1, 1, 3, 13, 13, 21, 8, 3, 23, 16, 14, 19, 9, 23, 14, 19, 19, 23, 4, 19, 19, 13, 13, 3, 18, 21, 9, 24, 3, 14, 13, 23]] %1 := 3 A[2] + 10 A[3] + 10 A[4] + 10 A[5] + 5 A[6] + A[7] + 3 A[16] + 2 A[17] + 4 A[18] + 3 A[19] + 2 A[20] + 3 A[21] + A[22] For example, C(100000), mudolo , 25, equals , 17 The congruence classes mod, 25, in the following set , {0, 2, 5, 6, 7, 10, 11, 12, 15, 17, 20, 22}, never show up! Theorem Number, 29, : Let C(n) be the constant term, in x, of n (2/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], 13 A[3], A[11], A[12], A[13]], [17 A[2], 20 A[2] + 22 A[3], 15 A[3] + 2 A[4], A[14], A[15]], [24 A[2], 19 A[3], 14 A[4], 9 A[5], A[16]], [9 A[2], 5 A[2] + 14 A[3], 10 A[3] + 19 A[4], 15 A[4] + 24 A[5], 20 A[5] + 4 A[6]], [18 A[2], 3 A[3], 13 A[4], 23 A[5], 8 A[6]], [22 A[2], 20 A[2] + 2 A[3], 15 A[3] + 7 A[4], 10 A[4] + 12 A[5], 5 A[5] + 17 A[6]], [24 A[2], 19 A[3], 14 A[4], 9 A[5], 4 A[6]], [19 A[2], 5 A[2] + 24 A[3], 10 A[3] + 4 A[4], 15 A[4] + 9 A[5], 20 A[5] + 14 A[6]], [16 A[2], 10 A[2] + 6 A[3], 20 A[3] + 21 A[4], 5 A[4] + 11 A[5], 15 A[5] + A[6]], [17 A[2], 2 A[3], 12 A[4], 22 A[5], 7 A[6]], [12 A[2], 15 A[2] + 2 A[3], 5 A[3] + 17 A[4], 20 A[4] + 7 A[5], 10 A[5] + 22 A[6]], [13 A[2], 3 A[3], 18 A[4], 8 A[5], 23 A[6]], [3 A[2], 10 A[2] + 13 A[3], 20 A[3] + 23 A[4], 5 A[4] + 8 A[5], 15 A[5] + 18 A[6]], [11 A[2], 20 A[2] + 6 A[3], 15 A[3] + A[4], 10 A[4] + 21 A[5], 5 A[5] + 16 A[6]]], [1, 1, 3, 17, 24, 9, 18, 22, 24, 19, 16, 17, 12, 13, 3, 11]] For example, C(100000), mudolo , 25, equals , 13 The congruence classes mod, 25, in the following set , {0, 2, 4, 5, 6, 7, 8, 10, 14, 15, 20, 21, 23}, never show up! Theorem Number, 30, : Let C(n) be the constant term, in x, of n (2/x + 3 + 3 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 23, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], A[11], A[12], A[13], A[14]], [21 A[2], A[15], A[16], A[9], A[18]], [10 A[2], 20 A[3], 5 A[4], 15 A[5], 0], [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]], [23 A[2], 10 A[3] + 3 A[7], A[19], A[20], A[21]], [6 A[2], 15 A[3] + A[7], A[8], A[22], A[23]], [5 A[2], 15 A[3], 0, 10 A[5], 20 A[6]], [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]], [14 A[2], 20 A[3] + 3 A[7] + 2 A[11], 4 A[8], 20 A[5] + 4 A[9], 15 A[6] + 4 A[10]], [ 23 A[2], 15 A[3] + 2 A[7] + 2 A[11], 15 A[4] + 2 A[8] + 2 A[12], 10 A[5] + 3 A[9], 15 A[6] + 3 A[10]], [15 A[2], 20 A[3] + 3 A[7] + 4 A[11], 0, 10 A[5] + A[9] + 3 A[13], 10 A[6]] , [10 A[2], 10 A[3] + 2 A[7] + A[11], 0, 20 A[5] + 4 A[9] + 2 A[13], 15 A[6]] , [13 A[2], 15 A[3] + 2 A[7] + 2 A[11], 15 A[4] + 3 A[8], 15 A[5] + A[9] + 4 A[13], 20 A[6] + 3 A[10]], [ 21 A[2], 5 A[3] + A[7], 2 A[8] + 4 A[16], 5 A[5] + 2 A[13], 10 A[6] + A[10] ], [5 A[2], 10 A[3] + A[7] + 3 A[11], 0, 10 A[5] + A[13] + 2 A[17], 20 A[6]], [ 20 A[2], 20 A[3] + 4 A[7] + 2 A[11], 0, 20 A[5] + 4 A[13] + 3 A[17], 5 A[6] ], [18 A[2], 5 A[3] + A[11], 15 A[4] + 2 A[16] + 2 A[19], 15 A[5] + 4 A[13] + A[17], 5 A[6] + 3 A[18]], [ 15 A[2], 20 A[3] + 3 A[7] + 4 A[11], 0, 10 A[5] + 2 A[17] + A[20], 10 A[6]] , [ 10 A[2], 10 A[3] + 2 A[7] + A[11], 0, 20 A[5] + 3 A[17] + 4 A[20], 15 A[6]] , [5 A[2], 10 A[3] + A[7] + 3 A[11], 0, 20 A[5] + 4 A[17] + 2 A[20], 20 A[6]] , [20 A[2], 20 A[3] + 4 A[7] + 2 A[11], 0, 10 A[5] + A[17] + 3 A[20], 5 A[6]] ], [1, 1, 3, 21, 10, 20, 23, 6, 5, 20, 14, 23, 15, 10, 13, 21, 5, 20, 18, 15, 10, 5, 20]] For example, C(100000), mudolo , 25, equals , 9 The congruence classes mod, 25, in the following set , {2, 4, 7, 8, 9, 11, 12, 16, 17, 19, 22, 24}, never show up! Theorem Number, 31, : Let C(n) be the constant term, in x, of n (3/x + x) For the record, the first 15 terms of the sequence are: 0, 6, 0, 54, 0, 540, 0, 5670, 0, 61236, 0, 673596, 0, 7505784, 0 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 22, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [0, 15 A[2], 5 A[3], A[11], A[12]], [6 A[2], A[13], A[14], A[9], A[16]], [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]], [4 A[2], A[17], A[18], A[19], A[20]], [0, 15 A[2], 5 A[3], 20 A[4], 10 A[5]], [11 A[2], A[7], 15 A[4] + A[8], A[21], A[22]], [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]], [19 A[2], 10 A[3] + 4 A[7], 5 A[4] + 4 A[8], 4 A[9], 20 A[6] + 4 A[10]], [20 A[2], 20 A[3], A[11], 20 A[5], 20 A[6]], 0, [0, 15 A[2], 20 A[3] + A[7] + 4 A[13], A[11], A[12]], [21 A[2], 2 A[7] + 4 A[13], 5 A[4] + A[8] + A[11], 5 A[5] + A[9] + A[12], 5 A[6] + A[10]], [0, 20 A[2], 20 A[3] + 3 A[7] + 2 A[13], 3 A[11], 3 A[12]], [4 A[2], 20 A[3] + 4 A[7], 20 A[4] + 4 A[8] + A[11], 20 A[5] + 4 A[9] + 4 A[12], 20 A[6] + 4 A[16]], [0, 10 A[2], 20 A[3] + 4 A[7] + 4 A[17], 4 A[11], 4 A[12]], [9 A[2], 20 A[3] + 3 A[7] + 4 A[17], 5 A[4] + 4 A[11] + A[18], 20 A[5] + 4 A[9] + 4 A[12], 10 A[6] + 4 A[16]], [0, 5 A[2], 10 A[3] + 2 A[7] + 2 A[17], 2 A[11], 5 A[5] + A[9] + A[19]], [21 A[2], 5 A[3] + 4 A[17], 5 A[4] + 4 A[18], 5 A[5] + 4 A[19], A[16]], [ 0, 