Mathematics  »   Cobweb posets  »   Sequences

(1) Cobweb Sequences Map

Let us recall [md5] the most important Cobweb sequences:

Fig.1 Families of Cobweb sequences

1. Natural numbers' sequence
   A = (1, 3, 5, 7, 9, ...);
2. Admissible sequence
   B = (1, 2, 2, 2, 1, 4, 1, 2, ...) = B_2,2 * B_2,3;
3. Tileable sequence
   C = (1, 2, 2, 1, 2, 2, 1, ...);
4. GCD-morphic sequence
   E = (1, 2, 3, 2, 1, 6, 1, ...) = B_2,2 * B_3,3;
5. Tileable and GCD-Morphic
   F = (1, 2, 1, 2, 1, 2, ...) = B_2,2;
6. Special Tileable sequence
   Natural numbers, Fibonacci numbers;
7. Special Tileable and Not GCD-morphic
   G = 1, 4, 12, 32, 80, 192, 448, 1024, ... (Example 4 in Section 5);

(2) Definitions

Let be natural numbers. Then a sequence , such that n-th element is equal

.

is called Tileable. It was shown that these numbers define a Cobweb Tileable sequence and F-Nomial coefficients are equal to the number of so-called F-bricks that form a tiling of hyper F-Box. See [md5] and [md6] for further reading.

Denote by the sequence , such that

where and . It is also the Cobweb Tileable sequence.

1. Natural Numbers
   T_{1,1} = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
2. Fibonacci Numbers
   Fib(1) = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
   Fib(2) = 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ...
   Fib(3) = 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, ...
   Fib(4) = 1, 4, 17, 72, 305, 1292, 5473, 23184, 98209, 416020, ...
   Fib(5) = 1, 5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, ...
   Fib(6) = 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, ...
   Fib(7) = 1, 7, 50, 357, 2549, 18200, 129949, 927843, 6624850, 47301793, ...
   Fib(8) = 1, 8, 65, 528, 4289, 34840, 283009, 2298912, 18674305, 151693352, ...
   Fib(9) = 1, 9, 82, 747, 6805, 61992, 564733, 5144589, 46866034, 426938895, ...
   Fib(10) = 1, 10, 101, 1020, 10301, 104030, 1050601, 10610040, 107151001, 1082120050, ...
   
3. Gaussian integers (p == 1)
   T_{1,2} = 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, ...
   T_{1,3} = 1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, ...
   T_{1,4} = 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, ...
   T_{1,5} = 1, 6, 31, 156, 781, 3906, 19531, 97656, 488281, 2441406, ...
   T_{1,6} = 1, 7, 43, 259, 1555, 9331, 55987, 335923, 2015539, 12093235, ...
   T_{1,7} = 1, 8, 57, 400, 2801, 19608, 137257, 960800, 6725601, 47079208, ...
   T_{1,8} = 1, 9, 73, 585, 4681, 37449, 299593, 2396745, 19173961, 153391689, ...
   T_{1,9} = 1, 10, 91, 820, 7381, 66430, 597871, 5380840, 48427561, 435848050, ...
   T_{1,10} = 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, ...
   
4. If (p == q)
   T_{1,1} = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
   T_{2,2} = 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120, ...
   T_{3,3} = 1, 6, 27, 108, 405, 1458, 5103, 17496, 59049, 196830, ...
   T_{4,4} = 1, 8, 48, 256, 1280, 6144, 28672, 131072, 589824, 2621440, ...
   T_{5,5} = 1, 10, 75, 500, 3125, 18750, 109375, 625000, 3515625, 19531250, ...
   T_{6,6} = 1, 12, 108, 864, 6480, 46656, 326592, 2239488, 15116544, 100776960, ...
   T_{7,7} = 1, 14, 147, 1372, 12005, 100842, 823543, 6588344, 51883209, 403536070, ...
   T_{8,8} = 1, 16, 192, 2048, 20480, 196608, 1835008, 16777216, 150994944, ...
   T_{9,9} = 1, 18, 243, 2916, 32805, 354294, 3720087, 38263752, 387420489, ...
   T_{10,10} = 1, 20, 300, 4000, 50000, 600000, 7000000, 80000000, 900000000, ...
   
