All the Finite Automata whose size is less than, 500, for determining the, Motzkin, numbers modulo prime and prime powers for the first, 4, primes and powers <=, 3 By Shalosh B. Ekhad Recall that the, Motzkin, numbers, let's call them A(n), may defined as the constant term of n 2 (1/x + 1 + x) (-x + 1) We are interested in generating automata for computing super-fast A(n) modul\ o primes, and modulo prime powers allowing automata with at most, 500, states. for the first, 3, primes and as high powers as we can afford with our limit of, 500, states. Thanks to the Chinese Remainder Theorem, this would enable us to determine A\ (n) modulo many composite numbers as well. -------------------------------------- For A(n) modulo, 2, there is an automata with 4, states Here it is: [[[2, 2], [3, 4], [3, 3], [0, 2]], [1, 1, 1, 0]] Just for fun, , A(100000000000000000000), modulo , 2, is , 1 and the next, 10, terms in the sequence A(n) modulo, 2, are 1, 0, 0, 1, 1, 1, 1, 1, 1, 0 -------------------------------------- For A(n) modulo, 4, there is an automata with 24, states Here it is: [[[3, 2], [4, 5], [4, 6], [7, 8], [9, 10], [11, 12], [7, 13], [7, 14], [0, 15], [16, 5], [17, 0], [18, 19], [20, 21], [20, 21], [17, 22], [20, 23], [17, 22], [ 20, 23], [11, 12], [20, 24], [7, 14], [17, 22], [20, 21], [7, 14]], [1, 1, 1, 1 , 0, 2, 1, 1, 0, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 1, 2, 3, 1]] Just for fun, , A(100000000000000000000), modulo , 4, is , 1 and the next, 10, terms in the sequence A(n) modulo, 4, are 1, 2, 0, 1, 1, 3, 3, 3, 3, 0 -------------------------------------- For A(n) modulo, 8, there is an automata with 128, states Here it is: [[[3, 2], [4, 5], [7, 6], [8, 9], [10, 11], [12, 13], [8, 14], [15, 16], [17, 18], [19, 20], [21, 22], [23, 24], [25, 26], [27, 28], [15, 29], [30, 31], [32, 33], [34, 35], [36, 37], [38, 39], [40, 41], [42, 11], [43, 44], [45, 46], [47, 48], [49, 50], [15, 51], [52, 53], [54, 55], [56, 57], [58, 59], [32, 60], [61, 62], [56, 57], [63, 64], [36, 65], [66, 67], [68, 69], [70, 71], [72, 73], [74, 75], [76, 77], [43, 78], [79, 80], [36, 65], [66, 67], [56, 81], [82, 83], [23, 24], [84, 85], [86, 87], [72, 88], [89, 90], [56, 57], [58, 59], [56, 91], [92, 93], [32, 33], [94, 95], [96, 97], [72, 88], [98, 99], [32, 33], [94, 95], [100 , 101], [36, 65], [100, 101], [68, 69], [102, 103], [43, 78], [104, 105], [72, 106], [107, 108], [72, 88], [109, 110], [36, 37], [38, 39], [111, 112], [68, 69 ], [113, 114], [115, 116], [72, 88], [109, 110], [47, 48], [49, 50], [56, 57], [58, 59], [117, 118], [15, 51], [119, 120], [121, 122], [15, 51], [123, 124], [ 56, 57], [63, 64], [72, 88], [98, 99], [15, 51], [119, 120], [36, 65], [100, 101], [43, 78], [104, 105], [68, 69], [113, 114], [125, 126], [32, 33], [127, 128], [15, 51], [119, 120], [68, 69], [113, 114], [43, 78], [104, 105], [15, 51 ], [123, 124], [32, 33], [127, 128], [72, 88], [89, 90], [15, 51], [123, 124], [72, 88], [89, 90], [32, 33], [127, 128], [56, 57], [63, 64]], [1, 1, 1, 1, 4, 2, 1, 1, 5, 4, 7, 2, 3, 1, 1, 3, 5, 3, 4, 6, 7, 4, 2, 4, 3, 2, 1, 7, 3, 3, 5, 5 , 7, 3, 5, 4, 4, 6, 2, 7, 7, 4, 2, 6, 4, 4, 3, 7, 2, 3, 3, 7, 1, 3, 5, 3, 1, 5, 3, 7, 7, 1, 5, 3, 4, 4, 4, 6, 2, 2, 6, 7, 5, 7, 1, 4, 6, 6, 6, 2, 1, 7, 1, 3, 2 , 3, 5, 5, 1, 7, 1, 1, 7, 3, 5, 7, 1, 1, 7, 4, 4, 2, 6, 6, 2, 5, 5, 3, 1, 7, 6, 2, 2, 6, 1, 7, 