Minimally Complex Sequences

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5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, 5, -5, ...

MCS62215 : A₀ = 5; A_N = - A_N-1 (score: 5)

5, 5, 5, -11, -11, -11, 21, 21, 21, -43, -43, -43, 85, 85, 85, -171, -171, -171, 341, 341, 341, -683, -683, -683, 1365, 1365, 1365, -2731, -2731, -2731, 5461, 5461, 5461, -10923, -10923, -10923, ...

MCS26744374 : A₀ = 1; A₁ = 1; A₂ = 1; A_N = - 2 A_N-3 - 1 (score: 5)

5, -5, 9, -8, 13, -11, 17, -14, 21, -17, 25, -20, 29, -23, 33, -26, 37, -29, 41, -32, 45, -35, 49, -38, 53, -41, 57, -44, 61, -47, 65, -50, 69, -53, 73, -56, 77, -59, 81, -62, 85, -65, 89, -68, ...

MCS13816582 : A₀ = 1; A₁ = 1; A_K+1 = - 2 A_K - A_K-1 + K (score: 5)

5, 5, -11, -11, 21, 21, -43, -43, 85, 85, -171, -171, 341, 341, -683, -683, 1365, 1365, -2731, -2731, 5461, 5461, -10923, -10923, 21845, 21845, -43691, -43691, 87381, 87381, -174763, -174763, ...

MCS877249 (alias MCS3508997, MCS3508998, MCS7017997)

A₀ = 0; A₁ = 0; A_N = - 2 A_N-2 - 1 (score: 4)

5, -5, -15, 51, 61, -461, -79, 4699, -3819, -52557, 102217, 633595, -2166835, -7970669, 44806881, 98665179, -949995899, -1023307661, 20973221561, 1539547003, -483923642883, 446974514835, ...

MCS3461174 : A₀ = 1; A₁ = 1; A_K+1 = - A_K - K A_K-1 + K (score: 5)

5, -5, 17, -25, 61, -109, 233, -449, 917, -1813, 3649, -7273, 14573, -29117, 58265, -116497, 233029, -466021, 932081, -1864121, 3728285, -7456525, 14913097, -29826145, 59652341, -119304629, ...

MCS1002320 : A₀ = 1; A_K+1 = - 2 A_K + 2 K + 1 (score: 5)

5, -5, -26, 29, 181, -233, -1630, 2329, 17929, -27949, -233078, 391285, 3496169, -6260561, -59434874, 112690097, 1129262605, -2253801941, -23714514706, 49583642701, 545433838237, ...

MCS109633 (alias MCS3508278) : A₀ = 0; A₁ = 1; A_K+1 = - K A_K-1 + A_K-1 - 1 (score: 5)

Example of a simple recurrence not in OEIS

5, -5, 49, -259, 1861, -14825, 133505, -1334951, 14684581, -176214829, 2290792945, -32071101035, 481066515749, ...

MCS7878 (alias MCS126056) : A₀ = 0; A_K+1 = - K A_K + K² - 1 (score: 5)

5, -6, -9, -20, 1, 6, 27, -16, -27, -70, 15, 36, 121, -50, -93, -264, 77, 162, 503, -180, -351, -1034, 331, 672, 2037, -694, -1377, -4108, 1353, 2718, 8179, -2744, -5475, -16398, 5447, 10908, ...

MCS1597793 : A₀ = 0; A₁ = 0; A₂ = 1; A_K+1 = - 2 A_K-2 - K (score: 5)

5, 6, 13, 16, 25, 30, 41, 48, 61, 70, 85, 96, 113, 126, 145, 160, 181, 198, 221, 240, 265, 286, 313, 336, 365, 390, 421, 448, 481, 510, 545, 576, 613, 646, 685, 720, 761, 798, 841, 880, 925, 966, ...