20 A[2], 15 A[3] + 3 A[7] + 3 A[17], 3 A[11], 20 A[5] + 4 A[9] + 4 A[19] ], [24 A[2], 20 A[3] + 3 A[7] + 4 A[17], 5 A[4] + 2 A[11] + A[18], 15 A[5] + 2 A[9] + 3 A[19], 15 A[6] + 4 A[16]]], [1, 1, 0, 6, 0, 4, 0, 11, 0, 19, 20, 0, 0, 21, 0, 4, 0, 9, 0, 21, 0, 24]] For example, C(100000), mudolo , 25, equals , 15 The congruence classes mod, 25, in the following set , {2, 3, 5, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 22, 23}, never show up! Theorem Number, 32, : Let C(n) be the constant term, in x, of n (3/x + 2 x) For the record, the first 15 terms of the sequence are: 0, 12, 0, 216, 0, 4320, 0, 90720, 0, 1959552, 0, 43110144, 0, 960740352, 0 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 22, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [0, 20 A[2], 15 A[3], A[11], A[12]], [12 A[2], A[13], A[14], A[15], A[16]], %1, [16 A[2], A[17], A[8], A[19], A[20]], %1, [2 A[2], 5 A[3] + 2 A[7], 10 A[4] + 2 A[8], A[21], A[22]], %1, [6 A[2], 10 A[3] + A[7], 15 A[4] + A[8], 20 A[5] + A[9], A[10]], [20 A[2], 20 A[3], 2 A[11], 20 A[5], 20 A[6]], 0, [0, 15 A[2], 4 A[7] + 3 A[13], 2 A[11], 2 A[12]], [19 A[2], 3 A[7] + 3 A[13], 20 A[4] + 4 A[8] + 4 A[11], 20 A[5] + 4 A[9], 5 A[6] + 4 A[10]], [0, 15 A[2], 4 A[7] + 3 A[13], 2 A[11], 2 A[12]], [ 17 A[2], 5 A[3] + 4 A[7] + 4 A[13], 10 A[4] + 2 A[8], 10 A[5] + 2 A[9] + 2 A[12], A[16]], [0, 20 A[2], 20 A[3] + 3 A[7] + 2 A[17], A[11], A[12]], [2 A[2], 10 A[3] + A[7] + A[17], 10 A[4] + 2 A[18], 10 A[5] + 2 A[9] + A[12], 20 A[6] + 2 A[10]], [0, 20 A[2], 20 A[3] + 3 A[7] + 2 A[17], A[11], 20 A[5] + 4 A[9] + A[19]], [16 A[2], 4 A[7] + 2 A[17], 5 A[4] + 2 A[11] + A[18], 5 A[5] + A[9], 15 A[6] + 3 A[16]], [0, 15 A[2], 20 A[3] + A[7] + 4 A[17], 2 A[11], 20 A[5] + 3 A[9] + 2 A[19]] , [7 A[2], 5 A[3] + 3 A[7] + 4 A[17], 10 A[4] + 4 A[11] + 2 A[18], 10 A[5] + 2 A[9], 15 A[6] + A[16]]], [1, 1, 0, 12, 0, 16, 0, 2, 0, 6, 20, 0, 0, 19, 0, 17, 0, 2, 0, 16, 0, 7]] %1 := [0, 20 A[2], 15 A[3], 10 A[4], 5 A[5]] For example, C(100000), mudolo , 25, equals , 15 The congruence classes mod, 25, in the following set , {3, 4, 5, 8, 9, 10, 11, 13, 14, 15, 18, 21, 22, 23, 24}, never show up! Theorem Number, 33, : Let C(n) be the constant term, in x, of n (3/x + 1) For the record, the first 15 terms of the sequence are: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[%6, [A[2], A[7], A[8], A[9], A[10]], [A[2], %4, A[11], A[12], A[13]], %6, [A[2], %4, %3, %2, A[13]], %1, %5, %6, %5, %1, %6, %5, %1, %5, %1, %1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] %1 := [A[2], 20 A[2] + 6 A[3], 15 A[3] + 11 A[4], 10 A[4] + 16 A[5], 5 A[5] + 21 A[6]] %2 := 20 A[4] + 6 A[5] %3 := 5 A[3] + 21 A[4] %4 := 15 A[2] + 11 A[3] %5 := [A[2], %4, %3, %2, 10 A[5] + 16 A[6]] %6 := [A[2], A[3], A[4], A[5], A[6]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, never show up! Theorem Number, 34, : Let C(n) be the constant term, in x, of n (3/x + 1 + x) For the record, the first 15 terms of the sequence are: 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 32, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], A[11], A[12], A[13], A[14]], [7 A[2], A[15], A[16], A[17], A[18]], [19 A[2], A[19], A[20], A[21], A[22]], [16 A[2], A[23], A[24], A[25], A[10]], [6 A[2], 5 A[2] + 10 A[3] + A[7], A[27], A[28], A[29]], [22 A[2], 20 A[2] + 5 A[3] + 2 A[7], 15 A[3] + 15 A[4] + 2 A[8], A[30], A[31]], [ 14 A[2], 20 A[2] + 15 A[3] + 4 A[7], 15 A[3] + 20 A[4] + 4 A[8], 10 A[4] + 4 A[9], A[32]], [16 A[2], 5 A[3] + A[7], 20 A[4] + A[8], 10 A[5] + A[9], A[10]], [16 A[2], 15 A[2] + 3 A[7] + 3 A[11], 15 A[2] + 20 A[3] + 3 A[7] + A[8] + 2 A[11], 15 A[4] + 5 A[5] + A[9], 20 A[5] + 10 A[6] + A[10]], [2 A[2], 20 A[2] + 10 A[3] + 2 A[7], 20 A[2] + 20 A[3] + 10 A[4] + 4 A[7] + A[8] + A[11] + A[12], 10 A[2] + 20 A[3] + 20 A[4] + 5 A[5] + 2 A[7] + 4 A[8] + 2 A[9] + 3 A[11] + A[12], 5 A[5] + 15 A[6] + 2 A[10]], [4 A[2], 10 A[2] + 20 A[3] + 2 A[7] + 2 A[11], 20 A[4] + 4 A[12], 15 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 3 A[7] + A[8] + A[9] + 2 A[11] + 4 A[12] + 3 A[13], 20 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 20 A[6] + 4 A[7] + 3 A[8] + 3 A[9] + 4 A[10] + A[11] + 2 A[12] + 2 A[13]], [A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 10 A[6] + 2 A[7] + 4 A[8] + 4 A[9] + 2 A[10] + 3 A[11] + A[12] + A[13] + 3 A[14], 5 A[3] + A[7], 5 A[4] + 20 A[5] + 10 A[6] + A[8] + 4 A[9] + 2 A[10] + A[13] + 3 A[14], 5 A[5] + 15 A[6] + A[9] + A[10] + 4 A[14], A[10]], [2 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 15 A[6] + A[7] + 2 A[8] + 2 A[9] + A[10] + 4 A[11] + 3 A[12] + 3 A[13] + 4 A[14], 10 A[3] + 20 A[4] + 20 A[5] + 10 A[6] + 4 A[8] + 4 A[9] + 2 A[10] + 2 A[11] + A[12] + A[13] + 3 A[14], 10 A[4] + 20 A[5] + 5 A[6] + A[9] + 3 A[10] + 2 A[12] + 4 A[13] + 2 A[14], 10 A[5] + 10 A[6] + 2 A[10] + 2 A[13] + 3 A[14], 20 A[6] + 2 A[14]], [ 4 A[2] + 20 A[3] + 15 A[4] + 4 A[7] + A[11] + 3 A[12] + A[16], 20 A[3] + 10 A[4] + A[7] + 