5, 3, 7, 1, 1, 7, 7, 1, 5, 3, 3, 5]] Just for fun, , A(100000000000000000000), modulo , 8, is , 1 and the next, 10, terms in the sequence A(n) modulo, 8, are 1, 2, 4, 1, 5, 3, 7, 3, 3, 4 -------------------------------------- For A(n) modulo, 3, there is an automata with 5, states Here it is: [[[2, 3, 4], [2, 2, 0], [2, 0, 3], [5, 0, 4], [5, 5, 0]], [1, 1, 1, 2, 2]] Just for fun, , A(100000000000000000000), modulo , 3, is , 0 and the next, 10, terms in the sequence A(n) modulo, 3, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For A(n) modulo, 9, there is an automata with 44, states Here it is: [[[4, 2, 3], [5, 6, 7], [8, 9, 10], [11, 12, 13], [11, 14, 15], [0, 16, 0], [5, 17, 18], [19, 20, 21], [22, 0, 0], [23, 24, 10], [11, 25, 26], [27, 28, 26], [ 29, 0, 0], [11, 30, 31], [29, 0, 0], [22, 16, 0], [29, 0, 0], [5, 17, 18], [19, 32, 33], [19, 34, 35], [22, 0, 0], [22, 16, 0], [36, 37, 38], [22, 0, 0], [39, 25, 40], [29, 0, 0], [27, 30, 40], [27, 28, 26], [29, 41, 0], [11, 30, 31], [29 , 0, 0], [42, 32, 43], [22, 0, 0], [19, 34, 35], [22, 0, 0], [36, 34, 43], [36, 44, 33], [22, 0, 0], [39, 28, 31], [29, 0, 0], [29, 41, 0], [42, 44, 35], [22, 0, 0], [36, 44, 33]], [1, 1, 2, 1, 1, 0, 1, 2, 3, 8, 1, 4, 6, 1, 6, 3, 6, 1, 2, 2, 3, 3, 8, 3, 7, 6, 4, 4, 6, 1, 6, 5, 3, 2, 3, 8, 8, 3, 7, 6, 6, 5, 3, 8]] Just for fun, , A(100000000000000000000), modulo , 9, is , 0 and the next, 10, terms in the sequence A(n) modulo, 9, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For A(n) modulo, 27, there is an automata with 465, states Here it is: [[[4, 2, 3], [5, 6, 7], [8, 9, 10], [13, 11, 12], [14, 15, 16], [17, 18, 19], [ 20, 21, 22], [23, 24, 25], [26, 27, 28], [29, 30, 31], [32, 33, 34], [35, 36, 37], [38, 39, 40], [38, 41, 42], [43, 44, 45], [46, 47, 48], [49, 50, 0], [51, 52, 53], [54, 0, 55], [56, 57, 16], [58, 59, 60], [61, 62, 63], [64, 65, 66], [ 67, 68, 69], [70, 71, 72], [73, 74, 75], [76, 77, 0], [78, 0, 79], [80, 81, 82] , [83, 84, 85], [86, 87, 31], [88, 89, 90], [91, 92, 93], [94, 95, 96], [97, 98 , 99], [76, 0, 100], [101, 0, 102], [38, 103, 104], [105, 106, 107], [94, 108, 109], [110, 111, 112], [113, 114, 115], [38, 116, 117], [118, 119, 120], [121, 122, 123], [124, 125, 126], [76, 0, 100], [127, 0, 128], [49, 129, 130], [76, 131, 132], [133, 74, 134], [135, 52, 136], [101, 0, 128], [49, 129, 130], [0, 0 , 0], [137, 138, 139], [140, 141, 142], [97, 98, 99], [76, 0, 100], [143, 0, 144], [145, 15, 146], [58, 59, 60], [147, 148, 149], [64, 150, 151], [152, 153, 154], [155, 156, 157], [158, 159, 151], [160, 161, 162], [163, 164, 165], [166, 167, 134], [168, 0, 132], [169, 0, 170], [73, 167, 171], [172, 173, 53], [127, 0, 174], [175, 50, 176], [168, 77, 100], [49, 129, 130], [0, 0, 0], [177, 178, 179], [180, 181, 182], [183, 184, 185], [166, 167, 134], [168, 0, 132], [78, 0, 79], [186, 187, 188], [189, 190, 191], [88, 192, 193], [194, 195, 196], [197, 198, 199], [200, 201, 193], [202, 203, 204], [205, 206, 207], [208, 209, 210], [76, 0, 100], [211, 0, 174], [97, 125, 210], [212, 213, 214], [215, 0, 170], [0 , 0, 0], [175, 50, 176], [0, 0, 0], [216, 217, 218], [205, 219, 220], [221, 222 , 223], [224, 225, 226], [227, 198, 228], [76, 0, 100], [211, 0, 174], [229, 230, 231], [232, 233, 234], [235, 236, 199], [208, 209, 210], [76, 0, 100], [ 211, 0, 174], [237, 238, 239], [240, 241, 207], [38, 