MCS505544 : A₀ = 1; A_K+1 = - A_K + K² - K - 1 (score: 5)

5, -6, 13, -16, 25, -42, 57, -92, 141, -206, 325, -488, 737, -1138, 1713, -2612, 3989, -6038, 9213, -14016, 21289, -32442, 49321, -75020, 114205, -173662, 264245, -402072, 611569, -930562, ...

MCS1581409 : A₀ = 0; A₁ = 0; A₂ = 1; A_N = A_N-2 - 2 A_N-3 (score: 5)

5, -6, -17, 4, 25, -18, -61, 24, 109, -62, -233, 108, 449, -234, -917, 448, 1813, -918, -3649, 1812, 7273, -3650, -14573, 7272, 29117, -14574, -58265, 29116, 116497, -58266, -233029, 116496, ...

MCS27445 (alias MCS1717633) : A₀ = 0; A₁ = 1; A_K+1 = - 2 A_K-1 - K - 1 (score: 5)

5, 6, 19, 23, 63, 76, 197, 237, 601, 722, 1815, 2179, 5459, 6552, 16393, 19673, 49197, 59038, 147611, 177135, 442855, 531428, 1328589, 1594309, 3985793, 4782954, 11957407, 14348891, 35872251, ...

MCS3347 : A₀ = 0; A₁ = 1; A_K+1 = 3 A_K-1 + K (score: 5)

5, -6, 21, -26, 77, -118, 285, -506, 1093, -2086, 4293, -8442, 17053, -33910, 68045, -135834, 271957, -543590, 1087541, -2174682, 4349805, -8699126, 17398781, -34796986, 69594597, -139188518, ...

MCS124296 : A₀ = 1; A_K+1 = - 2 A_K + K² (score: 4)

5, 6, 31, 43, 249, 388, 2491, 4269, 29893, 55498, 418503, 832471, 6696049, 14152008, 120528883, 268888153, 2410577661, 5646651214, 53032708543, 129872977923, ...

MCS868873 : A₀ = 0; A₁ = 0; A_K+1 = K A_K-1 + A_K-1 + 1 (score: 5)

5, 6, 31, 43, 249, 388, 2491, 4269, 29893, 55498, 418503, 832471, 6696049, 14152008, 120528883, 268888153, 2410577661, 5646651214, 53032708543, 129872977923, ...

MCS434193 : A₀ = -1; A₁ = 0; A_K+1 = K A_K-1 + 1 (score: 5)

5, 7, -3, -17, -11, 23, 45, -1, -91, -89, 93, 271, 85, -457, -627, 287, 1541, 967, -2115, -4049, 181, 8279, 7917, -8641, -24475, -7193, 41757, 56143, -27371, -139657, -84915, 194399, 364229, ...

MCS6725 (alias MCS792753, MCS3443462) : A₀ = 0; A₁ = 1; A_N = A_N-1 - 2 A_N-2 (score: 4)

Lucas and Lehmer numbers with parameters (1+-sqrt(-7))/2 (A107920)

Horadam sequence W_n(0, 1; -2, 1), Lucas sequence U_n(1, 2) ; Horadam sequence W_n(1, 1; -2, 1)

5, -7, -3, 17, -11, -23, 45, 1, -91, 89, 93, -271, 85, 457, -627, -287, 1541, -967, -2115, 4049, 181, -8279, 7917, 8641, -24475, 7193, 41757, -56143, -27371, 139657, -84915, -194399, 364229, ...

MCS3459846 (alias MCS3459845) : A₀ = 1; A₁ = 1; A_N = - A_N-1 - 2 A_N-2 (score: 4)

Horadam sequence W_n(1, 1; -2, -1) ; Horadam sequence W_n(0, -1; -2, -1)

5, 7, 5, -1, -9, -15, -15, -7, 7, 21, 27, 19, -3, -31, -51, -49, -19, 31, 79, 97, 65, -15, -113, -179, -165, -53, 125, 289, 341, 215, -75, -417, -633, -559, -143, 489, 1047, 1189, 699, -349, ...