5. If (p == 2)
   T_{2,3} = 1, 5, 19, 65, 211, 665, 2059, 6305, 19171, 58025, ...
   T_{2,4} = 1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, ...
   T_{2,5} = 1, 7, 39, 203, 1031, 5187, 25999, 130123, 650871, 3254867, ...
   T_{2,6} = 1, 8, 52, 320, 1936, 11648, 69952, 419840, 2519296, 15116288, ...
   T_{2,7} = 1, 9, 67, 477, 3355, 23517, 164683, 1152909, 8070619, 56494845, ...
   T_{2,8} = 1, 10, 84, 680, 5456, 43680, 349504, 2796160, 22369536, 178956800, ...
   T_{2,9} = 1, 11, 103, 935, 8431, 75911, 683263, 6149495, 55345711, 498111911, ...
   T_{2,10} = 1, 12, 124, 1248, 12496, 124992, 1249984, 12499968, 124999936, 1249999872, ...
   
6. If (p == 3)
   T_{3,2} = 1, 5, 19, 65, 211, 665, 2059, 6305, 19171, 58025, ...
   T_{3,3} = 1, 6, 27, 108, 405, 1458, 5103, 17496, 59049, 196830, ...
   T_{3,4} = 1, 7, 37, 175, 781, 3367, 14197, 58975, 242461, 989527, ...
   T_{3,5} = 1, 8, 49, 272, 1441, 7448, 37969, 192032, 966721, 4853288, ...
   T_{3,6} = 1, 9, 63, 405, 2511, 15309, 92583, 557685, 3352671, 20135709, ...
   T_{3,7} = 1, 10, 79, 580, 4141, 29230, 205339, 1439560, 10083481, 70604050, ...
   T_{3,8} = 1, 11, 97, 803, 6505, 52283, 418993, 3354131, 26839609, 214736555, ...
   T_{3,9} = 1, 12, 117, 1080, 9801, 88452, 796797, 7173360, 64566801, 581120892, ...
   T_{3,10} = 1, 13, 139, 1417, 14251, 142753, 1428259, 14284777, 142854331, 1428562993, ...
   
7. If (p == 4)
   T_{4,2} = 1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, ...
   T_{4,3} = 1, 7, 37, 175, 781, 3367, 14197, 58975, 242461, 989527, ...
   T_{4,4} = 1, 8, 48, 256, 1280, 6144, 28672, 131072, 589824, 2621440, ...
   T_{4,5} = 1, 9, 61, 369, 2101, 11529, 61741, 325089, 1690981, 8717049, ...
   T_{4,6} = 1, 10, 76, 520, 3376, 21280, 131776, 807040, 4907776, 29708800, ...
   T_{4,7} = 1, 11, 93, 715, 5261, 37851, 269053, 1899755, 13363821, 93808891, ...
   T_{4,8} = 1, 12, 112, 960, 7936, 64512, 520192, 4177920, 33488896, 268173312, ...
   T_{4,9} = 1, 13, 133, 1261, 11605, 105469, 953317, 8596237, 77431669, 697147165, ...
   T_{4,10} = 1, 14, 156, 1624, 16496, 165984, 1663936, 16655744, 166622976, 1666491904, ...
   
8. If (p == 5)
   T_{5,2} = 1, 7, 39, 203, 1031, 5187, 25999, 130123, 650871, 3254867, ...
   T_{5,3} = 1, 8, 49, 272, 1441, 7448, 37969, 192032, 966721, 4853288, ...
   T_{5,4} = 1, 9, 61, 369, 2101, 11529, 61741, 325089, 1690981, 8717049, ...
   T_{5,5} = 1, 10, 75, 500, 3125, 18750, 109375, 625000, 3515625, 19531250, ...
   T_{5,6} = 1, 11, 91, 671, 4651, 31031, 201811, 1288991, 8124571, 50700551, ...
   T_{5,7} = 1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, ...
   T_{5,8} = 1, 13, 129, 1157, 9881, 82173, 673009, 5462197, 44088201, 354658733, ...
   T_{5,9} = 1, 14, 151, 1484, 13981, 128954, 1176211, 10664024, 96366841, 869254694, ...
   T_{5,10} = 1, 15, 175, 1875, 19375, 196875, 1984375, 19921875, 199609375, 1998046875, ...
   