116, 117], [242, 243, 244] , [245, 246, 247], [124, 125, 126], [76, 0, 100], [127, 0, 128], [124, 209, 99] , [248, 249, 250], [169, 0, 251], [175, 50, 176], [0, 0, 0], [168, 77, 100], [0 , 0, 0], [76, 131, 132], [0, 0, 0], [133, 74, 134], [211, 0, 102], [73, 167, 171], [211, 0, 102], [137, 252, 117], [253, 254, 255], [256, 257, 258], [259, 252, 104], [260, 261, 262], [263, 264, 265], [175, 50, 176], [0, 0, 0], [259, 266, 267], [268, 269, 270], [145, 15, 146], [58, 59, 60], [147, 148, 149], [271 , 272, 273], [274, 275, 276], [277, 278, 279], [280, 281, 282], [283, 284, 285] , [133, 286, 171], [168, 0, 132], [287, 0, 251], [158, 288, 289], [290, 291, 292], [158, 159, 151], [293, 294, 295], [296, 297, 298], [166, 167, 134], [168, 0, 132], [169, 0, 170], [166, 286, 75], [51, 299, 300], [49, 129, 130], [49, 129, 130], [0, 0, 0], [101, 0, 128], [166, 286, 75], [172, 173, 53], [0, 0, 0], [175, 50, 176], [0, 0, 0], [177, 301, 302], [303, 304, 305], [306, 307, 308], [ 309, 301, 310], [311, 312, 313], [155, 314, 315], [133, 286, 171], [168, 0, 132 ], [287, 0, 251], [316, 317, 318], [319, 320, 321], [322, 323, 324], [166, 167, 134], [168, 0, 132], [78, 0, 79], [325, 326, 327], [235, 328, 228], [259, 252, 104], [242, 243, 244], [329, 330, 331], [97, 98, 99], [76, 0, 100], [101, 0, 102], [200, 89, 332], [333, 334, 335], [200, 201, 193], [202, 203, 204], [205, 206, 207], [208, 209, 210], [76, 0, 100], [211, 0, 174], [208, 98, 126], [336, 337, 338], [287, 0, 339], [175, 50, 176], [124, 209, 99], [212, 213, 214], [287 , 0, 339], [49, 129, 130], [340, 341, 342], [343, 344, 345], [197, 328, 346], [ 76, 0, 100], [211, 0, 174], [221, 230, 342], [347, 348, 349], [329, 350, 247], [229, 230, 231], [224, 225, 226], [197, 328, 346], [97, 98, 99], [101, 0, 102], [229, 341, 223], [351, 352, 353], [354, 246, 331], [340, 341, 342], [232, 233, 234], [227, 198, 228], [97, 98, 99], [76, 0, 100], [229, 230, 231], [232, 233, 234], [235, 236, 199], [208, 209, 210], [76, 0, 100], [38, 116, 117], [242, 243 , 244], [245, 246, 247], [124, 125, 126], [76, 0, 100], [127, 0, 128], [208, 98 , 126], [248, 249, 250], [215, 0, 170], [0, 0, 0], [355, 356, 357], [340, 341, 342], [343, 344, 345], [197, 328, 346], [208, 209, 210], [76, 0, 100], [211, 0, 174], [259, 116, 358], [259, 252, 104], [359, 360, 361], [354, 350, 362], [124, 125, 126], [76, 0, 100], [127, 0, 128], [363, 364, 365], [366, 367, 368], [124, 125, 126], [76, 0, 100], [127, 0, 128], [369, 370, 371], [372, 373, 374], [375, 376, 377], [133, 286, 171], [168, 0, 132], [287, 0, 251], [277, 370, 378], [379 , 380, 381], [296, 382, 383], [369, 370, 371], [280, 281, 282], [384, 385, 386] , [73, 74, 75], [168, 0, 132], [215, 0, 339], [135, 52, 136], [49, 129, 130], [ 387, 388, 389], [390, 391, 392], [393, 394, 378], [395, 396, 397], [384, 385, 386], [158, 159, 151], [293, 294, 295], [296, 297, 298], [166, 167, 134], [168, 0, 132], [169, 0, 170], [51, 299, 300], [127, 0, 174], [398, 399, 400], [384, 284, 377], [401, 150, 289], [293, 294, 295], [402, 382, 403], [73, 74, 75], [ 168, 0, 132], [215, 0, 339], [309, 404, 405], [375, 385, 285], [309, 301, 310], [406, 407, 408], [409, 275, 392], [168, 0, 132], [287, 0, 251], [316, 410, 310] , [411, 412, 413], [414, 415, 416], [177, 410, 405], [417, 418, 419], [420, 421 , 157], [133, 286, 171], [168, 0, 132], [287, 0, 251], [38, 116, 117], [422, 