MCS26871350 : A₀ = 1; A₁ = 1; A₂ = 1; A_N = A_N-1 - A_N-3 - 1 (score: 5)

5, -7, 9, -11, 13, -15, 17, -19, 21, -23, 25, -27, 29, -31, 33, -35, 37, -39, 41, -43, 45, -47, 49, -51, 53, -55, 57, -59, 61, -63, 65, -67, 69, -71, 73, -75, 77, -79, 81, -83, 85, -87, 89, -91, ...

MCS3453702 : A₀ = 1; A₁ = 1; A_N = - 2 A_N-1 - A_N-2 (score: 4)

Horadam sequence W_n(1, 1; -1, -2)

5, 7, 9, 15, 19, 23, 31, 37, 43, 53, 61, 69, 81, 91, 101, 115, 127, 139, 155, 169, 183, 201, 217, 233, 253, 271, 289, 311, 331, 351, 375, 397, 419, 445, 469, 493, 521, 547, 573, 603, 631, 659, ...

MCS12747830 : A₀ = 1; A₁ = 1; A₂ = 1; A_K+1 = A_K-2 + 2 K (score: 5)

5, -7, 9, -15, 21, -31, 45, -67, 97, -143, 209, -307, 449, -659, 965, -1415, 2073, -3039, 4453, -6527, 9565, -14019, 20545, -30111, 44129, -64675, 94785, -138915, 203589, -298375, 437289, -640879, ...

MCS26936886 : A₀ = 1; A₁ = 1; A₂ = 1; A_N = - A_N-1 - A_N-3 - 1 (score: 5)

5, 7, 13, 17, 25, 31, 41, 49, 61, 71, 85, 97, 113, 127, 145, 161, 181, 199, 221, 241, 265, 287, 313, 337, 365, 391, 421, 449, 481, 511, 545, 577, 613, 647, 685, 721, 761, 799, 841, 881, 925, 967, ...

MCS124616 (alias MCS249105) : A₀ = 1; A_K+1 = - A_K + K² - K (score: 4)

5, -7, 13, -23, 37, -63, 109, -183, 309, -527, 893, -1511, 2565, -4351, 7373, -12503, 21205, -35951, 60957, -103367, 175269, -297183, 503917, -854455, 1448821, -2456655, 4165565, -7063207, ...

MCS1622369 : A₀ = 0; A₁ = 0; A₂ = 1; A_N = - A_N-1 - 2 A_N-3 (score: 5)

5, 7, -21, -37, 153, 303, -1369, -3021, 15069, 36263, -195885, -507669, 2938289, 8122719, -49950897, -146208925, 949067061, 2924178519, -19930408261, -64331927397, 458399390025, ...

MCS3510326 : A₀ = 1; A₁ = 1; A_K+1 = - K A_K-1 + K - 1 (score: 5)

5, 7, 29, 47, 209, 383, 1889, 3839, 20789, 46079, 270269, 645119, 4054049, 10321919, 68918849, 185794559, 1309458149, 3715891199, 27498621149, 81749606399, 632468286449, ...

MCS53536 : A₀ = 0; A₁ = 1; A_K+1 = K A_K-1 + A_K-1 + K (score: 5)

5, 7, 41, 71, 493, 995, 7889, 17911, 157781, 394043, 3786745, 10245119, 106028861, 307353571, 3392923553, 10450021415, 122145247909, 397100813771, ...

MCS54274 : A₀ = 0; A₁ = 1; A_K+1 = 2 K A_K-1 + 1 (score: 5)

5, 7, -51, -85, 713, 1359, -12835, -27181, 282369, 652343, -7341595, -18265605, 220247849, 584499359, -7488426867, -21041976925, 284560220945, 841679076999, ...

MCS876589 : A₀ = 0; A₁ = 0; A_K+1 = - 2 K A_K-1 - 1 (score: 5)

5, 8, 11, 17, 25, 32, 39, 49, 61, 72, 83, 97, 113, 128, 143, 161, 181, 200, 219, 241, 265, 288, 311, 337, 365, 392, 419, 449, 481, 512, 543, 577, 613, 648, 683, 721, 761, 800, 839, 881, 925, 968, ...