9. If (p == 6)
   T_{6,2} = 1, 8, 52, 320, 1936, 11648, 69952, 419840, 2519296, 15116288, ...
   T_{6,3} = 1, 9, 63, 405, 2511, 15309, 92583, 557685, 3352671, 20135709, ...
   T_{6,4} = 1, 10, 76, 520, 3376, 21280, 131776, 807040, 4907776, 29708800, ...
   T_{6,5} = 1, 11, 91, 671, 4651, 31031, 201811, 1288991, 8124571, 50700551, ...
   T_{6,6} = 1, 12, 108, 864, 6480, 46656, 326592, 2239488, 15116544, 100776960, ...
   T_{6,7} = 1, 13, 127, 1105, 9031, 70993, 543607, 4085185, 30275911, 222009073, ...
   T_{6,8} = 1, 14, 148, 1400, 12496, 107744, 908608, 7548800, 62070016, 506637824, ...
   T_{6,9} = 1, 15, 171, 1755, 17091, 161595, 1501011, 13789035, 125780931, ...
   T_{6,10} = 1, 16, 196, 2176, 23056, 238336, 2430016, 24580096, 247480576, ...
   

1. Natural Numbers T_{1,1}:

   1;
   1, 1;
   1, 2, 1;
   1, 3, 3, 1;
   1, 4, 6, 4, 1;
   1, 5, 10, 10, 5, 1;
   1, 6, 15, 20, 15, 6, 1;
   1, 7, 21, 35, 35, 21, 7, 1;
   1, 8, 28, 56, 70, 56, 28, 8, 1;
   

2. Fibonacci Integers Fib(1):

   1;
   1, 1;
   1, 1, 1;
   1, 2, 2, 1;
   1, 3, 6, 3, 1;
   1, 5, 15, 15, 5, 1;
   1, 8, 40, 60, 40, 8, 1;
   1, 13, 104, 260, 260, 104, 13, 1;
   1, 21, 273, 1092, 1820, 1092, 273, 21, 1;
   1, 34, 714, 4641, 12376, 12376, 4641, 714, 34, 1;
   1, 55, 1870, 19635, 85085, 136136, 85085, 19635, 1870, 55, 1;
   

3. Fib(2):

   1;
   1, 1;
   1, 2, 1;
   1, 5, 5, 1;
   1, 12, 30, 12, 1;
   1, 29, 174, 174, 29, 1;
   1, 70, 1015, 2436, 1015, 70, 1;
   1, 169, 5915, 34307, 34307, 5915, 169, 1;
   1, 408, 34476, 482664, 1166438, 482664, 34476, 408, 1;
   

4. Fib(3):

   1;
   1, 1;
   1, 3, 1;
   1, 10, 10, 1;
   1, 33, 110, 33, 1;
   1, 109, 1199, 1199, 109, 1;
   1, 360, 13080, 43164, 13080, 360, 1;
   1, 1189, 142680, 1555212, 1555212, 142680, 1189, 1;
   1, 3927, 1556401, 56030436, 185070228, 56030436, 1556401, 3927, 1;
   

5. Fib(4):

   1;
   1, 1;
   1, 4, 1;
   1, 17, 17, 1;
   1, 72, 306, 72, 1;
   1, 305, 5490, 5490, 305, 1;
   1, 1292, 98515, 417240, 98515, 1292, 1;
   1, 5473, 1767779, 31716035, 31716035, 1767779, 5473, 1;
   

6. Fib(5):

   1;
   1, 1;
   1, 5, 1;
   1, 26, 26, 1;
   1, 135, 702, 135, 1;
   1, 701, 18927, 18927, 701, 1;
   1, 3640, 510328, 2649780, 510328, 3640, 1;
   1, 18901, 13759928, 370988828, 370988828, 13759928, 18901, 1;
   

7. Gaussian Integers T_{1,2}:

   1;
   1, 1;
   1, 3, 1;
   1, 7, 7, 1;
   1, 15, 35, 15, 1;
   1, 31, 155, 155, 31, 1;
   1, 63, 651, 1395, 651, 63, 1;
   1, 127, 2667, 11811, 11811, 2667, 127, 1;
   1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1;
   

8. T_{1,3}:

   1;
   1, 1;
   1, 4, 1;
   1, 13, 13, 1;
   1, 40, 130, 40, 1;
   1, 121, 1210, 1210, 121, 1;
   1, 364, 11011, 33880, 11011, 364, 1;
   1, 1093, 99463, 925771, 925771, 99463, 1093, 1;
   1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1;
   

9. T_{1,4}:

   1;
   1, 1;
   1, 5, 1;
   1, 21, 21, 1;
   1, 85, 357, 85, 1;
   1, 341, 5797, 5797, 341, 1;
   1, 1365, 93093, 376805, 93093, 1365, 1;
   1, 5461, 1490853, 24208613, 24208613, 1490853, 5461, 1;
   1, 21845, 23859109, 1550842085, 1926646245, 1550842085, 23859109, 21845, 1;
   

