Munafo's Classical Sequences

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6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, 6006, 8190, 10920, 14280, 18360, 23256, 29070, 35910, 43890, 53130, 63756, 75900, 89700, 105300, 122850, 142506, 164430, 188790, 215760, 245520, ...

MCS15512 : A₀ = 0; A_K+1 = A_K + K³ - K (score: 5)

6, 33, 202, 1419, 11358, 102229, 1022298, 11245287, 134943454, 1754264913, 24559708794, 368395631923, ...

MCS7764 (alias MCS504976) : A₀ = 0; A_K+1 = K A_K + 2 A_K + K (score: 5)

6, 39, 316, 3165, 37986, 531811, 8508984, 153161721, 3063234430, 67391157471, ...

MCS3858 (alias MCS61736, MCS123473) : A₀ = 0; A_K+1 = 2 K A_K + K (score: 4)

6, 42, 156, 420, 930, 1806, 3192, 5256, 8190, 12210, 17556, 24492, 33306, 44310, 57840, 74256, 93942, 117306, 144780, 176820, 213906, 256542, 305256, 360600, 423150, 493506, 572292, 660156, ...

MCS496267 : A₀ = 0; A_K+1 = A_K + 4 K³ + 2 K (score: 9)

A puzzle sequence from Terry Stickel (possibly same as A082986)

7, 9, 71, 109, 995, 1745, 17911, 34901, 394043, 837625, 10245119, 23453501, 307353571, 750512033, 10450021415, 27018433189, 397100813771, ...

MCS434197 : A₀ = 0; A₁ = 0; A_K+1 = 2 K A_K-1 + 1 (score: 5)

7, 11, 21, 29, 43, 55, 73, 89, 111, 131, 157, 181, 211, 239, 273, 305, 343, 379, 421, 461, 507, 551, 601, 649, 703, 755, 813, 869, 931, 991, 1057, 1121, 1191, 1259, 1333, 1405, 1483, 1559, 1641, ...

MCS124457 : A₀ = -1; A_K+1 = - A_K + 2 K² (score: 5)

7, 17, 25, 33, 47, 65, 81, 97, 119, 145, 169, 193, 223, 257, 289, 321, 359, 401, 441, 481, 527, 577, 625, 673, 727, 785, 841, 897, 959, 1025, 1089, 1153, 1223, 1297, 1369, 1441, 1519, 1601, 1681, ...

MCS1710406 : A₀ = 1; A₁ = 1; A_K+1 = - A_K-1 + 2 K² (score: 5)

7, 17, 31, 49, 71, 97, 127, 161, 199, 241, 287, 337, 391, 449, 511, 577, 647, 721, 799, 881, 967, 1057, 1151, 1249, 1351, 1457, 1567, 1681, 1799, 1921, 2047, 2177, 2311, 2449, 2591, 2737, 2887, ...

MCS7090 : A_K = 2 K² - 1 (score: 4)

7, 20, 44, 81, 135, 208, 304, 425, 575, 756, 972, 1225, 1519, 1856, 2240, 2673, 3159, 3700, 4300, 4961, 5687, 6480, 7344, 8281, 9295, 10388, 11564, 12825, 14175, 15616, 17152, 18785, 20519, 22356, ...

MCS3888 : A₀ = 0; A_K+1 = - A_K + K³ (score: 4)

7, 22, 50, 95, 161, 252, 372, 525, 715, 946, 1222, 1547, 1925, 2360, 2856, 3417, 4047, 4750, 5530, 6391, 7337, 8372, 9500, 10725, 12051, 13482, 15022, 16675, 18445, 20336, 22352, 24497, 26775, ...

MCS31026 : A₀ = 0; A_K+1 = A_K + 2 K² - K (score: 5)

7, 25, 61, 121, 211, 337, 505, 721, 991, 1321, 1717, 2185, 2731, 3361, 4081, 4897, 5815, 6841, 7981, 9241, 10627, 12145, 13801, 15601, 17551, 19657, 21925, 24361, 26971, 29761, 32737, 35905, ...

MCS14156 : A_K = K³ - K + 1 (score: 5)

7, 26, 57, 99, 159, 244, 353, 485, 647, 846, 1081, 1351, 1663, 2024, 2433, 2889, 3399, 3970, 4601, 5291, 6047, 6876, 7777, 8749, 9799, 10934, 12153, 13455, 14847, 16336, 17921, 19601, 21383, ...