MCS855173 : A₀ = 0; A₁ = -1; A_K+1 = - A_K-1 + K² (score: 5)

5, -8, 19, -34, 73, -140, 287, -566, 1141, -2272, 4555, -9098, 18209, -36404, 72823, -145630, 291277, -582536, 1165091, -2330162, 4660345, -9320668, 18641359, -37282694, 74565413, -149130800, ...

MCS124320 (alias MCS1010497) : A₀ = 1; A_K+1 = - 2 A_K + K (score: 3)

5, -8, 25, -53, 137, -320, 785, -1881, 4557, -10984, 26537, -64045, 154641, -373312, 901281, -2175857, 5253013, -12681864, 30616761, -73915365, 178447513, -430810368, 1040068273, -2510946889, ...

MCS13492 : A₀ = 0; A₁ = 1; A_K+1 = - 2 A_K + A_K-1 + K (score: 5)

5, -9, -6, 18, -13, -31, 48, 4, -111, 91, 118, -314, 63, 549, -692, -424, 1789, -961, -2638, 4538, 715, -9815, 8360, 11244, -27991, 5475, 50478, -61458, -39529, 162413, -83388, -241472, 408213, ...

MCS13855494 : A₀ = 1; A₁ = 1; A_K+1 = - A_K - 2 A_K-1 - K (score: 5)

5, 9, -7, -23, 5, 53, 13, -111, -79, 209, 269, -339, -747, 409, 1833, -71, -4075, -1691, 8221, 7457, -14751, -23135, 22045, 61021, -20955, -144087, -19111, 309129, 182309, -599147, -673747, ...

MCS25347638 : A₀ = 1; A₁ = 1; A₂ = 1; A_N = - 2 A_N-2 - A_N-3 (score: 5)

5, -9, 9, -14, 13, -19, 17, -24, 21, -29, 25, -34, 29, -39, 33, -44, 37, -49, 41, -54, 45, -59, 49, -64, 53, -69, 57, -74, 61, -79, 65, -84, 69, -89, 73, -94, 77, -99, 81, -104, 85, -109, 89, ...

MCS13818630 : A₀ = 1; A₁ = 1; A_K+1 = - 2 A_K - A_K-1 - K (score: 5)

5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, ...

MCS14155 (alias MCS251712) : A_K = 4 K + 1 (score: 5)

5, -9, 16, -27, 45, -74, 121, -197, 320, -519, 841, -1362, 2205, -3569, 5776, -9347, 15125, -24474, 39601, -64077, 103680, -167759, 271441, -439202, 710645, -1149849, 1860496, -3010347, 4870845, ...

MCS808993 : A₀ = 0; A₁ = 1; A₂ = 1; A_N = - 2 A_N-1 + A_N-3 (score: 5)

5, -9, 17, -31, 57, -105, 193, -355, 653, -1201, 2209, -4063, 7473, -13745, 25281, -46499, 85525, -157305, 289329, -532159, 978793, -1800281, 3311233, -6090307, 11201821, -20603361, 37895489, ...

MCS25986614 : A₀ = 1; A₁ = 1; A₂ = 1; A_N = - A_N-1 + A_N-2 - A_N-3 (score: 5)

5, -9, 29, -65, 181, -441, 1165, -2929, 7589, -19305, 49661, -126881, 325525, -833049, 2135149, -5467345, 14007941, -35877321, 91909085, -235418369, 603054709, -1544728185, 3956947021, ...

MCS13515 : A₀ = 0; A₁ = 1; A_N = - A_N-1 + 4 A_N-2 (score: 6)

(Not in OEIS)

Horadam sequence W_n(0, 1; 4, -1), Lucas sequence U_n(-1, -4)

5, 9, 29, 65, 181, 441, 1165, 2929, 7589, 19305, 49661, 126881, 325525, 833049, 2135149, 5467345, 14007941, 35877321, 91909085, 235418369, 603054709, 1544728185, 3956947021, 10135859761, ...