10. T_{1,5}:

    1;
    1, 1;
    1, 6, 1;
    1, 31, 31, 1;
    1, 156, 806, 156, 1;
    1, 781, 20306, 20306, 781, 1;
    1, 3906, 508431, 2558556, 508431, 3906, 1;
    1, 19531, 12714681, 320327931, 320327931, 12714681, 19531, 1;
    

11. T_{2,2}:

    1;
    1, 1;
    1, 4, 1;
    1, 12, 12, 1;
    1, 32, 96, 32, 1;
    1, 80, 640, 640, 80, 1;
    1, 192, 3840, 10240, 3840, 192, 1;
    1, 448, 21504, 143360, 143360, 21504, 448, 1;
    1, 1024, 114688, 1835008, 4587520, 1835008, 114688, 1024, 1;
    

12. T_{3,3}:

    1;
    1, 1;
    1, 6, 1;
    1, 27, 27, 1;
    1, 108, 486, 108, 1;
    1, 405, 7290, 7290, 405, 1;
    1, 1458, 98415, 393660, 98415, 1458, 1;
    1, 5103, 1240029, 18600435, 18600435, 1240029, 5103, 1;
    

13. T_{4,4}:

    1;
    1, 1;
    1, 8, 1;
    1, 48, 48, 1;
    1, 256, 1536, 256, 1;
    1, 1280, 40960, 40960, 1280, 1;
    1, 6144, 983040, 5242880, 983040, 6144, 1;
    1, 28672, 22020096, 587202560, 587202560, 22020096, 28672, 1;
    

14. T_{5,5}:

    1;
    1, 1;
    1, 10, 1;
    1, 75, 75, 1;
    1, 500, 3750, 500, 1;
    1, 3125, 156250, 156250, 3125, 1;
    1, 18750, 5859375, 39062500, 5859375, 18750, 1;
    

15. T_{2,3}:

    1;
    1, 1;
    1, 5, 1;
    1, 19, 19, 1;
    1, 65, 247, 65, 1;
    1, 211, 2743, 2743, 211, 1;
    1, 665, 28063, 96005, 28063, 665, 1;
    1, 2059, 273847, 3041143, 3041143, 273847, 2059, 1;
    1, 6305, 2596399, 90873965, 294990871, 90873965, 2596399, 6305, 1;
    

16. T_{2,4}:

    1;
    1, 1;
    1, 6, 1;
    1, 28, 28, 1;
    1, 120, 560, 120, 1;
    1, 496, 9920, 9920, 496, 1;
    1, 2016, 166656, 714240, 166656, 2016, 1;
    1, 8128, 2731008, 48377856, 48377856, 2731008, 8128, 1;
    

17. T_{2,5}:

    1;
    1, 1;
    1, 7, 1;
    1, 39, 39, 1;
    1, 203, 1131, 203, 1;
    1, 1031, 29899, 29899, 1031, 1;
    1, 5187, 763971, 3976567, 763971, 5187, 1;
    1, 25999, 19265259, 509294411, 509294411, 19265259, 25999, 1;
    

18. T_{3,4}:

    1;
    1, 1;
    1, 7, 1;
    1, 37, 37, 1;
    1, 175, 925, 175, 1;
    1, 781, 19525, 19525, 781, 1;
    1, 3367, 375661, 1776775, 375661, 3367, 1;
    1, 14197, 6828757, 144142141, 144142141, 6828757, 14197, 1;
    

19. T_{3,5}:

    1;
    1, 1;
    1, 8, 1;
    1, 49, 49, 1;
    1, 272, 1666, 272, 1;
    1, 1441, 48994, 48994, 1441, 1;
    1, 7448, 1341571, 7447088, 1341571, 7448, 1;
    1, 37969, 35349139, 1039553251, 1039553251, 35349139, 37969, 1;
    

20. T_{4,5}:

    1;
    1, 1;
    1, 9, 1;
    1, 61, 61, 1;
    1, 369, 2501, 369, 1;
    1, 2101, 86141, 86141, 2101, 1;
    1, 11529, 2691381, 16280649, 2691381, 11529, 1;
    

1. Natural Numbers T_{1,1}:

   I^sgn_F(n) = 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, ...
   