3 A[11] + A[12] + 2 A[16], 15 A[4] + 4 A[8] + 2 A[12] + 4 A[16], 20 A[5] + A[9] + 3 A[13], 20 A[5] + 10 A[6] + A[8] + A[9] + 4 A[10] + 3 A[12] + 4 A[13] + 3 A[16]], [ 3 A[2] + 20 A[3] + 10 A[4] + 3 A[7] + 2 A[11] + A[12] + 2 A[16], 15 A[3] + 10 A[4] + 3 A[11] + A[12] + 2 A[16], 4 A[8] + 2 A[16], 15 A[4] + 15 A[5] + 3 A[8] + 3 A[9] + 4 A[12] + 4 A[16], 15 A[4] + 10 A[5] + 2 A[8] + A[9] + 3 A[10] + A[12] + A[16] + 2 A[17]], [ 2 A[2] + 20 A[3] + 10 A[4] + 15 A[5] + 10 A[6] + A[7] + A[8] + 3 A[9] + 2 A[10] + 4 A[11] + 2 A[16] + A[17] + 4 A[18], 10 A[3] + 15 A[4] + 5 A[6] + 2 A[7] + 3 A[8] + 4 A[9] + A[10] + A[16] + 3 A[17] + 2 A[18], 10 A[4] + 10 A[5] + 5 A[6] + 2 A[8] + A[9] + 4 A[10] + 2 A[17] + 3 A[18], 10 A[5] + 2 A[9], 15 A[6] + 2 A[10]], [4 A[2] + 15 A[3] + 20 A[4] + 10 A[5] + 5 A[6] + 3 A[7] + 2 A[8] + A[9] + 4 A[10] + 4 A[16] + 2 A[17] + 3 A[18] + 3 A[19], 5 A[3] + 10 A[4] + 15 A[5] + 10 A[6] + A[8] + 3 A[9] + 2 A[10] + 2 A[16] + A[17] + 4 A[18] + A[19], 15 A[4] + 15 A[5] + 10 A[6] + 2 A[8] + 3 A[9] + 2 A[10] + A[16] + A[17] + 4 A[18], 15 A[5] + 10 A[6] + 2 A[9] + 2 A[10] + A[17] + 4 A[18], 20 A[6] + 2 A[10] + A[18]], [3 A[2], 15 A[3] + 2 A[7] + 3 A[8] + 3 A[16] + 4 A[19] + 4 A[20], 15 A[4] + 2 A[8] + 4 A[20], 15 A[4] + 10 A[5] + A[8] + A[9] + 3 A[16] + A[17] + 2 A[20], 4 A[18]], [ A[2] + 5 A[3] + 15 A[4] + A[7] + 2 A[8] + 2 A[16] + A[19] + A[20], 5 A[3] + 5 A[4] + 4 A[8] + 4 A[16] + 4 A[19] + 2 A[20], 3 A[8] + 4 A[16], 15 A[4] + 15 A[5] + A[8] + 3 A[16] + 4 A[17] + 2 A[20] + 2 A[21], 15 A[4] + 10 A[5] + 10 A[6] + 2 A[8] + A[16] + 3 A[17] + 3 A[18] + 4 A[20] + A[21] ], [4 A[2] + 10 A[3] + 10 A[4] + 15 A[5] + 20 A[6] + 2 A[7] + 2 A[16] + 4 A[17] + 4 A[18] + 2 A[19] + 4 A[20] + 3 A[21] + 3 A[22], 20 A[3] + 15 A[4] + 10 A[5] + 20 A[6] + 4 A[7] + 4 A[16] + 3 A[17] + 3 A[18] + 3 A[20] + A[21] + A[22], 15 A[4] + 10 A[5] + 3 A[16] + 2 A[17] + 2 A[18] + 2 A[20] + 4 A[21] + 4 A[22], 15 A[5] + 3 A[17] + 2 A[18] + 2 A[21] + 4 A[22], 15 A[6] + 3 A[18] + 2 A[22]], [A[2] + 15 A[3] + 10 A[4] + 5 A[5] + 3 A[7] + 3 A[16] + A[17] + A[18] + 3 A[19] + A[20] + 2 A[21] + 2 A[22], 5 A[3] + 10 A[4] + 15 A[5] + 20 A[6] + 2 A[16] + 4 A[17] + 4 A[18] + 4 A[19] + 4 A[20] + 3 A[21] + 3 A[22], 5 A[4] + 4 A[20], 5 A[5] + 20 A[6] + 4 A[18] + 4 A[21] + 3 A[22], 4 A[22]], [%1, 15 A[3] + 15 A[4] + 10 A[5] + 20 A[6] + 3 A[7] + 4 A[16] + 3 A[17] + 3 A[18] + A[19] + 3 A[20] + A[21] + A[22], 5 A[4] + A[16], 5 A[5] + A[17], 5 A[6] + A[18]], [4 A[2], 5 A[3] + 15 A[4] + 10 A[5] + 20 A[6] + 4 A[16] + 3 A[17] + 3 A[18] + A[19] + 3 A[20] + A[21] + A[22], 5 A[4] + 5 A[5] + A[17] + A[18] + A[20] + 2 A[21] + 2 A[22], 5 A[5] + 2 A[18] + A[21] + 4 A[22], 10 A[6] + A[22]], [10 A[4] + 10 A[3] + A[2] + 20 A[6] + 3 A[21] + 2 A[7] + 3 A[22] + 2 A[16] + 4 A[17] + 2 A[19] + 15 A[5] + 4 A[18] + 4 A[20], 5 A[3] + A[7], 10 A[4] + 20 A[6] + 3 A[21] + 3 A[22] + 2 A[16] + 4 A[17] + 15 A[5] + 4 A[18] + 3 A[20], 3 A[21] + 2 A[17] + 10 A[5], 15 A[6] + 3 A[22] + 2 A[18]], [%1, 15 A[3] + 15 A[4] + 10 A[5] + 20 A[6] + 3 A[7] + 4 A[16] + 3 A[17] + 3 A[18] + A[19] + 3 A[20] + A[21] + A[22], 5 A[4] + A[16], 5 A[5] + A[17], 5 A[6] + A[18]], [5 A[4] + 5 A[3] + 4 A[2] + 4 A[21] + A[7] + 4 A[22] + A[16] + 2 A[17] + A[19] + 10 A[5] + 2 A[18] + 2 A[20], 5 A[3] + A[19], 5 A[4] + A[20], A[21] + 5 A[5], 5 A[6] + A[22]], [5 A[4] + 5 A[3] + A[2] + 4 A[21] + A[7] + 4 A[22] + A[16] + 2 A[17] + A[19] + 10 A[5] + 2 A[18] + 2 A[20], 15 A[4] + 5 A[3] + 20 A[6] + A[21] + A[7] + A[22] + 4 A[16] + 3 A[17] + 10 A[5] + 3 A[18] + 3 A[20], 10 A[4] + 2 A[16] + 3 A[20], 3 A[21] + 4 A[22] + 2 A[17] + 10 A[5] + 2 A[18], 20 A[6] + 3 A[22] + 2 A[18]], [5 A[4] + 5 A[3] + 3 A[2] + 4 A[21] + A[7] + 4 A[22] + A[16] + 2 A[17] + A[19] + 10 A[5] + 2 A[18] + 2 A[20], 15 A[4] + 5 A[3] + 20 A[6] + A[21] + A[22] + 4 A[16] + 3 A[17] + 2 A[19] + 10 A[5] + 3 A[18] + 3 A[20], 5 A[4] + 2 A[21] + 2 A[22] + A[17] + 5 A[5] + A[18] + 2 A[20], 2 A[21] + 4 A[22] + 5 A[5] + 2 A[18], 10 A[6] + 2 A[22]], [%1, 15 A[4] + 10 A[3] + 20 A[6] + A[21] + 2 A[7] + A[22] + 4 A[16] + 3 A[17] + 10 A[5] + 3 A[18] + 3 A[20], 15 A[4] + 20 A[6] + 3 A[21] + 3 A[22] + 4 A[16] + 4 A[17] + 15 A[5] + 4 A[18] + A[20], A[21] + 4 A[22] + 4 A[17] + 15 A[5] + 2 A[18], 5 A[6] + A[22] + 4 A[18]], [ 10 A[4] + 15 A[3] + 4 A[2] + 2 A[21] + 3 A[7] + 2 A[22] + 3 A[16] + A[17] + 3 A[19] + 5 A[5] + A[18] + A[20], 20 A[3] + 4 A[7], 15 A[4] + 2 A[21] + 2 A[22] + 3 A[16] + A[17] + 5 A[5] + A[18] + 2 A[20], 2 A[21] + 3 A[17] + 15 A[5], 10 A[6] + 2 A[22] + 3 A[18]]], [1, 1, 1, 7, 19, 16, 6, 22, 14, 16, 16, 2, 4, 16, 22, 9, 13, 22, 14, 3, 21, 19, 11, 12, 4, 16, 12, 24, 21, 23, 12, 14]] %1 := 2 A[2] + 15 A[3] + 10 A[4] + 5 A[5] + 3 A[7] + 3 A[16] + A[17] + A[18] + 3 A[19] + A[20] + 2 A[21] + 2 A[22] For example, C(100000), mudolo , 25, equals , 17 The congruence classes mod, 25, in the following set , {0, 5, 8, 10, 15, 17, 18, 20}, never show up! Theorem Number, 35, : Let C(n) be the constant term, in