My integer sequences in OEIS

18 minute read

Published:

I have submitted few sequences to the OEIS, most of which are connected to my work on applying the Kauffman bracket polynomial to knot shadows.

0, 1, 2, 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, 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, 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

0, 1, 7, 19, 39, 71, 123, 211, 367, 655, 1203, 2267, 4359, 8503, 16747, 33187, 66015, 131615, 262755, 524971, 1049335, 2097991, 4195227, 8389619, 16778319, 33555631, 67110163, 134219131, 268436967, 536872535, 1073743563, 2147485507

1, 5, 19, 62, 184, 512, 1360, 3488, 8704, 21248, 50944, 120320, 280576, 647168, 1478656, 3350528, 7536640, 16842752, 37421056, 82706432, 181927936, 398458880, 869269504, 1889533952, 4093640704, 8841592832, 19042140160, 40902852608

0, 1,
0, 3,   4,    1,
0, 9,   24,   22,    8,     1,
0, 27,  108,  171,   136,   57,    12,    1,
0, 81,  432,  972,   1200,  886,   400,   108,   16,    1,
0, 243, 1620, 4725,  7920,  8430,  5944,  2810,  880,   175,  20,   1, 
0, 729, 5832, 20898, 44280, 61695, 59472, 40636, 19824, 6855, 1640, 258, 24, 1
0, 0,    1,
0, 1,    2,     1,
0, 4,    7,     4,     1, 
0, 12,   26,    19,    6,     1, 
0, 32,   88,    88,    39,    8,     1, 
0, 80,   272,   360,   230,   71,    10,    1, 
0, 192,  784,   1312,  1140,  532,   123,   12,    1, 
0, 448,  2144,  4368,  4872,  3164,  1162,  211,   14,   1, 
0, 1024, 5632,  13568, 18592, 15680, 8176,  2480,  367,  16,  1, 
0, 2304, 14336, 39936, 65088, 67872, 46368, 20304, 5262, 655, 18, 1
0, 0, 1,
0, 1, 2,  1,
0, 2, 6,  6,   2,   
0, 3, 13, 22,  18,  7,    1,
0, 4, 23, 56,  75,  60,   29,   8,    1,
0, 5, 36, 115, 215, 261,  215,  121,  45,   10,   1,
0, 6, 52, 206, 495, 806,  938,  798,  496,  220,  66,   12,  1,
0, 7, 71, 336, 987, 2016, 3031, 3452, 3010, 2003, 1001, 364, 91, 14, 1
  • A300401: (square array read by antidiagonals)
0, 
1,  1, 
2,  2,  2, 
3,  4,  4,  3, 
4,  7,  8,  7,   4, 
5,  11, 14, 14,  11,  5, 
6,  16, 22, 24,  22,  16,  6, 
7,  22, 32, 37,  37,  32,  22,  7, 
8,  29, 44, 53,  56,  53,  44,  29,  8, 
9,  37, 58, 72,  79,  79,  72,  58,  37,  9, 
10, 46, 74, 94,  106, 110, 106, 94,  74,  46, 10, 
11, 56, 92, 119, 137, 146, 146, 137, 119, 92, 56, 11

1, 2, 7, 26, 88, 272, 784, 2144, 5632, 14336, 35584, 86528, 206848, 487424, 1134592, 2613248, 5963776, 13500416, 30343168, 67764224, 150470656, 332398592, 730857472, 1600126976, 3489660928, 7583301632, 16424894464, 35467034624, 76369887232, 164014063616

0, 0,  1,
0, 1,  1, 
0, 2,  2, 
0, 3,  4,  1, 
0, 4,  7,  4,   1, 
0, 5,  11, 10,  5,   1, 
0, 6,  16, 20,  15,  6,   1, 
0, 7,  22, 35,  35,  21,  7,   1, 
0, 8,  29, 56,  70,  56,  28,  8,   1, 
0, 9,  37, 84,  126, 126, 84,  36,  9,   1, 
0, 10, 46, 120, 210, 252, 210, 120, 45,  10, 1, 
0, 11, 56, 165, 330, 462, 462, 330, 165, 55, 11, 1
0, 1,  2,  1,
0, 3,  4,  1,
0, 5,  8,  3,
0, 7,  14, 9,   2,
0, 9,  22, 21,  10,  2,
0, 11, 32, 41,  30,  12,  2,
0, 13, 44, 71,  70,  42,  14,  2,
0, 15, 58, 113, 140, 112, 56,  16,  2,
0, 17, 74, 169, 252, 252, 168, 72,  18, 2,
0, 19, 92, 241, 420, 504, 420, 240, 90, 20, 2