MCS13451 : A₀ = 0; A₁ = 1; A_N = A_N-1 + 4 A_N-2 (score: 6)

A6131

Horadam sequence W_n(0, 1; 4, 1), Lucas sequence U_n(1, -4)

5, 9, -32, -58, 265, 490, -2708, -5125, 32986, 63917, -466929, -925769, 7534781, 15271144, -136551827, -282616955, 2746307684, 5798404228, -60701386003, -130616989560, ...

MCS6296374 : A₀ = 1; A₁ = 1; A₂ = 1; A_K+1 = - K A_K-1 + A_K-2 (score: 5)

5, -9, 79, -517, 4185, -37601, 376091, -4136901, 49642933, -645357985, 9035011959, -135525179189, ...

MCS7782 : A₀ = 0; A_K+1 = - K A_K - A_K + K² (score: 5)

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, ...

MCS1767 : A_K = 5 K (score: 5)

5, 10, 16, 23, 31, 40, 50, 61, 73, 86, 100, 115, 131, 148, 166, 185, 205, 226, 248, 271, 295, 320, 346, 373, 401, 430, 460, 491, 523, 556, 590, 625, 661, 698, 736, 775, 815, 856, 898, 941, 985, ...

MCS1015072 : A₀ = 1; A_K+1 = A_K + K - 3 (score: 5)

5, 10, 17, 30, 46, 66, 94, 127, 166, 215, 271, 335, 411, 496, 591, 700, 820, 952, 1100, 1261, 1436, 1629, 1837, 2061, 2305, 2566, 2845, 3146, 3466, 3806, 4170, 4555, 4962, 5395, 5851, 6331, 6839, ...

MCS6300726 : A₀ = 1; A₁ = 1; A₂ = 1; A_K+1 = A_K-2 + K² (score: 5)

5, 10, 18, 27, 39, 52, 68, 85, 105, 126, 150, 175, 203, 232, 264, 297, 333, 370, 410, 451, 495, 540, 588, 637, 689, 742, 798, 855, 915, 976, 1040, 1105, 1173, 1242, 1314, 1387, 1463, 1540, 1620, ...

MCS31154 : A₀ = 0; A_K+1 = - A_K + 2 K² - K (score: 5)

5, 10, 21, 35, 57, 84, 121, 165, 221, 286, 365, 455, 561, 680, 817, 969, 1141, 1330, 1541, 1771, 2025, 2300, 2601, 2925, 3277, 3654, 4061, 4495, 4961, 5456, 5985, 6545, 7141, 7770, 8437, 9139, ...

MCS3336 : A₀ = 0; A₁ = 1; A_K+1 = A_K-1 + K² (score: 4)

5, 11, 15, 19, 27, 37, 45, 53, 65, 79, 91, 103, 119, 137, 153, 169, 189, 211, 231, 251, 275, 301, 325, 349, 377, 407, 435, 463, 495, 529, 561, 593, 629, 667, 703, 739, 779, 821, 861, 901, 945, ...

MCS3427462 : A₀ = 1; A₁ = 1; A_K+1 = - A_K-1 + K² + K (score: 5)

5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639, ...

MCS7092 (alias MCS7094, MCS14158, MCS124065, MCS1006160)

A_K = K² + K - 1 (score: 4)

Same as Fibonacci polynomial N²-N-1 (A28387)

5, -11, 21, -43, 85, -171, 341, -683, 1365, -2731, 5461, -10923, 21845, -43691, 87381, -174763, 349525, -699051, 1398101, -2796203, 5592405, -11184811, 22369621, -44739243, 89478485, -178956971, ...

MCS1973 (alias MCS126304, MCS252609, MCS1729797)

A₀ = 0; A_N = - 2 A_N-1 - 1 (score: 3)

Horadam sequence W_n(0, -1; 2, -1)

5, -11, 25, -55, 121, -267, 589, -1299, 2865, -6319, 13937, -30739, 67797, -149531, 329801, -727399, 1604329, -3538459, 7804317, -17212963, 37964385, -83733087, 184679137, -407322659, 898378405, ...