423, 424], [354, 350, 362], [76, 0, 100], [124, 125, 126], [76, 0, 100], [127, 0, 128], [227, 236, 346], [137, 103, 358], [359, 360, 361], [245, 246, 247], [ 97, 125, 210], [336, 337, 338], [169, 0, 251], [0, 0, 0], [340, 222, 231], [425 , 426, 427], [245, 330, 362], [221, 222, 223], [343, 344, 345], [235, 236, 199] , [101, 0, 102], [428, 192, 332], [202, 203, 204], [240, 219, 429], [76, 0, 100 ], [200, 201, 193], [430, 431, 432], [433, 241, 220], [124, 125, 126], [221, 222, 223], [224, 225, 226], [227, 198, 228], [433, 206, 429], [259, 252, 104], [359, 360, 361], [354, 350, 362], [127, 0, 128], [221, 222, 223], [224, 225, 226], [227, 198, 228], [208, 209, 210], [76, 0, 100], [211, 0, 174], [369, 394, 279], [434, 435, 436], [402, 437, 298], [393, 394, 378], [372, 373, 374], [283, 284, 285], [73, 74, 75], [168, 0, 132], [215, 0, 339], [438, 297, 403], [177, 410, 405], [406, 407, 408], [390, 439, 276], [168, 0, 132], [169, 0, 170], [73, 74, 75], [168, 0, 132], [215, 0, 339], [277, 278, 279], [280, 281, 282], [283, 284, 285], [133, 286, 171], [168, 0, 132], [287, 0, 251], [393, 278, 371], [440 , 441, 442], [277, 278, 279], [395, 396, 397], [375, 376, 377], [158, 159, 151] , [443, 444, 445], [438, 437, 383], [401, 159, 446], [166, 167, 134], [169, 0, 170], [447, 448, 449], [283, 376, 386], [309, 301, 310], [406, 407, 408], [409, 275, 392], [133, 286, 171], [450, 451, 452], [64, 288, 446], [453, 454, 455], [ 296, 297, 298], [73, 74, 75], [168, 0, 132], [215, 0, 339], [177, 410, 405], [ 456, 457, 458], [274, 391, 459], [133, 286, 171], [168, 0, 132], [137, 103, 358 ], [422, 423, 424], [329, 330, 331], [88, 89, 90], [460, 461, 462], [205, 206, 207], [428, 201, 90], [211, 0, 174], [88, 89, 90], [430, 431, 432], [240, 219, 429], [208, 209, 210], [309, 301, 310], [463, 464, 465], [274, 391, 459], [168, 0, 132], [166, 167, 134], [168, 0, 132], [316, 404, 302], [456, 457, 458], [409 , 275, 392], [64, 288, 446], [443, 444, 445], [402, 382, 403], [409, 439, 459], [64, 288, 446], [453, 454, 455], [296, 297, 298], [401, 150, 289], [293, 294, 295], [402, 382, 403], [401, 150, 289], [453, 454, 455], [438, 437, 383], [177, 410, 405], [456, 457, 458], [274, 391, 459], [287, 0, 251], [428, 192, 332], [ 460, 461, 462], [433, 241, 220], [316, 404, 302], [463, 464, 465], [390, 439, 276]], [1, 1, 2, 1, 1, 9, 19, 2, 21, 26, 4, 24, 1, 1, 1, 15, 9, 12, 9, 19, 24, 10, 2, 20, 3, 21, 18, 9, 26, 3, 17, 4, 13, 6, 24, 18, 18, 1, 25, 6, 16, 6, 1, 1 , 15, 15, 18, 18, 9, 18, 12, 21, 18, 9, 0, 19, 10, 24, 18, 18, 10, 24, 10, 2, 5 , 12, 20, 20, 3, 3, 9, 9, 21, 3, 18, 18, 9, 9, 0, 26, 8, 12, 3, 9, 9, 17, 3, 4, 10, 24, 13, 13, 6, 6, 18, 18, 24, 15, 9, 0, 18, 0, 7, 6, 25, 16, 24, 18, 18, 16 , 7, 24, 6, 18, 18, 16, 6, 1, 1, 15, 15, 18, 18, 15, 6, 9, 18, 0, 9, 0, 18, 0, 12, 18, 21, 18, 19, 7, 6, 10, 10, 15, 18, 0, 10, 15, 10, 24, 10, 14, 12, 5, 14, 21, 12, 9, 9, 20, 23, 20, 20, 3, 3, 9, 9, 3, 12, 9, 9, 0, 18, 3, 3, 0, 18, 0, 26, 11, 21, 8, 8, 12, 12, 9, 9, 17, 26, 12, 3, 9, 9, 1, 24, 10, 1, 15, 24, 18, 18, 13, 19, 13, 13, 6, 6, 18, 18, 6, 24, 9, 18, 15, 15, 9, 9, 7, 25, 24, 18, 18 , 25, 22, 15, 16, 16, 24, 24, 18, 16, 13, 15, 7, 7, 24, 24, 18, 16, 7, 24, 6, 18, 1, 1, 15, 15, 18, 18, 6, 6, 9, 0, 25, 7, 25, 24, 6, 18, 18, 10, 10, 10, 15, 15, 18, 18, 25, 6, 15, 18, 18, 14, 23, 21, 12, 9, 9, 5, 26, 3, 14, 14, 21, 21, 9, 9, 21, 9, 5, 12, 23, 5, 