1, 8, 16, 25, 35, 46, 58, 71, 85, 100, 116, 133, 151, 170, 190, 211, 233, 256, 280, 305, 331, 358, 386, 415, 445, 476, 508, 541, 575, 610, 646, 683, 721, 760, 800, 841, 883, 926, 970, 1015, 1061, 1108, 1156, 1205, 1255, 1306, 1358, 1411, 1465, 1520, 1576

  • A303273: (array read by antidiagonals)
1, 
1, 1, 
1, 2,  2, 
1, 3,  4,  4, 
1, 4,  6,  7,  7, 
1, 5,  8,  10, 11, 11, 
1, 6,  10, 13, 15, 16, 16, 
1, 7,  12, 16, 19, 21, 22, 22, 
1, 8,  14, 19, 23, 26, 28, 29, 29, 
1, 9,  16, 22, 27, 31, 34, 36, 37, 37, 
1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 
1, 11, 20, 28, 35, 41, 46, 50, 53, 55, 56, 56

12, 168, 2784, 48912, 874416, 15709488, 282620784, 5086424112, 91551884016, 1647915162288, 29662379171184, 533922356331312, 9610600070213616, 172990789545095088, 3113834153217961584, 56049014464954558512, 1008882258904338303216, 18159880652953870707888

1, 
1, 1, 
1, 2, 1, 
1, 2, 3,  1, 
1, 3, 4,  4,   1, 
1, 3, 8,  7,   5,   1,
1, 4, 10, 17,  11,  6,   1, 
1, 4, 16, 26,  31,  16,  7,   1,
1, 5, 19, 47,  56,  51,  22,  8,   1,
1, 5, 27, 65,  112, 106, 78,  29,  9,   1,
1, 6, 31, 101, 176, 232, 183, 113, 37,  10, 1, 
1, 6, 41, 131, 296, 407, 435, 295, 157, 46, 11, 1
0, 0,    0,     1,
0, 1,    2,     1, 
0, 5,    8,     3, 
0, 16,   30,    16,    2, 
0, 45,   104,   81,    24,    2, 
0, 121,  340,   356,   170,   35,    2, 
0, 320,  1068,  1411,  932,   315,   48,    2, 
0, 841,  3262,  5209,  4396,  2079,  532,   63,   2, 
0, 2205, 9760,  18281, 18784, 11440, 4144,  840,  80,   2, 
0, 5776, 28746, 61786, 74838, 55809, 26226, 7602, 1260, 99, 2
0, 1,  3,   3,   1, 
0, 7,  14,  9,   2, 
0, 13, 37,  43,  26,   8,    1, 
0, 19, 72,  129, 141,  98,   42,   10,   1, 
0, 25, 119, 291, 463,  504,  378,  192,  63,   12,   1, 
0, 31, 178, 553, 1156, 1716, 1848, 1452, 825,  330,  88,   14,  1, 
0, 37, 249, 939, 2432, 4576, 6435, 6864, 5577, 3432, 1573, 520, 117, 16, 1

2, 3, 0, 1, 4, 5, 6, 7, 10, 11, 8, 9, 12, 13, 14, 15, 18, 19, 16, 17, 20, 21, 22, 23, 26, 27, 24, 25, 28, 29, 30, 31, 34, 35, 32, 33, 36, 37, 38, 39, 42, 43, 40, 41, 44, 45, 46, 47, 50, 51, 48, 49, 52, 53, 54, 55, 58, 59, 56, 57, 60, 61, 62, 63, 66, 67, 64

0, 1,
0, 1,
0, 1,  1,
0, 3, 
0, 3,  3,  1, 
0, 9,  11, 1, 
0, 17, 22, 9, 1
0,   0,     1,
1,
0,   0,     1,   0,    1, 
2,   0,     1, 
1,   0,     3,   0,    2,  0, 1, 
9,   0,     8,   0,    3,  0, 1, 
11,  0,    21,   0,   12,  0, 4,   0, 1, 
54,  0,    68,   0,   32,  0, 1,   0, 1, 
148, 228,   0, 124,    0, 44, 7,   0, 1, 
619, 0,   900,    0, 461,  0, 162, 0, 34