   1;
   -1, 1;
   1, -2, 1;
   -1, 3, -3, 1;
   1, -4, 6, -4, 1;
   -1, 5, -10, 10, -5, 1;
   1, -6, 15, -20, 15, -6, 1;
   -1, 7, -21, 35, -35, 21, -7, 1;
   1, -8, 28, -56, 70, -56, 28, -8, 1;
   

2. Fibonacci Numbers Fib(1):

   I^sgn_F(n) = 1, -1, 0, 1, -1, -6, 35, 181, -6056, ...
   

   1;
   -1, 1;
   0, -1, 1;
   1, 0, -2, 1;
   -1, 3, 0, -3, 1;
   -6, -5, 15, 0, -5, 1;
   35, -48, -40, 60, 0, -8, 1;
   181, 455, -624, -260, 260, 0, -13, 1;
   -6056, 3801, 9555, -6552, -1820, 1092, 0, -21, 1;
   

3. Fib(2):

   I^sgn_F(n) = 1, -1, 1, -1, -7, 231, -7575, 182519, ...
   

   1;
   -1, 1;
   1, -2, 1;
   -1, 5, -5, 1;
   -7, -12, 30, -12, 1;
   231, -203, -174, 174, -29, 1;
   -7575, 16170, -7105, -2436, 1015, -70, 1;
   182519, -1280175, 1366365, -240149, -34307, 5915, -169, 1;
   

4. Fib(3):

   I^sgn_F(n) = 1, -1, 2, -11, 175, -8176, 1103363, -400685867, ...
   

   1;
   -1, 1;
   2, -3, 1;
   -11, 20, -10, 1;
   175, -363, 220, -33, 1;
   -8176, 19075, -13189, 2398, -109, 1;
   1103363, -2943360, 2289000, -474804, 26160, -360, 1;
   -400685867, 1311898607, -1166551680, 272162100, -17107332, 285360, -1189, 1;
   

5. Fib(4):

   I^sgn_F(n) = 1, -1, 3, -35, 1673, -334281, 281384603, -311783547, ...
   

   1;
   -1, 1;
   3, -4, 1;
   -35, 51, -17, 1;
   1673, -2520, 918, -72, 1;
   -334281, 510265, -192150, 16470, -305, 1;
   281384603, -431891052, 164815595, -14603400, 295545, -1292, 1;
   -311783547, -1875327045, 1770554949, 1521319003, -1110061225, 5303337, -5473, 1;
   

6. Fib(5):

   I^sgn_F(n) = 1, -1, 4, -79, 7991, -4181466, -1535101749, 1887305821, ...
   

   1;
   -1, 1;
   4, -5, 1;
   -79, 104, -26, 1;
   7991, -10665, 2808, -135, 1;
   -4181466, 5601691, -1495233, 75708, -701, 1;
   -1535101749, 1959332944, -216936248, -209332620, 2041312, -3640, 1;
   1887305821, 1840893927, -1289197232, 1044290308, 756653660, 55039712, -18901, 1;
   

7. Gaussian Integers T_{1,2}:

   I^sgn_F(n) = 1, -1, 2, -8, 64, -1024, 32768, -2097152, ...
   

   1;
   -1, 1;
   2, -3, 1;
   -8, 14, -7, 1;
   64, -120, 70, -15, 1;
   -1024, 1984, -1240, 310, -31, 1;
   32768, -64512, 41664, -11160, 1302, -63, 1;
   -2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1;
   

8. T_{1,3}:

   I^sgn_F(n) = 1, -1, 3, -27, 729, -59049, 14348907, -1870418611, ...
   

   1;
   -1, 1;
   3, -4, 1;
   -27, 39, -13, 1;
   729, -1080, 390, -40, 1;
   -59049, 88209, -32670, 3630, -121, 1;
   14348907, -21493836, 8027019, -914760, 33033, -364, 1;
   -1870418611, -1496513833, -1578223391, 674887059, -24995817, 298389, -1093, 1;
   

9. T_{1,4}:

   I^sgn_F(n) = 1, -1, 4, -64, 4096, -1048576, 1073741824, ...
   

   1;
   -1, 1;
   4, -5, 1;
   -64, 84, -21, 1;
   4096, -5440, 1428, -85, 1;
   -1048576, 1396736, -371008, 23188, -341, 1;
   1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1;
   

10. T_{1,5}:

    I^sgn_F(n) = 1, -1, 5, -125, 15625, -9765625, 452807053, ...
    