MCS25888310 : A₀ = 1; A₁ = 1; A₂ = 1; A_N = - 2 A_N-1 - A_N-3 (score: 5)

5, 11, -26, -67, 181, 535, -1630, -5351, 17929, 64211, -233078, -898955, 3496169, 14383279, -59434874, -258899023, 1129262605, 5177980459, -23714514706, -113915570099, 545433838237, ...

MCS876598 : A₀ = 1; A₁ = 1; A_K+1 = - K A_K-1 - 1 (score: 4)

5, 11, -29, -76, 233, 685, -2329, -7534, 27949, 97943, -391285, -1469144, 6260561, 24975449, -112690097, -474533530, 2253801941, 9965204131, -49583642701, -229199695012, ...

MCS3476534 : A₀ = 1; A₁ = 1; A_K+1 = - K A_K-1 - A_K-1 + 1 (score: 5)

5, 11, -29, -76, 233, 685, -2329, -7534, 27949, 97943, -391285, -1469144, 6260561, 24975449, -112690097, -474533530, 2253801941, 9965204131, -49583642701, -229199695012, ...

MCS27137 : A₀ = 0; A₁ = 1; A_K+1 = - K A_K-1 + 1 (score: 4)

5, 11, -29, -76, 233, 685, -2329, -7534, 27949, 97943, -391285, -1469144, 6260561, 24975449, -112690097, -474533530, 2253801941, 9965204131, -49583642701, -229199695012, ...

MCS868877 : A₀ = 0; A₁ = 0; A_K+1 = - K A_K-1 + A_K-1 + 1 (score: 5)

5, -11, 51, -161, 773, -3027, 15395, -69881, 377781, -1915163, 10981907, -60776145, 368269541, -2191553891, 13976179203, -88489011497, 591631462805, ...

MCS54018 : A₀ = 0; A₁ = 1; A_K+1 = - A_K + 2 K A_K-1 (score: 5)

5, 11, 51, 161, 773, 3027, 15395, 69881, 377781, 1915163, 10981907, 60776145, 368269541, 2191553891, 13976179203, 88489011497, 591631462805, ...

MCS53762 : A₀ = 0; A₁ = 1; A_K+1 = A_K + 2 K A_K-1 (score: 5)

5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, ...

MCS7827 : A₀ = 0; A_K+1 = A_K + 3 K + 1 (score: 5)

5, -12, 29, -70, 169, -408, 985, -2378, 5741, -13860, 33461, -80782, 195025, -470832, 1136689, -2744210, 6625109, -15994428, 38613965, -93222358, 225058681, -543339720, 1311738121, -3166815962, ...

MCS3372 (alias MCS13813782) : A₀ = 0; A₁ = 1; A_N = - 2 A_N-1 + A_N-2 (score: 4)

Pell numbers with signs (A77985)

Horadam sequence W_n(0, 1; 1, -2), Lucas sequence U_n(-2, -1) ; Horadam sequence W_n(1, 0; 1, -2)

5, 13, 24, 36, 48, 61, 77, 97, 120, 144, 168, 193, 221, 253, 288, 324, 360, 397, 437, 481, 528, 576, 624, 673, 725, 781, 840, 900, 960, 1021, 1085, 1153, 1224, 1296, 1368, 1441, 1517, 1597, 1680, ...

MCS860961 : A₀ = 0; A₁ = 0; A_K+1 = A_K - A_K-1 + K² (score: 5)

5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, ...

MCS248176 : A₀ = 1; A_K+1 = A_K + 4 K (score: 5)

Centered square numbers 2N(N+1)+1 or N²+(N+1)² (A1844)

5, 13, 29, 61, 125, 253, 509, 1021, 2045, 4093, 8189, 16381, 32765, 65533, 131069, 262141, 524285, 1048573, 2097149, 4194301, 8388605, 16777213, 33554429, 67108861, 134217725, 268435453, ...