MCS6680 : A₀ = 0; A₁ = 1; A_K+1 = - A_K-1 + K³ (score: 5)

7, 26, 63, 124, 215, 342, 511, 728, 999, 1330, 1727, 2196, 2743, 3374, 4095, 4912, 5831, 6858, 7999, 9260, 10647, 12166, 13823, 15624, 17575, 19682, 21951, 24388, 26999, 29790, 32767, 35936, ...

MCS3544 : A_K = K³ - 1 (score: 4)

7, 27, 57, 98, 159, 245, 353, 484, 647, 847, 1081, 1350, 1663, 2025, 2433, 2888, 3399, 3971, 4601, 5290, 6047, 6877, 7777, 8748, 9799, 10935, 12153, 13454, 14847, 16337, 17921, 19600, 21383, ...

MCS855110 : A₀ = 1; A₁ = 1; A_K+1 = - A_K-1 + K³ (score: 5)

7, 27, 127, 695, 4431, 32415, 268191, 2478559, 25321983, 283498943, 3452631295, 45451204735, 643222128895, ...

MCS27694341 : A₀ = 0; A₁ = -1; A_K+1 = K A_K - A_K + 2 A_K-1 + K (score: 8)

Simplest continuation of 7, 27, 127 (see 695)

7, 30, 106, 343, 1065, 3244, 9796, 29469, 88507, 265642, 797070, 2391379, 7174333, 21523224, 64569928, 193710073, 581130543, 1743391990, 5230176370, 15690529551, 47071589137, 141214767940, ...

MCS3896 : A₀ = 0; A_K+1 = 3 A_K + K² (score: 5)

7, 31, 91, 211, 421, 757, 1261, 1981, 2971, 4291, 6007, 8191, 10921, 14281, 18361, 23257, 29071, 35911, 43891, 53131, 63757, 75901, 89701, 105301, 122851, 142507, 164431, 188791, 215761, 245521, ...

MCS124100 : A₀ = 1; A_K+1 = A_K + K³ - K (score: 5)

7, 33, 96, 220, 435, 777, 1288, 2016, 3015, 4345, 6072, 8268, 11011, 14385, 18480, 23392, 29223, 36081, 44080, 53340, 63987, 76153, 89976, 105600, 123175, 142857, 164808, 189196, 216195, 245985, ...

MCS126212 : A₀ = 1; A_K+1 = A_K + K³ - 1 (score: 5)

7, 37, 201, 1231, 8653, 69273, 623521, 6235291, 68588301, 823059733, 10699776673, 149796873591, ...

MCS7748 : A₀ = 0; A_K+1 = K A_K + A_K + K² (score: 5)

7, 52, 472, 5197, 67567, 1013512, 17229712, 327364537, 6874655287, 158117071612, ...

MCS15506 : A₀ = 0; A_K+1 = 2 K A_K + A_K + K (score: 5)

8, 19, 45, 80, 136, 207, 305, 424, 576, 755, 973, 1224, 1520, 1855, 2241, 2672, 3160, 3699, 4301, 4960, 5688, 6479, 7345, 8280, 9296, 10387, 11565, 12824, 14176, 15615, 17153, 18784, 20520, 22355, ...

MCS62217 : A₀ = -1; A_K+1 = - A_K + K³ (score: 5)

8, 20, 45, 81, 136, 208, 305, 425, 576, 756, 973, 1225, 1520, 1856, 2241, 2673, 3160, 3700, 4301, 4961, 5688, 6480, 7345, 8281, 9296, 10388, 11565, 12825, 14176, 15616, 17153, 18785, 20520, 22356, ...

MCS15664 : A₀ = 0; A_K+1 = - A_K + K³ + 1 (score: 5)

8, 25, 56, 105, 176, 273, 400, 561, 760, 1001, 1288, 1625, 2016, 2465, 2976, 3553, 4200, 4921, 5720, 6601, 7568, 8625, 9776, 11025, 12376, 13833, 15400, 17081, 18880, 20801, 22848, 25025, 27336, ...

MCS252436 : A₀ = 1; A_K+1 = A_K + 2 K² - 1 (score: 5)

8, 26, 56, 99, 160, 244, 352, 485, 648, 846, 1080, 1351, 1664, 2024, 2432, 2889, 3400, 3970, 4600, 5291, 6048, 6876, 7776, 8749, 9800, 10934, 12152, 13455, 14848, 16336, 17920, 19601, 21384, ...

MCS213777 : A₀ = 0; A₁ = 0; A_K+1 = - A_K-1 + K³ (score: 5)

8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, ...