    1;
    -1, 1;
    5, -6, 1;
    -125, 155, -31, 1;
    15625, -19500, 4030, -156, 1;
    -9765625, 12203125, -2538250, 101530, -781, 1;
    452807053, 510174414, -645700217, -319819500, 2542155, -3906, 1;
    

11. T_{2,2}:

    I^sgn_F(n) = 1, -1, 3, -25, 543, -29281, 3781503, ...
    

    1;
    -1, 1;
    3, -4, 1;
    -25, 36, -12, 1;
    543, -800, 288, -32, 1;
    -29281, 43440, -16000, 1920, -80, 1;
    3781503, -5621952, 2085120, -256000, 11520, -192, 1;
    

12. T_{3,3}:

    I^sgn_F(n) = 1, -1, 5, -109, 9449, -3068281, -708918611, ...
    

    1;
    -1, 1;
    5, -6, 1;
    -109, 135, -27, 1;
    9449, -11772, 2430, -108, 1;
    -3068281, 3826845, -794610, 36450, -405, 1;
    -708918611, -178586402, 929923335, -42908940, 492075, -1458, 1;
    

13. T_{4,4}:

    I^sgn_F(n) = 1, -1, 7, -289, 63487, -69711361, -1962528769, ...
    

    1;
    -1, 1;
    7, -8, 1;
    -289, 336, -48, 1;
    63487, -73984, 10752, -256, 1;
    -69711361, 81263360, -11837440, 286720, -1280, 1;
    -1962528769, 1190127616, -2014248960, -1515192320, 6881280, -6144, 1;
    

14. T_{5,5}:

    I^sgn_F(n) = 1, -1, 9, -601, 267249, -742650001, -191458119, ...
    

    1;
    -1, 1;
    9, -10, 1;
    -601, 675, -75, 1;
    267249, -300500, 33750, -500, 1;
    -742650001, 835153125, -93906250, 1406250, -3125, 1;
    -191458119, -403545118, -1750953665, -2001726020, 52734375, -18750, 1;
    

15. T_{2,3}:

    I^sgn_F(n) = 1, -1, 4, -58, 2846, -452174, 226285114, ...
    

    1;
    -1, 1;
    4, -5, 1;
    -58, 76, -19, 1;
    2846, -3770, 988, -65, 1;
    -452174, 600506, -159094, 10972, -211, 1;
    226285114, -300695710, 79867298, -5568290, 112252, -665, 1;
    

16. T_{2,4}:

    I^sgn_F(n) = 1, -1, 5, -113, 10879, -4324129, -1605663297, ...
    

    1;
    -1, 1;
    5, -6, 1;
    -113, 140, -28, 1;
    10879, -13560, 2800, -120, 1;
    -4324129, 5395984, -1120960, 49600, -496, 1;
    -1605663297, -127509472, 1813050624, -80709120, 833280, -2016, 1;
    

17. T_{2,5}:

    I^sgn_F(n) = 1, -1, 6, -196, 33204, -28551484, -1049568668, ...
    

    1;
    -1, 1;
    6, -7, 1;
    -196, 234, -39, 1;
    33204, -39788, 6786, -203, 1;
    -28551484, 34233324, -5860204, 179394, -1031, 1;
    -1049568668, -2067659444, -402910692, -779407132, 4583826, -5187, 1;
    

18. T_{3,4}:

    I^sgn_F(n) = 1, -1, 6, -186, 27174, -17707614, -1798053678, ...
    

    1;
    -1, 1;
    6, -7, 1;
    -186, 222, -37, 1;
    27174, -32550, 5550, -175, 1;
    -17707614, 21222894, -3631650, 117150, -781, 1;
    -1798053678, 508005806, 1618277422, -330480150, 2253966, -3367, 1;
    

19. T_{3,5}:

    I^sgn_F(n) = 1, -1, 7, -295, 68849, -85099697, -1816617705, ...
    

    1;
    -1, 1;
    7, -8, 1;
    -295, 343, -49, 1;
    68849, -80240, 11662, -272, 1;
    -85099697, 99211409, -14453230, 342958, -1441, 1;
    -1816617705, 1832616552, -2123458733, 2098076336, 9390997, -7448, 1;
    

20. T_{4,5}:

    I^sgn_F(n) = 1, -1, 8, -428, 138292, -254370172, -1006593020, ...
    

    1;
    -1, 1;
    8, -9, 1;
    -428, 488, -61, 1;
    138292, -157932, 20008, -369, 1;
    -254370172, 290551492, -36868348, 689128, -2101, 1;
    -1006593020, 828950180, -1465693500, 1621816820, 21531048, -11529, 1;