x, of n (3/x + 1 + 2 x) For the record, the first 15 terms of the sequence are: 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 32, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], A[11], A[12], A[13], A[14]], [13 A[2], A[15], A[16], A[17], A[18]], [12 A[2], A[19], A[20], A[21], A[22]], [14 A[2], A[23], A[24], A[25], A[26]], [A[2], 10 A[2] + 5 A[3] + A[7], A[12], A[13], A[14]], [3 A[2], 15 A[2] + 3 A[7], 5 A[3] + 3 A[8], A[30], A[31]], [7 A[2], 20 A[2] + 10 A[3] + 2 A[7], 15 A[3] + 15 A[4] + 2 A[8], 10 A[4] + 20 A[5] + 2 A[9], A[32]], [19 A[2], 10 A[3] + 4 A[7], 5 A[4] + 4 A[8], 4 A[9], 20 A[6] + 4 A[10]], [ 16 A[2], 5 A[2] + 3 A[7] + 3 A[11], 10 A[2] + 20 A[3] + A[7] + A[8] + 4 A[11], 5 A[4] + 5 A[5] + A[9], 15 A[5] + 10 A[6] + A[10]], [3 A[2], 20 A[2] + 15 A[3] + A[7] + 2 A[11], 5 A[2] + 20 A[3] + 15 A[4] + 3 A[7] + 2 A[8] + 2 A[11] + A[12], 5 A[2] + 20 A[3] + 20 A[4] + 3 A[7] + 3 A[8] + 3 A[9] + 2 A[11] + 2 A[12], 10 A[5] + 3 A[10]], [2 A[2], 5 A[2] + 5 A[3] + 3 A[7] + 4 A[11], 20 A[2] + 20 A[3] + 10 A[4] + 2 A[7] + 2 A[8] + 3 A[11], 10 A[5] + 2 A[13], 10 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 20 A[6] + A[7] + A[8] + 3 A[9] + 2 A[10] + 4 A[11] + 4 A[12] + 2 A[13]], [4 A[2], 20 A[3] + 4 A[7], 20 A[4] + 20 A[5] + 4 A[8] + A[9] + 4 A[10] + 4 A[13] + A[14], 20 A[5] + 4 A[9] + 4 A[10] + A[14], 5 A[6] + 4 A[10]], [3 A[2], 15 A[3] + 20 A[4] + 20 A[5] + 2 A[8] + A[9] + 4 A[10] + 3 A[11] + 3 A[12] + 4 A[13] + A[14], 15 A[4] + 20 A[5] + 10 A[6] + 3 A[9] + 2 A[10] + 3 A[12] + 2 A[13] + 3 A[14], 15 A[5] + 15 A[6] + A[10] + 3 A[13] + 4 A[14], 3 A[14]], [ 4 A[2] + 20 A[3] + 10 A[4] + A[7] + 2 A[8] + 4 A[11] + A[16], 20 A[3] + 20 A[4] + 2 A[7] + 4 A[8] + 2 A[11] + 2 A[16], 20 A[4] + 2 A[8] + 2 A[12], 20 A[4] + 20 A[5] + 2 A[8] + A[9] + 2 A[12] + 3 A[13] + 2 A[16], 15 A[4] + 20 A[5] + 15 A[6] + 4 A[8] + 3 A[9] + 4 A[10] + 4 A[12] + 2 A[13] + 4 A[16]], [A[2], 5 A[3] + 10 A[4] + 2 A[8] + A[11] + A[16], 4 A[8] + 4 A[16], 5 A[5] + A[13], 10 A[5] + 10 A[6] + A[9] + A[10] + A[13] + A[17]], [ 2 A[2] + 20 A[3] + 10 A[4] + A[7] + 2 A[8] + 4 A[11] + A[16], 10 A[3] + 10 A[4] + 2 A[7] + A[8] + 3 A[16], 10 A[4] + 2 A[8], 10 A[5] + 10 A[6] + 2 A[9] + 2 A[10] + A[18], 5 A[6] + 2 A[10]], [ 2 A[2] + 20 A[4] + 4 A[7] + 3 A[8] + 4 A[16] + 3 A[19], 5 A[3] + A[19], 20 A[4] + 10 A[5] + 3 A[8] + 2 A[9] + 3 A[16] + A[17], 20 A[5] + 10 A[6] + 3 A[9] + A[10] + 3 A[17] + 3 A[18], 10 A[6] + 3 A[10] + 3 A[18]], [ A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 4 A[8] + 2 A[16] + 4 A[19], 20 A[4] + 3 A[7] + 4 A[8] + 2 A[16] + 4 A[19], 5 A[4] + 2 A[16], 5 A[5] + 2 A[17], 5 A[6] + 2 A[18]], [ 4 A[2] + 10 A[3] + 10 A[4] + A[7] + 2 A[8] + A[16] + 2 A[19], 10 A[3] + 20 A[4] + 4 A[8] + 2 A[16] + 2 A[19], 3 A[8] + 3 A[16] + A[20], 5 A[4] + 10 A[5] + 3 A[8] + 3 A[16] + 3 A[17] + 4 A[20], 5 A[4] + 20 A[5] + 20 A[6] + 4 A[8] + 4 A[16] + 4 A[17] + 3 A[18] + 2 A[20] + 4 A[21]], [ 3 A[2] + 10 A[3] + 20 A[4] + 15 A[5] + 5 A[6] + A[7] + 4 A[16] + 3 A[17] + A[18] + 2 A[19] + 4 A[20] + 3 A[21] + A[22], 15 A[3] + 10 A[4] + 20 A[5] + 15 A[6] + 3 A[7] + 2 A[16] + 4 A[17] + 3 A[18] + 2 A[20] + 4 A[21] + 3 A[22], 10 A[4] + 15 A[5] + 5 A[6] + 2 A[16] + 3 A[17] + A[18] + A[20] + 3 A[21] + A[22], 10 A[5] + 2 A[17] + A[21], 20 A[6] + 2 A[18] + A[22]], [4 A[2] + 20 A[3] + 15 A[4] + 5 A[5] + 10 A[6] + 2 A[7] + 3 A[16] + A[17] + 2 A[18] + 4 A[19] + 3 A[20] + A[21] + 2 A[22], 10 A[3] + 15 A[4] + 5 A[5] + 10 A[6] + 3 A[16] + A[17] + 2 A[18] + 2 A[19] + 3 A[20] + A[21] + 2 A[22], 10 A[4] + 20 A[5] + 15 A[6] + 4 A[17] + 3 A[18] + 2 A[20] + 4 A[21] + 3 A[22], 10 A[5] + 10 A[6] + 2 A[18] + 2 A[21] + 2 A[22], 2 A[22]], [2 A[2] + 5 A[4] + 10 A[5] + 20 A[6] + 4 A[7] + A[16] + 2 A[17] + 4 A[18] + 3 A[19] + A[20] + 2 A[21] + 4 A[22], 15 A[3] + 5 A[4] + 10 A[5] + 20 A[6] + A[7] + A[16] + 2 A[17] + 4 A[18] + 3 A[19] + A[20] + 2 A[21] + 4 A[22], 10 A[4] + 4 A[16], 10 A[5] + 4 A[17], 10 A[6] + 4 A[18]], [3 A[2] + 10 A[3] + 20 A[4] + 15 A[5] + 5 A[6] + A[7] + 4 A[16] + 3 A[17] + A[18] + 2 A[19] + 4 A[20] + 3 A[21] + A[22], 15 A[3] + 20 A[4] + 15 A[5] + 5 A[6] + 4 A[16] + 3 A[17] + A[18] + 4 A[19] + 4 A[20] + 3 A[21] + A[22], 15 A[4] + 5 A[5] + 10 A[6] + A[17] + 2 A[18] + 4 A[20] + A[21] + 2 A[22], 15 A[5] + 10 A[6] + 2 A[18] + 4 A[21] + 2 A[22], 20 A[6] + 4 A[22]], [ 5 A[4] + A[2] + 20 A[6] + 2 A[21] + 4 A[7] + 4 A[22] + A[16] + 2 A[17] + 3 A[19] + 10 A[5] + 4 A[18] + A[20], 15 A[4] + 5 A[3] + 10 A[6] + A[21] + A[7] + 2 A[22] + 3 A[16] + A[17] + 5 A[5] + 2 A[18] + 3 A[20], 15 A[4] + 20 A[6] + 2 A[21] + 4 A[22] + 4 A[16] + 2 A[17] + 10 A[5] + 4 A[18] + 2 A[20], 10 A[6] + 2 A[21] + 2 A[22] + 4 A[17] + 15 A[5] + 2 A[18], 15 A[6] + 2 A[22] + 4 A[18]], [3 A[2], 10 A[4] + 15 A[6] + 4 A[21] + 4 A[7] + 3 A[22] + 2 A[16] + 4 A[17] + 2 A[19] + 20 A[5] + 3 A[18] + 2 A[20], 5 A[4] + 20 A[6] + 2 A[21] + 4 A[22] + A[16] + 2 A[17] + 10 A[5] + 4 A[18], 20 A[6] + 4 A[22] + A[17] + 5 A[5] + 4 A[18], 15 