21, 20, 20, 3, 3, 9, 9, 12, 18, 20, 21, 11, 20, 3, 21, 9, 9, 8, 21, 8, 8, 12, 9, 9, 17, 2, 21, 26, 26, 12, 12, 9, 9, 1, 19, 15, 18 , 15, 18, 18, 24, 19, 10, 15, 24, 24, 9, 0, 7, 4, 15, 25, 25, 24, 18, 22, 13, 6 , 18, 13, 4, 6, 15, 25, 16, 24, 6, 10, 10, 15, 18, 25, 16, 24, 6, 18, 18, 14, 8 , 3, 23, 23, 21, 21, 9, 9, 3, 26, 8, 12, 9, 9, 21, 9, 9, 5, 14, 21, 12, 9, 9, 23, 17, 5, 5, 21, 20, 2, 3, 11, 3, 9, 2, 21, 8, 8, 12, 12, 11, 2, 11, 3, 21, 9, 9, 26, 26, 12, 12, 9, 19, 19, 15, 4, 22, 6, 22, 18, 4, 4, 6, 6, 8, 17, 12, 9, 3 , 9, 17, 26, 12, 2, 2, 3, 12, 2, 11, 3, 11, 20, 3, 11, 11, 3, 26, 26, 12, 9, 22 , 22, 6, 17, 17, 12]] Just for fun, , A(100000000000000000000), modulo , 27, is , 0 and the next, 10, terms in the sequence A(n) modulo, 27, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For A(n) modulo, 5, there is an automata with 9, states Here it is: [[[2, 2, 3, 5, 4], [2, 2, 6, 3, 7], [3, 3, 2, 7, 6], [7, 0, 7, 7, 8], [7, 6, 0, 3, 9], [6, 6, 7, 2, 3], [7, 7, 3, 6, 2], [3, 6, 6, 0, 4], [0, 7, 6, 7, 5]], [1, 1, 2, 4, 4, 3, 4, 2, 0]] Just for fun, , A(100000000000000000000), modulo , 5, is , 1 and the next, 10, terms in the sequence A(n) modulo, 5, are 1, 2, 4, 4, 1, 1, 2, 3, 0, 3 -------------------------------------- For A(n) modulo, 25, there is an automata with 415, states Here it is: [[[6, 2, 3, 4, 5], [7, 8, 9, 11, 10], [12, 13, 14, 15, 16], [17, 18, 19, 20, 21 ], [22, 23, 24, 25, 26], [7, 27, 28, 29, 30], [7, 31, 32, 33, 34], [7, 35, 36, 37, 38], [39, 40, 41, 42, 43], [44, 45, 46, 47, 48], [49, 50, 51, 52, 53], [12, 54, 55, 56, 57], [12, 58, 59, 52, 60], [61, 62, 63, 64, 65], [66, 67, 68, 69, 70], [39, 71, 72, 73, 74], [66, 75, 76, 77, 78], [39, 79, 80, 81, 82], [83, 84, 85, 86, 87], [88, 89, 90, 92, 91], [93, 94, 95, 96, 97], [44, 98, 99, 101, 100] , [83, 102, 103, 104, 105], [44, 106, 107, 108, 109], [44, 110, 111, 112, 113], [114, 115, 116, 117, 118], [119, 120, 121, 122, 123], [124, 125, 126, 127, 128] , [129, 130, 131, 132, 60], [133, 134, 135, 136, 137], [7, 35, 36, 37, 38], [ 138, 139, 140, 141, 142], [88, 58, 143, 144, 145], [133, 134, 135, 136, 137], [ 146, 147, 148, 149, 150], [138, 139, 140, 141, 142], [12, 151, 152, 153, 154], [66, 155, 156, 157, 158], [39, 159, 160, 161, 162], [124, 163, 164, 165, 166], [167, 168, 169, 170, 171], [119, 172, 148, 37, 173], [88, 174, 55, 175, 176], [ 44, 177, 135, 178, 158], [66, 179, 180, 181, 182], [129, 183, 184, 185, 186], [ 187, 163, 188, 189, 190], [146, 191, 32, 192, 193], [49, 174, 194, 195, 196], [ 49, 130, 152, 144, 197], [119, 198, 199, 200, 201], [66, 67, 68, 69, 70], [124, 202, 203, 204, 162], [12, 58, 59, 52, 60], [205, 206, 207, 208, 209], [210, 179 , 211, 212, 213], [124, 202, 203, 204, 162], [88, 214, 51, 132, 154], [205, 206 , 207, 208, 209], [187, 159, 215, 216, 217], [61, 218, 219, 220, 193], [146, 147, 148, 149, 150], [221, 222, 223, 224, 225], [129, 130, 131, 132, 60], [167, 226, 227, 178, 228], [66, 45, 229, 136, 228], [66, 179, 180, 181, 182], [12, 230, 231, 232, 233], [39, 234, 235, 165, 236], [205, 218, 237, 238, 34], [124, 163, 164, 165, 166], [44, 239, 240, 241, 242], [7, 243, 121, 244, 150], [12, 245, 246, 247, 196], [44, 248, 68, 212, 249], [250, 251, 252, 253, 254], [221, 255, 256, 257, 166], [119, 31, 258, 220, 259], [187, 260, 235, 257, 261], [66, 262, 263, 264, 265], [119, 172, 148, 37, 173], [129, 266, 