MCS126624 : A₀ = 1; A_N = 2 A_N-1 + 3 (score: 5)

5, -13, 41, -121, 365, -1093, 3281, -9841, 29525, -88573, 265721, -797161, 2391485, -7174453, 21523361, -64570081, 193710245, -581130733, 1743392201, -5230176601, 15690529805, -47071589413, ...

MCS502752 : A₀ = 1; A_N = - 3 A_N-1 + 2 (score: 5)

5, -14, 23, -52, 97, -202, 395, -800, 1589, -3190, 6367, -12748, 25481, -50978, 101939, -203896, 407773, -815566, 1631111, -3262244, 6524465, -13048954, 26097883, -52195792, 104391557, -208783142, ...

MCS1010529 (alias MCS2012592) : A₀ = -1; A_K+1 = - 2 A_K - K - 1 (score: 5)

5, 14, 27, 44, 65, 90, 119, 152, 189, 230, 275, 324, 377, 434, 495, 560, 629, 702, 779, 860, 945, 1034, 1127, 1224, 1325, 1430, 1539, 1652, 1769, 1890, 2015, 2144, 2277, 2414, 2555, 2700, 2849, ...

MCS28378 : A_K = 2 K² - K - 1 (score: 5)

5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, 7714, 8555, 9455, 10416, 11440, 12529, 13685, 14910, ...

MCS1936 : A₀ = 0; A_K+1 = A_K + K² (score: 3)

Square Pyramidal numbers (A0330)

5, 15, -21, -133, 27, 1215, 935, -12441, -23673, 138047, 469455, -1601265, -9112561, 18108927, 182135007, -161934625, -3804634785, -404007681, 83297957567, 92590134207, ...

MCS861709 : A₀ = 0; A₁ = 0; A_K+1 = A_K - K A_K-1 - K (score: 5)

5, 15, -31, -97, 209, 767, -1891, -7681, 20789, 92159, -270271, -1290241, 4054049, 20643839, -68918851, -371589121, 1309458149, 7431782399, -27498621151, -163499212801, 632468286449, ...

MCS3512374 : A₀ = 1; A₁ = 1; A_K+1 = - K A_K-1 - K - 1 (score: 5)

5, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, 4010, 4641, 5335, 6095, 6924, 7825, 8801, 9855, 10990, 12209, 13515, 14911, 16400, 17985, 19669, 21455, ...

MCS26973 : A₀ = 0; A₁ = 1; A_K+1 = 2 A_K - A_K-1 + 3 K (score: 7)

Magic numbers of magic squares N(N²+1)/2 (A6003)

5, 15, 36, 76, 148, 273, 485, 839, 1424, 2384, 3952, 6505, 10653, 17383, 28292, 45964, 74580, 120905, 195885, 317231, 513600, 831360, 1345536, 2177521, 3523733, 5701983, 9226500, 14929324, ...

MCS860705 : A₀ = 0; A₁ = 0; A_K+1 = A_K + A_K-1 + K² (score: 5)

5, 15, 37, 83, 177, 367, 749, 1515, 3049, 6119, 12261, 24547, 49121, 98271, 196573, 393179, 786393, 1572823, 3145685, 6291411, 12582865, 25165775, 50331597, 100663243, 201326537, 402653127, ...

MCS15658 : A₀ = 0; A_K+1 = 2 A_K + 2 K + 1 (score: 5)

5, -16, 14, -31, 34, -62, 78, -128, 173, -269, 378, -571, 819, -1219, 1767, -2610, 3804, -5597, 8180, -12012, 17580, -25790, 37771, -55383, 81140, -118945, 174293, -255469, 374377, -548708, ...