A[6] + A[18]], [2 A[2], 15 A[4] + 5 A[3] + 10 A[6] + A[21] + 2 A[22] + 3 A[16] + A[17] + A[19] + 5 A[5] + 2 A[18] + 3 A[20], 5 A[4] + 20 A[6] + 2 A[21] + 4 A[22] + 2 A[17] + 10 A[5] + 4 A[18] + A[20], 20 A[6] + A[21] + 4 A[22] + 5 A[5] + 4 A[18], 15 A[6] + A[22]], [4 A[2], 20 A[3] + 4 A[7], 15 A[4] + A[16] + 3 A[20], 10 A[6] + 3 A[21] + 2 A[22] + A[17] + 15 A[5] + 2 A[18], 3 A[22] + A[18]], [15 A[4] + 20 A[3] + A[2] + 10 A[6] + A[21] + 2 A[7] + 2 A[22] + 3 A[16] + A[17] + 4 A[19] + 5 A[5] + 2 A[18] + 3 A[20], 15 A[4] + 10 A[3] + 10 A[6] + A[21] + 2 A[22] + 3 A[16] + A[17] + 3 A[19] + 5 A[5] + 2 A[18] + 3 A[20], 10 A[4] + 20 A[6] + 2 A[21] + 4 A[22] + 2 A[17] + 10 A[5] + 4 A[18] + 3 A[20], 20 A[6] + 3 A[21] + 4 A[22] + 10 A[5] + 4 A[18], 20 A[6] + 3 A[22]], [2 A[2], 10 A[3] + 2 A[7], 20 A[4] + 3 A[16] + 4 A[20], 5 A[6] + 4 A[21] + A[22] + 3 A[17] + 20 A[5] + A[18], 4 A[22] + 3 A[18]], [ 10 A[4] + 15 A[3] + 3 A[2] + 15 A[6] + 4 A[21] + 3 A[7] + 3 A[22] + 2 A[16] + 4 A[17] + A[19] + 20 A[5] + 3 A[18] + 2 A[20], 5 A[4] + 15 A[3] + 20 A[6] + 2 A[21] + 3 A[7] + 4 A[22] + A[16] + 2 A[17] + 10 A[5] + 4 A[18] + A[20], 10 A[4] + 15 A[6] + 4 A[21] + 3 A[22] + 2 A[16] + 4 A[17] + 20 A[5] + 3 A[18] + A[20], 10 A[6] + A[21] + 2 A[22] + 2 A[17] + 10 A[5] + 2 A[18], A[22] + 2 A[18]]], [1, 1, 1, 13, 12, 14, 1, 3, 7, 19, 16, 3, 2, 4, 3, 19, 1, 17, 12, 6, 19, 18, 9, 12, 18, 11, 3, 2, 4, 6, 2, 23]] For example, C(100000), mudolo , 25, equals , 13 The congruence classes mod, 25, in the following set , {0, 5, 8, 10, 15, 20, 21, 22, 24}, never show up! Theorem Number, 36, : Let C(n) be the constant term, in x, of n (3/x + 1 + 3 x) For the record, the first 15 terms of the sequence are: 1, 19, 55, 595, 2611, 22141, 119449, 902035, 5420035, 38712169, 246360709, 1714206781, 11255897485, 77419522675, 517370395015 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 15, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [A[2], 5 A[2] + 11 A[3], A[11], A[12], A[13]], [19 A[2], 20 A[2] + 4 A[3], 15 A[3] + 14 A[4], A[14], A[15]], [5 A[2], 10 A[3], 15 A[4], 20 A[5], 0], %1, [11 A[2], 5 A[2] + 21 A[3], 10 A[3] + 6 A[4], 15 A[4] + 16 A[5], 20 A[5] + A[6]], [19 A[2], 20 A[2] + 4 A[3], 15 A[3] + 14 A[4], 10 A[4] + 24 A[5], 5 A[5] + 9 A[6]], [15 A[2], 20 A[3], 0, 5 A[5], 10 A[6]], %1, [9 A[2], 20 A[2] + 19 A[3], 15 A[3] + 4 A[4], 10 A[4] + 14 A[5], 5 A[5] + 24 A[6]], [15 A[2], 20 A[3], 0, 5 A[5], 10 A[6]], %1, [10 A[2], 5 A[3], 0, 20 A[5], 15 A[6]], [5 A[2], 15 A[3], 0, 10 A[5], 20 A[6]]], [1, 1, 1, 19, 5, 20, 11, 19, 15, 20, 9, 15, 20, 10, 5]] %1 := [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]] For example, C(100000), mudolo , 25, equals , 9 The congruence classes mod, 25, in the following set , {2, 3, 4, 6, 7, 8, 12, 13, 14, 16, 17, 18, 21, 22, 23, 24}, never show up! Theorem Number, 37, : Let C(n) be the constant term, in x, of n (3/x + 2) For the record, the first 15 terms of the sequence are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], 20 A[2] + 22 A[3], A[11], A[12], A[13]], [4 A[2], 20 A[2] + 4 A[3], 15 A[3] + 4 A[4], A[14], A[15]], [8 A[2], 5 A[2] + 13 A[3], 10 A[3] + 18 A[4], 15 A[4] + 23 A[5], A[16]], [ 16 A[2], 20 A[2] + 21 A[3], 15 A[3] + A[4], 10 A[4] + 6 A[5], 5 A[5] + 11 A[6]], [7 A[2], 20 A[2] + 2 A[3], 15 A[3] + 22 A[4], 10 A[4] + 17 A[5], 5 A[5] + 12 A[6]], [24 A[2], 20 A[2] + 24 A[3], 15 A[3] + 24 A[4], 10 A[4] + 24 A[5], 5 A[5] + 24 A[6]], [18 A[2], 5 A[2] + 23 A[3], 10 A[3] + 3 A[4], 15 A[4] + 8 A[5], 20 A[5] + 13 A[6]], [ A[2], 20 A[2] + 6 A[3], 15 A[3] + 11 A[4], 10 A[4] + 16 A[5], 5 A[5] + 21 A[6]], [23 A[2], 15 A[2] + 23 A[3], 5 A[3] + 23 A[4], 20 A[4] + 23 A[5], 10 A[5] + 23 A[6]], [11 A[2], 10 A[2] + 21 A[3], 20 A[3] + 6 A[4], 5 A[4] + 16 A[5], 15 A[5] + A[6]], [2 A[2], 15 A[2] + 12 A[3], 5 A[3] + 22 A[4], 20 A[4] + 7 A[5], 10 A[5] + 17 A[6]], [22 A[2], 20 A[2] + 17 A[3], 15 A[3] + 12 A[4], 10 A[4] + 7 A[5], 5 A[5] + 2 A[6]], [4 A[2], 5 A[2] + 24 A[3], 10 A[3] + 19 A[4], 15 A[4] + 14 A[5], 20 A[5] + 9 A[6]], [8 A[2], 10 A[2] + 23 A[3], 20 A[3] + 13 A[4], 5 A[4] + 3 A[5], 15 A[5] + 18 A[6]]], [1, 1, 2, 4, 8, 16, 7, 24, 18, 1, 23, 11, 2, 22, 4, 8]] For example, C(100000), mudolo , 25, equals , 1 The congruence classes mod, 25, in the following set , {0, 3, 5, 6, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21}, never show up! Theorem Number, 38, : Let C(n) be the constant term, in x, of n (3/x + 2 + x) For the record, the first 15 terms of the sequence are: 2, 10, 44, 214, 1052, 5284, 26840, 137638, 710828, 3692140, 19266920, 100932220, 530479640, 2795917960, 14771797424 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 28, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], A[11], A[12], A[13], A[14]], [10 A[2], 5 A[2] + 5 A[3], 10 A[3], A[15], A[16]], [19 A[2], A[17], A[18], A[19], A[20]], [14 A[2], A[21], A[22], A[23], A[24]], [2 A[2], 10 A[2] + 10 A[3] + 2 A[7], A[12], A[13], A[14]], [15 A[2], 5 A[2] + 10 A[3], 10 