0, 0, 1, 0, 2, 1, 5, 8, 22, 51, 182, 562

1,
1, 1,
1, 2,  1,
1, 3,  3,  1,
1, 4,  4,  4,  1,
1, 5,  5,  5,  5,  1,
1, 6,  6,  6,  6,  6,  1,
1, 7,  7,  7,  7,  7,  7,  1, 
1, 8,  8,  8,  8,  8,  8,  8,  1,
1, 9,  9,  9,  9,  9,  9,  9,  9,  1,
1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1,
1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 
1, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 1
1, 
1,           1, 
2,           1,         1, 
5,           2,         1,     1, 
14,          5,         2,     1,   1, 
51,          15,        5,     2,   1,  1, 
267,         67,        15,    5,   2,  1,  1, 
2328,        504,       77,    15,  5,  2,  1, 1, 
56092,       9310,      684,   83,  15, 5,  2, 1, 1, 
10494213,    1396077,   34297, 860, 87, 15, 5, 2, 1, 1, 
49487367289, 5937876645,
1, 
1, 1, 
1, 1,  1, 
1, 3,  3,  1, 
1, 1,  3,  1,   1, 
1, 5,  5,  5,   5,   1, 
1, 3,  15, 5,   15,  3,  1, 
1, 7,  21, 35,  35,  21,  7,   1, 
1, 1,  7,  7,   35,  7,   7,   1,   1, 
1, 9,  9,  21,  63,  63,  21,  9,   9,   1, 
1, 5,  45, 15,  105, 63,  105, 15,  45,  5,  1, 
1, 11, 55, 165, 165, 231, 231, 165, 165, 55, 11, 1
1,
2,   2,
4,   2,   4,
8,   8,   8,   8,
16,  4,   8,   4,   16,
32,  32,  16,  16,  32,  32,
64,  32,  64,  16,  64,  32,  64,
128, 128, 128, 128, 128, 128, 128, 128,
256, 32,  64,  32,  128, 32,  64,  32,  256,
512, 512, 128, 128, 256, 256, 128, 128, 512, 512

5, 5, 2, 2, 8, 4, 7, 4, 9, 8, 3, 0, 7, 9, 3, 3, 9, 8, 4, 0, 2, 2, 5, 1, 6, 3, 2, 2, 7, 9, 5, 9, 7, 4, 3, 8, 0, 9, 2, 8, 9, 5, 8, 3, 3, 8, 3, 5, 9, 3, 0, 7, 6, 4, 2, 3, 5, 5, 7, 2, 9, 8, 3, 9, 8, 7, 6, 4, 3, 3, 0, 4, 6, 1, 6, 1, 4, 2, 7, 1, 8, 4, 6, 7, 1, 8, 3

 0,
-1,  1,
-1,  0,   1,
-3, -3,   3,    3, 
-1, -1,   0,    1,   1,
-5, -15, -5,    5,   15,  5,
-3, -3,  -15,   0,   15,  3,  3,
-7, -35, -63,  -35,  35,  63, 35,  7, 
-1, -3,  -7,   -7,   0,   7,  7,   3,  1, 
-9, -63, -45,  -63, -63,  63, 63,  45, 63,  9, 
-5, -5,  -135, -15, -105, 0,  105, 15, 135, 5, 5
1,
1,   1,
1,   1,   1, 
4,   4,   4,  4, 
2,   1,   1,  1,   2, 
16,  16,  8,  8,   16,  16, 
16,  4,   16, 1,   16,  4,   16, 
64,  64,  64, 64,  64,  64,  64,  64, 
16,  8,   8,  8,   1,   8,   8,   8,   16, 
256, 256, 64, 64,  128, 128, 64,  64,  256, 256, 
256, 32, 256, 16,  128, 1,   128, 16,  256, 32,  256
1,
1, -1, 
1,  0,  1, 
1,  1,  0,  -1, 
1,  2,  0,   0,  1, 
1,  3,  1,   0,  0,  -1, 
1,  4,  3,   0,  0,   0, 1, 
1,  5,  6,   1,  0,   0, 0, -1,
1,  6,  10,  4,  0,   0, 0,  0, 1, 
1,  7,  15,  10, 1,   0, 0,  0, 0, -1, 
1,  8,  21,  20, 5,   0, 0,  0, 0,  0, 1, 
1,  9,  28,  35, 15,  1, 0,  0, 0,  0, 0, -1, 
1,  10, 36,  56, 35,  6, 0,  0, 0,  0, 0,  0, 1
  • A320530: (square array read by antidiagonals)
1,
1, 0, 
0, 1,  1, 
0, 2,  2,    1, 
0, 4,  7,    3,     1, 
0, 7,  26,   16,    4,     1, 
0, 11, 88,   90,    29,    5,     1, 
0, 16, 272,  459,   220,   46,    6,    1, 
0, 22, 784,  2133,  1504,  440,   67,   7,    1, 
0, 29, 2144, 9234,  9344,  3775,  774,  92,   8,   1, 
0, 37, 5632, 37908, 54016, 29375, 7992, 1246, 121, 9, 1
  • A320531: (square array read by antidiagonals)
0,
1, 0, 
0, 1, 0, 
0, 2, 1, 0, 
0, 3, 4, 1, 0, 
0, 4, 12, 6, 1, 0, 
0, 5, 32, 27, 8, 1, 0, 
0, 6, 80, 108, 48, 10, 1, 0, 
0, 7, 192, 405, 256, 75, 12, 1, 0, 
0, 8, 448, 1458, 1280, 500, 108, 14, 1, 0, 
0, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 0