194, 56, 267], [83, 268, 269, 271, 270], [272, 273, 274, 275, 276], [277, 278, 279, 280, 281], [83, 268, 269, 271, 270], [83, 268, 269, 271, 270], [88, 266, 282, 247, 283], [129, 151, 143, 284, 53], [7, 285, 286, 287, 288], [138, 289, 290, 161, 291], [44, 292, 293, 181, 294], [272, 295, 297, 296, 298], [66, 299, 300, 301, 302], [187, 303, 304, 305, 306], [210, 307, 308, 309, 310], [311, 312, 313, 314, 315], [66, 179, 180, 181, 182], [12, 230, 231, 232, 233], [7, 316, 219, 317, 318], [39, 234, 235, 165, 236], [319, 320, 321, 323, 322], [319, 320, 321, 323, 322], [319 , 320, 321, 323, 322], [83, 268, 269, 271, 270], [167, 292, 324, 69, 213], [49, 325, 326, 327, 328], [138, 260, 164, 329, 330], [61, 331, 332, 33, 333], [133, 67, 211, 334, 294], [88, 335, 336, 337, 338], [187, 163, 188, 189, 190], [119, 31, 258, 220, 259], [129, 339, 340, 341, 342], [221, 343, 344, 345, 346], [124, 347, 348, 349, 350], [277, 351, 352, 353, 354], [355, 356, 357, 358, 26], [119, 191, 332, 238, 318], [61, 243, 359, 360, 173], [124, 125, 126, 127, 128], [49, 50, 51, 52, 53], [44, 45, 46, 47, 48], [124, 71, 215, 361, 291], [138, 234, 256 , 362, 190], [133, 363, 364, 365, 366], [7, 243, 121, 244, 150], [49, 54, 367, 368, 283], [129, 369, 246, 368, 176], [129, 151, 143, 284, 53], [61, 62, 63, 64 , 65], [210, 179, 211, 212, 213], [133, 226, 156, 47, 370], [210, 371, 293, 372 , 70], [12, 230, 231, 232, 233], [138, 260, 164, 329, 330], [205, 218, 237, 238 , 34], [138, 202, 72, 216, 373], [221, 374, 188, 329, 236], [210, 375, 376, 377 , 378], [205, 147, 36, 360, 123], [12, 245, 246, 247, 196], [146, 379, 380, 381 , 382], [44, 292, 293, 181, 294], [39, 71, 72, 73, 74], [146, 331, 237, 317, 259], [205, 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182], [39, 71, 72, 73, 74], [187, 387, 290, 204, 74], [66, 262, 263, 264, 265], [119, 172, 148 , 37, 173], [250, 369, 282, 195, 57], [119, 120, 121, 122, 123], [49, 50, 51, 52, 53], [44, 45, 46, 47, 48], [7, 285, 286, 287, 288], [133, 248, 324, 372, 182], [39, 71, 72, 73, 74], [221, 387, 160, 361, 373], [7, 35, 36, 37, 38], [39 , 40, 41, 42, 43], [49, 50, 51, 52, 53], [210, 177, 229, 385, 370], [138, 234, 256, 362, 190], [66, 262, 263, 264, 265], [205, 147, 36, 360, 123], [205, 316, 258, 192, 333], [61, 243, 359, 360, 173], [138, 139, 140, 141, 142], [250, 214, 59, 284, 197], [66, 155, 156, 157, 158], [210, 134, 227, 157, 48], [88, 335, 336, 337, 338], [221, 255, 256, 257, 166], [119, 31, 258, 220, 259], [250, 50, 131, 153, 145], [210, 375, 376, 377, 378], [119, 172, 148, 37, 173], [49, 54, 367, 368, 283], [61, 243, 359, 360, 173], [187, 388, 389, 390, 391], [12, 151, 152, 153, 154], [221, 289, 203, 73, 217], [187, 260, 235, 257, 261], [44, 239, 240, 241, 242], [61, 120, 383, 149, 38], [250, 369, 282, 195, 57], [133, 67, 211, 334, 294], [49, 325, 326, 327, 328], [7, 316, 219, 317, 318], [250, 251, 252, 253, 254], [250, 50, 131, 153, 145], [7, 285, 286, 287, 288], [66, 67, 68, 69, 70], [221, 387, 160, 361, 373], [39, 255, 386, 189, 330], [167, 168, 169, 170, 171], [129, 266, 194, 56, 267], [39, 40, 41, 42, 43], [129, 130, 131, 132, 60], [66, 179, 180, 181, 182], [49, 325, 326, 327, 328], [221, 255, 256, 257, 166], [205, 218, 237, 238, 34], [7, 35, 36, 37, 38], [250, 214, 59, 284, 197], [250, 50, 131, 153, 145], [146, 379, 380, 381, 382], [167, 371, 180, 334, 249], [133, 67, 211, 