A[3] + 5 A[4], 15 A[4], 20 A[5] + 20 A[6]], [24 A[2], 20 A[2] + 10 A[3] + 4 A[7], 15 A[3] + 4 A[8], 10 A[4] + 15 A[5] + 4 A[9], A[28]], [24 A[2], 20 A[3] + 4 A[7], 20 A[4] + 4 A[8], 20 A[5] + 4 A[9], 20 A[6] + 4 A[10]], [9 A[2], 10 A[2] + 15 A[3] + 2 A[7] + A[11], 20 A[2] + 20 A[4] + 4 A[7] + 4 A[8] + 3 A[11], 10 A[4] + 15 A[5] + 4 A[9], 5 A[5] + 10 A[6] + 4 A[10]], [20 A[2], 15 A[2] + 10 A[3] + A[7] + 2 A[11], 10 A[2] + 20 A[3] + 2 A[7] + 4 A[11], 5 A[2] + 10 A[3] + 10 A[4] + 15 A[5] + A[7] + A[8] + 2 A[11] + 2 A[12], 15 A[5] + 5 A[6]], [3 A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 2 A[8] + 4 A[9] + 4 A[11] + 4 A[12] + 3 A[13], 15 A[3] + 15 A[5] + 4 A[8] + 3 A[9] + 4 A[11] + 3 A[12] + A[13], 15 A[4] + 4 A[12], 5 A[5] + 4 A[13], 10 A[5] + 10 A[6] + 3 A[10]], [3 A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 2 A[8] + 4 A[9] + 4 A[11] + 4 A[12] + 3 A[13], 15 A[3] + 3 A[7], 15 A[4] + 3 A[8], 15 A[5] + 3 A[9], 15 A[6] + 3 A[10]], [ 15 A[5] + 4 A[7] + 4 A[8] + 3 A[9] + 3 A[11] + 3 A[12] + A[13], 20 A[4] + 2 A[8] + 4 A[9] + 4 A[12] + 3 A[13], 15 A[5] + 3 A[9] + A[13], 15 A[6] + A[10] + 2 A[14], 5 A[6]], [ 20 A[3] + 20 A[4] + 2 A[7] + 2 A[8] + 4 A[9] + 4 A[11] + 4 A[12] + 3 A[13], 10 A[4] + 20 A[5] + A[8] + 2 A[9] + 2 A[12] + 4 A[13], 4 A[9] + 3 A[13], 20 A[6] + 2 A[16], 15 A[6]], [3 A[2] + 3 A[7] + 3 A[11] + 4 A[17], 15 A[3] + 3 A[11] + 3 A[17], 10 A[3] + 15 A[4] + 3 A[8] + 3 A[11] + A[17], 15 A[3] + 15 A[5] + 20 A[6] + 4 A[8] + 3 A[9] + 4 A[11] + 3 A[12] + 2 A[16] + 3 A[17], 3 A[10] + 3 A[16]], [15 A[3] + 2 A[7] + 2 A[11] + A[17], 15 A[3] + A[7] + 3 A[11] + 2 A[17], 15 A[3] + 4 A[11] + 3 A[17], 10 A[3] + 10 A[4] + 15 A[6] + 2 A[8] + 2 A[11] + 4 A[16] + 4 A[17] + 2 A[18], 4 A[16]], [A[2] + 15 A[3] + 10 A[4] + 20 A[5] + 10 A[6] + 3 A[7] + 2 A[8] + 4 A[9] + A[16] + 3 A[17] + 2 A[18] + 4 A[19], 5 A[3] + 10 A[4] + 20 A[5] + 10 A[6] + 2 A[8] + 4 A[9] + A[16] + 4 A[17] + 2 A[18] + 4 A[19], 5 A[4] + 15 A[5] + 20 A[6] + 3 A[9] + 2 A[16] + 4 A[18] + 3 A[19], 5 A[5] + 15 A[6] + 4 A[16] + 4 A[19], 10 A[6] + A[10] + A[16]], [A[2] + 10 A[3] + 15 A[4] + 5 A[5] + 15 A[6] + 2 A[7] + 3 A[8] + A[9] + 4 A[16] + 2 A[17] + 3 A[18] + A[19], 5 A[3] + 20 A[4] + 15 A[5] + 20 A[6] + A[7] + 4 A[8] + 3 A[9] + 2 A[16] + 4 A[18] + 3 A[19], 5 A[4] + 5 A[5] + 15 A[6] + A[8] + A[9] + 4 A[16] + A[19], 5 A[5] + 5 A[6] + A[9] + 3 A[16], 15 A[6] + 4 A[16] + 4 A[20]], [3 A[2], 10 A[3] + A[7] + 2 A[17] + A[21], 15 A[3] + 10 A[4] + 3 A[7] + A[8] + 3 A[18] + 3 A[21], 5 A[5] + 20 A[6] + 2 A[16] + 2 A[19], 5 A[6] + 2 A[20]], [ 15 A[3] + 2 A[7] + 3 A[17] + 4 A[21], 5 A[3] + A[7] + A[17], 15 A[3] + 5 A[4] + 3 A[7] + 2 A[18] + 3 A[21] + 3 A[22], 15 A[3] + 5 A[4] + 3 A[7] + A[18] + 3 A[21] + 4 A[22], 20 A[6] + 4 A[16]], [%1, 4 A[7] + 4 A[17] + 4 A[21], 20 A[3] + 5 A[4] + 4 A[7] + 2 A[18] + 4 A[21] + 2 A[22], 5 A[5] + 15 A[6] + 4 A[16] + 4 A[19], 10 A[6] + 4 A[20]], [%1, 5 A[3] + 4 A[21], 5 A[4] + 4 A[22], 15 A[3] + 5 A[4] + 5 A[5] + 3 A[7] + A[18] + 4 A[19] + 3 A[21] + 4 A[22], 10 A[6] + 4 A[16] + 4 A[20]], [20 A[3] + 4 A[7] + A[17] + 3 A[21], 5 A[3] + 3 A[17] + 2 A[21], 5 A[3] + A[7] + A[21], 20 A[3] + 5 A[4] + 20 A[6] + 4 A[7] + 2 A[16] + 3 A[18] + 4 A[21] + 2 A[22] , 2 A[16]], [15 A[3] + 3 A[2] + A[21] + 3 A[7] + 2 A[17], 5 A[3] + 2 A[17], 5 A[4] + 2 A[18], 2 A[19] + 5 A[5], 5 A[6] + 2 A[20]], [ 15 A[3] + 3 A[2] + A[21] + 3 A[7] + 2 A[17], 15 A[3] + 3 A[7], 5 A[4] + 15 A[3] + 3 A[21] + 3 A[7] + A[22] + A[18], 5 A[4] + 20 A[3] + 5 A[6] + 4 A[21] + 4 A[7] + 2 A[22] + 3 A[16] + 2 A[19] + 5 A[5] + 3 A[18], 5 A[6] + 2 A[16] + 2 A[20]], [%1, 5 A[3] + 4 A[21], 5 A[4] + 4 A[22], 15 A[3] + 5 A[4] + 5 A[5] + 3 A[7] + A[18] + 4 A[19] + 3 A[21] + 4 A[22], 10 A[6] + 4 A[16] + 4 A[20]]], [1, 1, 2, 10, 19, 14, 2, 15, 24, 24, 9, 20, 18, 18, 5, 15, 13, 15, 11, 16, 3, 10, 11, 11, 20, 18, 18, 11]] %1 := A[2] + 15 A[3] + 2 A[7] + 3 A[17] + 4 A[21] For example, C(100000), mudolo , 25, equals , 0 The congruence classes mod, 25, in the following set , {0, 4, 6, 7, 8, 12, 17, 21, 22, 23}, never show up! Theorem Number, 39, : Let C(n) be the constant term, in x, of n (3/x + 2 + 2 x) For the record, the first 15 terms of the sequence are: 2, 16, 80, 520, 3152, 20224, 129152, 838240, 5462720, 35846656, 236191232, 1562588416, 10370408960, 69019648000, 460456939520 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 24, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], A[11], A[12], A[13], A[14]], [16 A[2], A[7], A[16], A[17], A[18]], [5 A[2], 20 A[3], 10 A[4], 0, A[19]], %3, [2 A[2], 10 A[3] + 2 A[7], A[12], A[13], A[14]], [6 A[2], 15 A[3] + A[7], A[8], A[23], A[24]], %3, %3, [24 A[2], 20 A[3] + A[7] + 4 A[11], 10 A[4] + 4 A[8], 5 A[5] + 4 A[9], 4 A[10]], [ 7 A[2], 15 A[3] + A[7] + 3 A[11], 5 A[4] + 4 A[8] + 4 A[12], 15 A[5] + 2 A[9], 10 A[6] + 2 A[10]], [15 A[2], %2, 0, 5 A[5], 10 A[6]], [15 A[2], %2, 0, 5 A[5], 10 A[6]], [2 A[2], 10 A[3] + 2 A[7], 5 A[4] + 4 A[8] + 4 A[12], 5 A[5] + 2 