12, 72, 672, 6048, 55488, 511872, 4738560, 43943424, 407890944, 3787941888, 35186122752, 326885842944, 3037038034944, 28217571901440, 262178452930560, 2436006721486848, 22634041833160704, 210303674768424960, 1954034324430913536, 18155901427591938048

0, 
1,   1, 
3,   10,  3, 
4,   11,  11,  4, 
11,  32,  26,  32,  11, 
15,  43,  37,  37,  43,  15, 
41,  118, 100, 106, 100, 118, 41, 
56,  161, 137, 143, 143, 137, 161, 56, 
153, 440, 374, 392, 386, 392, 374, 440, 153, 
209, 601, 511, 535, 529, 529, 535, 511, 601, 209

4, 2, 8, 10, 28, 38, 104, 142, 388, 530, 1448, 1978, 5404, 7382, 20168, 27550, 75268, 102818, 280904, 383722, 1048348, 1432070, 3912488, 5344558, 14601604, 19946162, 54493928, 74440090, 203374108, 277814198, 759002504, 1036816702, 2832635908, 3869452610

3, 9, 4, 3, 3, 7, 5, 6, 7, 2, 9, 7, 4, 0, 6, 4, 4, 1, 1, 2, 7, 2, 8, 7, 1, 9, 5, 1, 2, 5, 4, 8, 9, 3, 6, 3, 9, 1, 1, 9, 0, 0, 4, 3, 7, 8, 1, 7, 5, 3, 1, 7, 1, 9, 0, 0, 4, 6, 5, 0, 5, 8, 1, 6, 2, 0, 9, 9, 4, 4, 1, 8, 0, 7, 5, 7, 3, 3, 3, 3, 6, 4, 2, 3, 4, 2, 8

0,
1,   1, 
1,   4,    1, 
3,   9,    9,    3, 
13,  44,   30,   44,   13, 
35,  115,  90,   90,   115,  35, 
16,  53,   40,   46,   40,   53,   16, 
131, 433,  330,  366,  366,  330,  433,  131, 
179, 592,  450,  504,  486,  504,  450,  592,  179, 
163, 539,  410,  458,  446,  446,  458,  410,  539,  163, 
668, 2209, 1680, 1878, 1824, 1842, 1824, 1878, 1680, 2209, 668

4, 2, 3, 8, 36, 96, 44, 360, 492, 448, 1836, 5016, 2284, 18720, 25572, 23288, 95436, 260736, 118724, 973080, 1329252, 1210528, 4960836, 13553256, 6171364, 50581440, 69095532, 62924168, 257868036, 704508576, 320792204, 2629261800, 3591638412, 3270846208

2, 7, 17, 33, 57, 93, 149, 241, 401, 693, 1245, 2313, 4409, 8557, 16805, 33249, 66081, 131685, 262829, 525049, 1049417, 2098077, 4195317, 8389713, 16778417, 33555733, 67110269, 134219241, 268437081, 536872653, 1073743685, 2147485633, 4294969409, 8589936837

1, 5, 13, 33, 73, 141, 245, 393, 593, 853, 1181, 1585, 2073, 2653, 3333, 4121, 5025, 6053, 7213, 8513, 9961, 11565, 13333, 15273, 17393, 19701, 22205, 24913, 27833, 30973, 34341, 37945, 41793, 45893, 50253, 54881, 59785, 64973, 70453, 76233, 82321, 88725