334, 294], [61, 331, 332, 33, 333], [250, 245, 367, 175, 267], [ 129, 151, 143, 284, 53], [205, 206, 207, 208, 209], [167, 371, 180, 334, 249], [187, 159, 215, 216, 217], [221, 374, 188, 329, 236], [44, 239, 240, 241, 242], [146, 35, 359, 122, 384], [221, 222, 223, 224, 225], [66, 155, 156, 157, 158], [138, 234, 256, 362, 190], [88, 174, 55, 175, 176], [167, 292, 324, 69, 213], [ 12, 230, 231, 232, 233], [187, 163, 188, 189, 190], [61, 331, 332, 33, 333], [ 88, 214, 51, 132, 154], [138, 289, 290, 161, 291], [83, 268, 269, 271, 270], [ 277, 278, 279, 280, 281], [272, 273, 274, 275, 276], [319, 320, 321, 323, 322], [272, 273, 274, 275, 276], [272, 273, 274, 275, 276], [319, 320, 321, 323, 322] , [277, 278, 279, 280, 281], [83, 268, 269, 271, 270], [277, 278, 279, 280, 281 ], [277, 278, 279, 280, 281], [272, 273, 274, 275, 276], [83, 268, 269, 271, 270], [319, 320, 321, 323, 322], [119, 198, 199, 200, 201], [187, 159, 215, 216 , 217], [133, 248, 324, 372, 182], [205, 172, 383, 244, 384], [124, 125, 126, 127, 128], [12, 151, 152, 153, 154], [44, 45, 46, 47, 48], [221, 374, 188, 329, 236], [167, 168, 169, 170, 171], [129, 266, 194, 56, 267], [44, 248, 68, 212, 249], [49, 325, 326, 327, 328], [7, 316, 219, 317, 318], [319, 320, 321, 323, 322], [319, 320, 321, 323, 322], [319, 320, 321, 323, 322], [277, 278, 279, 280 , 281], [66, 179, 180, 181, 182], [129, 183, 184, 185, 186], [187, 163, 188, 189, 190], [119, 31, 258, 220, 259], [138, 234, 256, 362, 190], [66, 262, 263, 264, 265], [205, 147, 36, 360, 123], [49, 54, 367, 368, 283], [66, 179, 180, 181, 182], [49, 325, 326, 327, 328], [221, 255, 256, 257, 166], [7, 316, 219, 317, 318], [44, 392, 393, 394, 395], [221, 396, 397, 398, 399], [83, 84, 85, 86 , 87], [49, 400, 401, 402, 403], [93, 94, 95, 96, 97], [205, 172, 383, 244, 384 ], [250, 214, 59, 284, 197], [167, 226, 227, 178, 228], [319, 320, 321, 323, 322], [319, 320, 321, 323, 322], [83, 268, 269, 271, 270], [277, 278, 279, 280, 281], [272, 273, 274, 275, 276], [129, 183, 184, 185, 186], [12, 58, 59, 52, 60 ], [119, 198, 199, 200, 201], [44, 292, 293, 181, 294], [138, 289, 290, 161, 291], [61, 120, 383, 149, 38], [49, 54, 367, 368, 283], [146, 147, 148, 149, 150], [124, 125, 126, 127, 128], [210, 177, 229, 385, 370], [124, 374, 386, 362 , 261], [49, 130, 152, 144, 197], [146, 379, 380, 381, 382], [210, 179, 211, 212, 213], [124, 202, 203, 204, 162], [49, 130, 152, 144, 197], [146, 379, 380, 381, 382], [210, 179, 211, 212, 213], [39, 71, 72, 73, 74], [39, 255, 386, 189, 330], [133, 363, 364, 365, 366], [61, 120, 383, 149, 38], [49, 54, 367, 368, 283], [39, 255, 386, 189, 330], [133, 363, 364, 365, 366], [61, 120, 383, 149, 38], [129, 266, 194, 56, 267], [272, 273, 274, 275, 276], [277, 278, 279, 280, 281], [83, 268, 269, 271, 270], [272, 273, 274, 275, 276], [66, 404, 405, 407, 406], [83, 102, 103, 104, 105], [133, 408, 409, 410, 411], [210, 412, 413, 414, 415], [187, 388, 389, 390, 391], [129, 130, 131, 132, 60], [146, 35, 359, 122, 384], [205, 147, 36, 360, 123], [210, 371, 293, 372, 70], [129, 183, 184, 185, 186], [124, 374, 386, 362, 261], [119, 31, 258, 220, 259], [61, 62, 63, 64, 65] , [66, 67, 68, 69, 70], [129, 151, 143, 284, 53], [61, 331, 332, 33, 333], [167 , 292, 324, 69, 213], [138, 260, 164, 329, 330], [88, 174, 55, 175, 176], [124, 163, 164, 165, 166], [44, 248, 68, 212, 249], [88, 