A[9], 15 A[6] + 2 A[10]], [ 16 A[2], 3 A[7] + 4 A[11], 3 A[8] + 3 A[16], 20 A[5] + A[9], 5 A[6] + A[10] ], [20 A[2], %1, 0, 20 A[5] + 3 A[9] + 2 A[17], 5 A[6]], [20 A[2], %1, 0, 20 A[5] + 3 A[9] + 2 A[17], 5 A[6]], 0, [7 A[2], 15 A[3] + A[7] + 3 A[11], 20 A[4] + 3 A[16] + 2 A[20], 5 A[5] + 3 A[9] + 4 A[17], 2 A[18]], [15 A[2], %2, 0, 4 A[17] + 3 A[21], 10 A[6]], [15 A[2], %2, 0, 4 A[17] + 3 A[21], 10 A[6]], [20 A[2], %1, 0, 20 A[5] + 2 A[17] + 4 A[21], 5 A[6]], [20 A[2], %1, 0, 20 A[5] + 2 A[17] + 4 A[21], 5 A[6]]], [1, 1, 2, 16, 5, 20, 2, 6, 20, 20, 24, 7, 15, 15, 2, 16, 20, 20, 0, 7, 15, 15, 20, 20]] %1 := 10 A[3] + A[7] + 2 A[11] %2 := 20 A[3] + 2 A[7] + 4 A[11] %3 := [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]] For example, C(100000), mudolo , 25, equals , 9 The congruence classes mod, 25, in the following set , {3, 4, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 21, 22, 23}, never show up! Theorem Number, 40, : Let C(n) be the constant term, in x, of n (3/x + 2 + 3 x) For the record, the first 15 terms of the sequence are: 2, 22, 116, 934, 6332, 48124, 352424, 2669062, 20107628, 153277972, 1170192344, 8981891164, 69111416792, 533463087928, 4126851588176 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 16, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [2 A[2], 5 A[2] + 22 A[3], A[11], A[12], A[13]], [22 A[2], 10 A[2] + 7 A[3], 20 A[3] + 17 A[4], A[14], A[15]], [16 A[2], 15 A[2] + 11 A[3], 5 A[3] + 6 A[4], 20 A[4] + A[5], A[16]], [ 9 A[2], 5 A[2] + 24 A[3], 10 A[3] + 14 A[4], 15 A[4] + 4 A[5], 20 A[5] + 19 A[6]], [7 A[2], 5 A[2] + 2 A[3], 10 A[3] + 22 A[4], 15 A[4] + 17 A[5], 20 A[5] + 12 A[6]], [22 A[2], 10 A[2] + 7 A[3], 20 A[3] + 17 A[4], 5 A[4] + 2 A[5], 15 A[5] + 12 A[6]], [A[2], 15 A[2] + 21 A[3], 5 A[3] + 16 A[4], 20 A[4] + 11 A[5], 10 A[5] + 6 A[6]], [19 A[2], 5 A[2] + 9 A[3], 10 A[3] + 24 A[4], 15 A[4] + 14 A[5], 20 A[5] + 4 A[6]], [19 A[2], 20 A[2] + 14 A[3], 15 A[3] + 9 A[4], 10 A[4] + 4 A[5], 5 A[5] + 24 A[6]], [22 A[2], 5 A[2] + 12 A[3], 10 A[3] + 2 A[4], 15 A[4] + 17 A[5], 20 A[5] + 7 A[6]], [8 A[2], 10 A[2] + 13 A[3], 20 A[3] + 18 A[4], 5 A[4] + 23 A[5], 15 A[5] + 3 A[6]], [17 A[2], 5 A[2] + 7 A[3], 10 A[3] + 22 A[4], 15 A[4] + 12 A[5], 20 A[5] + 2 A[6]], [23 A[2], 10 A[2] + 3 A[3], 20 A[3] + 8 A[4], 5 A[4] + 13 A[5], 15 A[5] + 18 A[6]], [24 A[2], 5 A[2] + 14 A[3], 10 A[3] + 4 A[4], 15 A[4] + 19 A[5], 20 A[5] + 9 A[6]]], [1, 1, 2, 22, 16, 9, 7, 22, 1, 19, 19, 22, 8, 17, 23, 24]] For example, C(100000), mudolo , 25, equals , 13 The congruence classes mod, 25, in the following set , {0, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 18, 20, 21}, never show up! Theorem Number, 41, : Let C(n) be the constant term, in x, of n (3/x + 3 + x) For the record, the first 15 terms of the sequence are: 3, 15, 81, 459, 2673, 15849, 95175, 576963, 3523257, 21640365, 133549155, 827418645, 5143397535, 32063180535, 200367960201 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 27, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], A[11], A[12], A[13], A[14]], [15 A[2], 20 A[2] + 10 A[3], 15 A[3] + 5 A[4], 10 A[4], A[15]], [6 A[2], A[16], A[17], A[18], A[19]], [9 A[2], A[20], A[21], A[22], A[23]], [23 A[2], 10 A[2] + 10 A[3] + 3 A[7], A[24], A[25], A[26]], [15 A[2], 20 A[2] + 10 A[3], 15 A[3] + 5 A[4], 10 A[4], 5 A[5] + 20 A[6]], [A[2], 20 A[2] + 10 A[3] + A[7], 15 A[3] + 20 A[4] + A[8], 10 A[4] + 5 A[5] + A[9], A[27]], [24 A[2], 20 A[3] + 4 A[7], 20 A[4] + 4 A[8], 20 A[5] + 4 A[9], 20 A[6] + 4 A[10]], [24 A[2], 15 A[2] + 20 A[3] + 2 A[7] + 4 A[11], 5 A[2] + 20 A[3] + 10 A[4] + 4 A[7] + 4 A[8] + 2 A[11], 15 A[4] + 5 A[5] + 4 A[9], 20 A[5] + 4 A[10]], [5 A[2], 5 A[2] + 10 A[3] + A[7] + 3 A[11], 10 A[2] + 20 A[3] + 3 A[7] + 4 A[11], 10 A[2] + 20 A[3] + 20 A[4] + 10 A[5] + 3 A[7] + 3 A[8] + 4 A[11] + 4 A[12] , 15 A[5] + 20 A[6]], [23 A[2], 15 A[2] + 15 A[3] + 2 A[7] + 2 A[11], 15 A[2] + 10 A[3] + 15 A[4] + 2 A[7] + 2 A[8] + A[11] + 2 A[12], 15 A[2] + 10 A[3] + 10 A[4] + 15 A[5] + 2 A[7] + 2 A[8] + 2 A[9] + A[11] + A[12] + 2 A[13], 10 A[2] + 20 A[3] + 20 A[4] + 20 A[5] + 15 A[6] + 3 A[7] + 3 A[8] + 3 A[9] + 3 A[10] + 4 A[11] + 4 A[12] + 4 A[13]], [2 A[2], 10 A[3] + 20 A[4] + 20 A[5] + 20 A[6] + 2 A[7] + 4 A[8] + 4 A[9] + 4 A[10] + 2 A[12] + 2 A[13] + 2 A[14], 10 A[4] + 10 A[5] + 10 A[6] + 2 A[8] + A[9] + A[10] + 3 A[13] + 3 A[14], 10 A[5] + 10 A[6] + 2 A[9] + 2 A[10] + A[14], 2 A[10]], [10 A[3] + 10 A[4] + 10 A[5] + 10 A[6] + 2 A[7] + 2 A[8] + 2 A[9] + 2 A[10] + A[11] + A[12] + A[13] + A[14], 10 A[4] + 10 A[5] + 10 A[6] + A[8] + A[9] + A[10] + 3 A[12] + 3 A[13] + 3 A[14], 20 A[5] + 20 A[6] + 4 A[9] + 4 A[10] + 2 A[13] + 2 A[14], 20 A[6] + 3 A[10] + 4 A[14], 10 A[6]], [ 3 A[2] + 5 A[3] + 2 A[7] + 3 A[11] + 4 A[16], 15 A[3] + 4 A[11] + A[16], 20 A[3] + 15 A[4] + 3 A[8] + 2 A[11] + 4 A[16], 20 A[3] + 10 A[4] + 15 A[5] + A[8] + A[9] + 4 A[11] + 3 A[12] + 4 A[13] + 3 A[16], 20 A[3] + 20 A[4] + 20 A[5] + 