1,
1, 1,
1, 2, 1, 
1, 1, 1, 1, 
1, 1, 3, 1, 1,
1, 1, 2, 2, 1, 1,
1, 1, 2, 1, 2, 1, 1,
1, 1, 2, 1, 1, 2, 1, 1, 
1, 1, 2, 1, 1, 1, 2, 1, 1, 
1, 1, 2, 1, 1, 1, 1, 2, 1, 1,
1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1,
1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 
1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1
  • A321126: (square array read by antidiagonals)
1, 
2,  2, 
3,  2,  3, 
4,  3,  3,  4, 
5,  4,  3,  4,  5, 
6,  5,  4,  4,  5,  6, 
7,  6,  5,  5,  5,  6,  7, 
8,  7,  6,  6,  6,  6,  7,  8, 
9,  8,  7,  7,  7,  7,  7,  8,  9, 
10, 9,  8,  8,  8,  8,  8,  8,  9,  10, 
11, 10, 9,  9,  9,  9,  9,  9,  9,  10, 11, 
12, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 
13, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13
  • A321127: (square array read by antidiagonals)
0, 1, 
0, 2,  2, 
0, 5,  8,   3, 
0, 10, 24,  21,  8,    1, 
0, 17, 56,  80,  64,   30,   8,    1, 
0, 26, 110, 220, 270,  220,  122,  45,   10,   1, 
0, 37, 192, 495, 820,  952,  804,  497,  220,  66,   12,  1, 
0, 50, 308, 973, 2030, 3059, 3472, 3017, 2004, 1001, 364, 91, 14, 1

1, 5, 13, 33, 73, 141, 245, 393, 593, 853, 1181, 1585, 2073, 2653, 3333, 4121, 5025, 6053, 7213, 8513, 9961, 11565, 13333, 15273, 17393, 19701, 22205, 24913, 27833, 30973, 34341, 37945, 41793, 45893, 50253, 54881, 59785, 64973, 70453, 76233, 82321, 88725

1, 5, 25, 69, 145, 261, 425, 645, 929, 1285, 1721, 2245, 2865, 3589, 4425, 5381, 6465, 7685, 9049, 10565, 12241, 14085, 16105, 18309, 20705, 23301, 26105, 29125, 32369, 35845, 39561, 43525, 47745, 52229, 56985, 62021, 67345, 72965, 78889, 85125, 91681, 98565

3, 11, 21, 35, 55, 83, 121, 171, 235, 315, 413, 531, 671, 835, 1025, 1243, 1491, 1771, 2085, 2435, 2823, 3251, 3721, 4235, 4795, 5403, 6061, 6771, 7535, 8355, 9233, 10171, 11171, 12235, 13365, 14563, 15831, 17171, 18585, 20075, 21643, 23291, 25021, 26835

  • A322596: (square array read by antidiagonals)
1,
1, 1,
1, 1, 1,
1, 2, 1,  1,
1, 2, 2,  1,  1,
1, 3, 4,  3,  1,  1,
1, 3, 5,  5,  3,  1,   1,
1, 4, 7,  9,  7,  4,   1,   1,
1, 4, 10, 14, 14, 10,  4,   1,  1,
1, 5, 12, 21, 26, 21,  12,  5,  1, 1,
1, 5, 15, 30, 42, 42,  30,  15, 5, 1, 1, 
1, 6, 19, 42, 66, 77,  66,  42, 19, 6, 1, 1, 
1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1

1, 7, 17, 39, 81, 151, 257, 407, 609, 871, 1201, 1607, 2097, 2679, 3361, 4151, 5057, 6087, 7249, 8551, 10001, 11607, 13377, 15319, 17441, 19751, 22257, 24967, 27889, 31031, 34401, 38007, 41857, 45959, 50321, 54951, 59857, 65047, 70529, 76311, 82401, 88807

1, 
3,     1, 
11,    2,      1, 
25,    35,     5,      1, 
137,   15,     17,     3,      1, 
49,    203,    49,     35,     7,      1, 
363,   469,    967,    28,     23,     4,    1, 
761,   29531,  801,    1069,   27,     39,   9,     1, 
7129,  6515,   4523,   285,    3013,   75,   145,   5,  1, 
7381,  177133, 84095,  341693, 8591,   7513, 605,   44, 11, 1, 
83711, 190553, 341747, 139381, 242537, 1903, 10831, 33, 35, 6, 1
1, 
2,     1, 
6,     1,     1, 
12,    12,    2,     1, 
60,    4,     4,     1,     1, 
20,    45,    8,     6,     2,    1, 
140,   90,    120,   3,     3,    1,   1, 
280,   5040,  80,    80,    2,    4,   2,   1, 
2520,  1008,  378,   16,    144,  4,   12,  1, 1, 
2520,  25200, 6048,  15120, 288,  240, 24,  3, 2, 1, 
27720, 25200, 21600, 5040,  6048, 40,  240, 1, 2, 1, 1