335, 336, 337, 338], [138, 260, 164, 329, 330], [7, 316, 219, 317, 318], [119, 120, 121, 122, 123], [187, 388, 389, 390, 391], [88, 58, 143, 144, 145], [133, 134, 135, 136, 137], [221, 222, 223, 224, 225], [167, 226, 227, 178, 228], [221, 255, 256, 257, 166], [133 , 363, 364, 365, 366], [187, 260, 235, 257, 261], [39, 255, 386, 189, 330], [66 , 262, 263, 264, 265], [146, 35, 359, 122, 384], [129, 266, 194, 56, 267], [210 , 371, 293, 372, 70], [12, 230, 231, 232, 233], [138, 260, 164, 329, 330], [146 , 191, 32, 192, 193], [138, 234, 256, 362, 190], [210, 375, 376, 377, 378], [61 , 120, 383, 149, 38], [12, 245, 246, 247, 196], [88, 214, 51, 132, 154], [205, 206, 207, 208, 209], [66, 67, 68, 69, 70], [221, 387, 160, 361, 373], [167, 292 , 324, 69, 213], [250, 251, 252, 253, 254], [205, 218, 237, 238, 34], [124, 374 , 386, 362, 261], [44, 248, 68, 212, 249], [129, 183, 184, 185, 186], [39, 234, 235, 165, 236], [7, 316, 219, 317, 318], [167, 292, 324, 69, 213], [129, 183, 184, 185, 186], [221, 255, 256, 257, 166], [146, 191, 32, 192, 193]], [1, 1, 2, 4, 9, 1, 1, 1, 23, 9, 17, 2, 2, 11, 4, 23, 4, 23, 10, 7, 15, 9, 10, 9, 9, 22, 21, 13, 22, 19, 1, 3, 7, 19, 16, 3, 2, 4, 23, 13, 24, 21, 7, 9, 4, 22, 8, 16, 17, 17, 21, 4, 13, 2, 6, 14, 13, 7, 6, 8, 11, 16, 18, 22, 24, 4, 4, 2, 23, 6, 13, 9, 1, 2, 9, 12, 18, 21, 8, 4, 21, 22, 10, 15, 5, 10, 10, 7, 22, 1, 3, 9, 15 , 4, 8, 14, 9, 4, 2, 1, 23, 20, 20, 20, 10, 24, 17, 3, 11, 19, 7, 8, 21, 22, 18 , 13, 5, 4, 21, 11, 13, 17, 9, 13, 3, 19, 1, 17, 22, 22, 11, 14, 19, 14, 2, 3, 6, 3, 18, 14, 6, 2, 16, 9, 23, 16, 6, 23, 7, 14, 2, 1, 24, 3, 24, 7, 23, 21, 23 , 19, 11, 12, 8, 14, 1, 2, 24, 19, 12, 23, 16, 21, 19, 17, 9, 18, 9, 13, 14, 12 , 8, 16, 7, 11, 19, 23, 8, 4, 21, 12, 21, 17, 9, 1, 19, 23, 18, 1, 23, 17, 14, 3, 4, 6, 6, 11, 3, 12, 4, 14, 7, 18, 21, 12, 14, 21, 17, 11, 8, 2, 18, 8, 9, 11 , 12, 19, 17, 1, 12, 12, 1, 4, 18, 23, 24, 22, 23, 22, 4, 17, 18, 6, 1, 12, 12, 16, 24, 19, 11, 12, 22, 6, 24, 8, 18, 9, 16, 18, 4, 3, 7, 24, 2, 8, 11, 7, 3, 10, 5, 15, 20, 15, 15, 20, 5, 10, 5, 5, 15, 10, 20, 21, 8, 19, 6, 13, 2, 9, 18, 24, 22, 9, 17, 1, 20, 20, 20, 5, 4, 22, 8, 21, 3, 4, 6, 17, 4, 17, 18, 1, 9, 18 , 10, 17, 15, 6, 12, 24, 20, 20, 10, 5, 15, 22, 2, 21, 9, 3, 11, 17, 16, 13, 14 , 13, 17, 16, 14, 13, 17, 16, 14, 23, 23, 19, 11, 17, 23, 19, 11, 22, 15, 5, 10 , 15, 4, 10, 19, 14, 8, 22, 16, 6, 14, 22, 13, 21, 11, 4, 22, 11, 24, 3, 7, 13, 9, 7, 3, 1, 21, 8, 7, 19, 18, 24, 18, 19, 8, 23, 4, 16, 22, 14, 2, 3, 16, 3, 14 , 11, 2, 7, 6, 4, 18, 24, 12, 6, 13, 9, 22, 23, 1, 24, 22, 18, 16]] Just for fun, , A(100000000000000000000), modulo , 25, is , 1 and the next, 10, terms in the sequence A(n) modulo, 25, are 1, 2, 4, 9, 21, 1, 2, 23, 10, 13 -------------------------------------- For A(n) modulo, 7, there is an automata with 9, states Here it is: [[[2, 2, 3, 4, 3, 5, 6], [2, 2, 7, 0, 8, 3, 2], [3, 3, 9, 0, 7, 4, 3], [4, 4, 8 , 0, 9, 2, 4], [0, 9, 8, 2, 8, 4, 5], [3, 7, 2, 9, 8, 0, 6], [7, 7, 3, 0, 2, 9, 7], [8, 8, 2, 0, 4, 7, 8], [9, 9, 4, 0, 3, 8, 9]], [1, 1, 2, 4, 0, 2, 3, 5, 6]] Just for fun, , A(100000000000000000000), modulo , 7, is , 0 and the next, 10, terms in the sequence A(n) modulo, 7, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ------------------------------------------------ This ends this thrilling article, that took, 10.301, seconds. to produce .