20 A[6] + 3 A[8] + 3 A[9] + 3 A[10] + 2 A[11] + 4 A[12] + 4 A[13] + 4 A[16]], [ 20 A[3] + 20 A[4] + 2 A[7] + 3 A[8] + 3 A[16] + 2 A[17], 20 A[3] + 20 A[4] + 4 A[7] + 4 A[8] + A[16] + A[17], 20 A[4] + 4 A[8] + A[17], 20 A[5] + 4 A[9] + 2 A[13], 10 A[4] + 10 A[5] + 15 A[6] + A[9] + 3 A[12] + 3 A[13] + A[17]], [ A[2] + 20 A[3] + 20 A[4] + 2 A[7] + 3 A[8] + 3 A[16] + 2 A[17], 5 A[3] + A[16], 5 A[4] + A[17], 5 A[5] + A[18], 10 A[4] + 20 A[5] + 2 A[9] + A[10] + A[12] + 2 A[17] + 3 A[18]], [ 4 A[2] + 20 A[3] + 20 A[4] + A[7] + 4 A[8] + 4 A[16] + A[17], 20 A[3] + 20 A[4] + 4 A[7] + A[8] + 4 A[17], 20 A[4] + 20 A[5] + 10 A[6] + 4 A[8] + 4 A[9] + 2 A[10] + A[18] + 3 A[19], 20 A[5] + 5 A[6] + 4 A[9] + 3 A[10] + 2 A[19], 5 A[6] + 4 A[10]], [ 2 A[2] + 2 A[7] + 4 A[16] + A[20], 10 A[3] + 2 A[16], 10 A[4] + 20 A[5] + 5 A[6] + A[9] + 3 A[10] + 2 A[17] + 4 A[18] + 2 A[19], 10 A[5] + 4 A[10] + 2 A[18] + A[19], 5 A[6] + 2 A[19]], [ 5 A[3] + 4 A[7] + 3 A[16] + 2 A[20], 5 A[3] + 3 A[7] + 4 A[16] + 2 A[20], 5 A[3] + 15 A[4] + A[7] + 3 A[17] + A[20] + 3 A[21], 20 A[3] + 5 A[4] + 4 A[7] + A[17] + 4 A[20] + A[21], 15 A[3] + 10 A[4] + 20 A[5] + 5 A[6] + 3 A[7] + 3 A[9] + 2 A[17] + 2 A[18] + 3 A[20] + 2 A[21]], [4 A[2] + 2 A[7] + 4 A[16] + A[20], A[7] + 4 A[16] + A[20], 20 A[3] + 5 A[4] + 4 A[7] + 4 A[20] + A[21], 5 A[3] + 20 A[4] + 20 A[5] + A[7] + A[9] + 4 A[17] + 3 A[18] + A[20] + 4 A[21], 10 A[6] + 4 A[19]], [ A[2] + 15 A[3] + A[7] + 2 A[16] + 3 A[20], 5 A[3] + 4 A[20], 5 A[4] + 4 A[21], 20 A[3] + 5 A[4] + 5 A[5] + 4 A[7] + A[17] + A[18] + 4 A[20] + A[21], 5 A[6] + 4 A[23]], [2 A[7] + 4 A[16] + A[20], 15 A[3] + 3 A[16] + 3 A[20], 15 A[3] + 3 A[7] + 3 A[20], 15 A[3] + 10 A[4] + 15 A[6] + 3 A[7] + 2 A[17] + 2 A[19] + 3 A[20] + 2 A[21] + 2 A[23], 5 A[6] + 3 A[19] + 3 A[23]], [ 3 A[2] + 15 A[3] + A[7] + 2 A[16] + 3 A[20], 10 A[3] + 4 A[7] + 3 A[16] + 4 A[20], 5 A[3] + 15 A[4] + A[7] + 2 A[17] + A[20] + 4 A[21], 15 A[5] + 15 A[6] + 3 A[18] + 2 A[19] + 2 A[23], 3 A[19]], [ 2 A[2] + 2 A[7] + 4 A[16] + A[20], 5 A[3] + 3 A[20], 5 A[4] + 3 A[21], 10 A[4] + 15 A[3] + 2 A[21] + 3 A[7] + 2 A[17] + 10 A[5] + 2 A[18] + 3 A[20], 5 A[6] + 3 A[23]], [ 5 A[3] + 4 A[2] + 4 A[7] + 3 A[16] + 2 A[20], 5 A[3] + A[20], 5 A[4] + A[21], 20 A[4] + 5 A[3] + 4 A[21] + A[7] + 4 A[17] + 20 A[5] + 4 A[18] + A[20], 5 A[6] + A[23]]], [1, 1, 3, 15, 6, 9, 23, 15, 1, 24, 24, 5, 23, 2, 10, 13, 10, 11, 9, 7, 10, 9, 16, 5, 18, 7, 14]] For example, C(100000), mudolo , 25, equals , 0 The congruence classes mod, 25, in the following set , {0, 4, 8, 12, 17, 19, 20, 21, 22}, never show up! Theorem Number, 42, : Let C(n) be the constant term, in x, of n (3/x + 3 + 2 x) For the record, the first 15 terms of the sequence are: 3, 21, 135, 945, 6723, 48789, 358263, 2655585, 19825155, 148853781, 1122869223, 8503237521, 64604559555, 492221474325, 3759348384855 We are interested in C(n) modulo , 25 Then there is a Linear Scheme to compute it in linear-time (in bit-size, i.e\ . log-time in n) with , 23, states . Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3 A[2], A[11], A[12], A[13], A[14]], [21 A[2], A[15], A[16], A[9], A[18]], [10 A[2], 20 A[3], 5 A[4], 15 A[5], 0], [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]], [23 A[2], 10 A[3] + 3 A[7], A[19], A[20], A[21]], [6 A[2], 15 A[3] + A[7], A[8], A[22], A[23]], [5 A[2], 15 A[3], 0, 10 A[5], 20 A[6]], [20 A[2], 10 A[3], 0, 15 A[5], 5 A[6]], [14 A[2], 20 A[3] + 3 A[7] + 2 A[11], 4 A[8], 20 A[5] + 4 A[9], 15 A[6] + 4 A[10]], [ 23 A[2], 15 A[3] + 2 A[7] + 2 A[11], 15 A[4] + 2 A[8] + 2 A[12], 10 A[5] + 3 A[9], 15 A[6] + 3 A[10]], [15 A[2], 20 A[3] + 3 A[7] + 4 A[11], 0, 10 A[5] + A[9] + 3 A[13], 10 A[6]] , [10 A[2], 10 A[3] + 2 A[7] + A[11], 0, 20 A[5] + 4 A[9] + 2 A[13], 15 A[6]] , [13 A[2], 15 A[3] + 2 A[7] + 2 A[11], 15 A[4] + 3 A[8], 15 A[5] + A[9] + 4 A[13], 20 A[6] + 3 A[10]], [ 21 A[2], 5 A[3] + A[7], 2 A[8] + 4 A[16], 5 A[5] + 2 A[13], 10 A[6] + A[10] ], [5 A[2], 10 A[3] + A[7] + 3 A[11], 0, 10 A[5] + A[13] + 2 A[17], 20 A[6]], [ 20 A[2], 20 A[3] + 4 A[7] + 2 A[11], 0, 20 A[5] + 4 A[13] + 3 A[17], 5 A[6] ], [18 A[2], 5 A[3] + A[11], 15 A[4] + 2 A[16] + 2 A[19], 15 A[5] + 4 A[13] + A[17], 5 A[6] + 3 A[18]], [ 15 A[2], 20 A[3] + 3 A[7] + 4 A[11], 0, 10 A[5] + 2 A[17] + A[20], 10 A[6]] , [ 10 A[2], 10 A[3] + 2 A[7] + A[11], 0, 20 A[5] + 3 A[17] + 4 A[20], 15 A[6]] , [5 A[2], 10 A[3] + A[7] + 3 A[11], 0, 20 A[5] + 4 A[17] + 2 A[20], 20 A[6]] , [20 A[2], 20 A[3] + 4 A[7] + 2 A[11], 0, 10 A[5] + A[17] + 3 A[20], 5 A[6]] ], [1, 1, 3, 21, 10, 20, 23, 6, 5, 20, 14, 23, 15, 10, 13, 21, 5, 20, 18, 15, 10, 5, 20]] For example, C(100000), mudolo , 25, equals , 9 The congruence classes mod, 25, in the following set , {2, 4, 7, 8, 9, 11, 12, 16, 17, 19, 22, 24}, never show up! ------------------------------------------ This ends this fascinating book that took, 13.406, to generate. ----------------------------------------- The whole thing took, 39.966, seconds.