MCS880 : A_K = K³ (score: 3)

Perfect cubes : N×N×N (A000578)

8, 28, 72, 153, 288, 496, 800, 1225, 1800, 2556, 3528, 4753, 6272, 8128, 10368, 13041, 16200, 19900, 24200, 29161, 34848, 41328, 48672, 56953, 66248, 76636, 88200, 101025, 115200, 130816, 147968, ...

MCS213521 : A₀ = 0; A₁ = 0; A_K+1 = A_K-1 + K³ (score: 5)

8, 57, 472, 4745, 56976, 797713, 12763472, 229742577, 4594851640, 101086736201, ...

MCS3850 (alias MCS61608) : A₀ = 0; A_K+1 = 2 K A_K + K² (score: 5)

8, 75, 904, 13565, 244176, 5127703, 123064880, 3322751769, 99682553080, ...

MCS3859 (alias MCS61752) : A₀ = 0; A_K+1 = 3 K A_K + K (score: 5)

9, 28, 73, 153, 289, 496, 801, 1225, 1801, 2556, 3529, 4753, 6273, 8128, 10369, 13041, 16201, 19900, 24201, 29161, 34849, 41328, 48673, 56953, 66249, 76636, 88201, 101025, 115201, 130816, 147969, ...

MCS6672 : A₀ = 0; A₁ = 1; A_K+1 = A_K-1 + K³ (score: 5)

9, 29, 67, 129, 221, 349, 519, 737, 1009, 1341, 1739, 2209, 2757, 3389, 4111, 4929, 5849, 6877, 8019, 9281, 10669, 12189, 13847, 15649, 17601, 19709, 21979, 24417, 27029, 29821, 32799, 35969, ...

MCS14184 : A_K = K³ + K - 1 (score: 5)

9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, 11025, 14400, 18496, 23409, 29241, 36100, 44100, 53361, 64009, 76176, 90000, 105625, 123201, 142884, 164836, 189225, 216225, 246016, ...

MCS3872 : A₀ = 0; A_K+1 = A_K + K³ (score: 4)

10, 47, 158, 441, 1098, 2539, 5590, 11909, 24818, 50967, 103662, 209521, 421786, 846947, 1697990, 3400893, 6807618, 13622095, 27252190, 54513641, 109037930, 218088027, 436189878, 872395381, ...

MCS7760 : A₀ = 0; A_K+1 = 2 A_K + K³ (score: 5)

10, 57, 292, 1585, 9726, 68425, 547912, 4931937, 49320370, 542525401, 6510306540, 84633987217, ...

MCS3848 (alias MCS30788) : A₀ = 0; A_K+1 = K A_K + K³ (score: 5)

14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, 5474, 6735, 8176, 9809, 11646, 13699, 15980, 18501, 21274, 24311, 27624, 31225, 35126, 39339, 43876, 48749, 53970, 59551, 65504, ...

MCS14130 : A_K = 2 K³ - K (score: 5)

15, 53, 127, 249, 431, 685, 1023, 1457, 1999, 2661, 3455, 4393, 5487, 6749, 8191, 9825, 11663, 13717, 15999, 18521, 21295, 24333, 27647, 31249, 35151, 39365, 43903, 48777, 53999, 59581, 65535, ...

MCS14178 : A_K = 2 K³ - 1 (score: 5)

Statistics:

Complexity level 1 : 1 sequence and 0 aliases.

Complexity level 2 : 8 sequences and 5 aliases.

Complexity level 3 : 43 sequences and 30 aliases.

Complexity level 4 : 207 sequences and 120 aliases.

Complexity level 5 : 862 sequences and 395 aliases.

Higher complexity levels: 14 sequences.

There are 1135 sequences in the entire catalog.

References

[1] Neil J. A. Sloane, A Handbook of Integer Sequences, Academic Press (1973), ISBN 0-12-648550-X.

This book encouraged my developing interest in integer sequences, something that was already a hobby at age 9 after beginning to memorize the powers of 2, 3, 5, 6 and 7. It established many of the guidelines I still follow in my catalogs of sequences (notably this project and most of my work documented in nu-sequences), showing how to put sequences in a definitive order and other important ideas.

[2] Neil J.A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press (1995), ISBN 0-12-558630-2.

Footnotes

1 : recurrence relation : I use a broader definition of recurrence relation than some other authors.

For example, the OEIS has an index that includes many linear recurrence relations (go to this page and see the entries starting with "recurrence, linear"). These are sequences that can be described by formulas of the type

A_N = JA_N-1 + KA_N-2 + LA_N-3 + ...

for integer constants J, K, L, ... The precise name for this type is of sequence definition is a linear homogeneous recurrence relation with constant coefficients; the sequence generated by it is called a linear recursive sequence.

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