MCS6513037 : A₀ = 0; A₁ = 0; A₂ = 0; A_K+1 = - A_K - A_K-2 - K (score: 5)

5, -16, 33, -87, 201, -496, 1185, -2875, 6925, -16736, 40385, -97519, 235409, -568352, 1372097, -3312563, 7997205, -19306992, 46611169, -112529351, 271669849, -655869072, 1583407969, -3822685035, ...

MCS13494 : A₀ = 0; A₁ = 1; A_K+1 = - 2 A_K + A_K-1 - K (score: 5)

5, 16, 35, 64, 105, 160, 231, 320, 429, 560, 715, 896, 1105, 1344, 1615, 1920, 2261, 2640, 3059, 3520, 4025, 4576, 5175, 5824, 6525, 7280, 8091, 8960, 9889, 10880, 11935, 13056, 14245, 15504, ...

MCS31560 (alias MCS505032) : A₀ = 0; A_K+1 = A_K + K² + K - 1 (score: 5)

5, -17, 25, -61, 109, -233, 449, -917, 1813, -3649, 7273, -14573, 29117, -58265, 116497, -233029, 466021, -932081, 1864121, -3728285, 7456525, -14913097, 29826145, -59652341, 119304629, ...

MCS2004656 : A₀ = 1; A_K+1 = - 2 A_K - 2 K + 1 (score: 5)

5, 17, 53, 161, 485, 1457, 4373, 13121, 39365, 118097, 354293, 1062881, 3188645, 9565937, 28697813, 86093441, 258280325, 774840977, 2324522933, 6973568801, 20920706405, 62762119217, 188286357653, ...

MCS125664 : A₀ = 1; A_N = 3 A_N-1 + 2 (score: 5)

5, -17, 89, -529, 3709, -29665, 266993, -2669921, 29369141, -352429681, 4581585865, -64142202097, 962133031469, ...

MCS249137 : A₀ = -1; A_K+1 = - K A_K - A_K + K (score: 5)

5, -17, 89, -529, 3709, -29665, 266993, -2669921, 29369141, -352429681, 4581585865, -64142202097, 962133031469, ...

MCS7881 (alias MCS126104, MCS252209) : A₀ = 0; A_K+1 = - K A_K + K - 1 (score: 4)

5, 18, 58, 179, 543, 1636, 4916, 14757, 44281, 132854, 398574, 1195735, 3587219, 10761672, 32285032, 96855113, 290565357, 871696090, 2615088290, 7845264891, 23535794695, 70607384108, 211822152348, ...

MCS974 (alias MCS3918) : A₀ = 0; A_K+1 = 3 A_K + K (score: 4)

5, -18, 85, -492, 3359, -26380, 234061, -2314230, 25222469, -300355398, 3879397705, -54011212472, 806288789375, ...

MCS13517 : A₀ = 0; A₁ = 1; A_K+1 = - K A_K - A_K - A_K-1 (score: 5)

5, -18, 85, -492, 3359, -26380, 234061, -2314230, 25222469, -300355398, 3879397705, -54011212472, 806288789375, ...

MCS427713 : A₀ = -1; A₁ = 0; A_K+1 = - K A_K - A_K-1 (score: 5)

5, 19, 41, 79, 131, 205, 299, 421, 569, 751, 965, 1219, 1511, 1849, 2231, 2665, 3149, 3691, 4289, 4951, 5675, 6469, 7331, 8269, 9281, 10375, 11549, 12811, 14159, 15601, 17135, 18769, 20501, 22339, ...

MCS124612 : A₀ = 1; A_K+1 = - A_K + K³ - K (score: 5)

5, 19, 73, 317, 1621, 9775, 68489, 547993, 4932037, 49320491, 542525545, 6510306709, 84633987413, ...

MCS7780 (alias MCS124488) : A₀ = 0; A_K+1 = K A_K - A_K + K² (score: 5)

5, -20, 106, -655, 4692, -38190, 348403, -3522219, 39092813, -472635974, 6183360476, -87039682637, ...

MCS868961 : A₀ = 0; A₁ = 0; A_K+1 = - K A_K + A_K-1 + 1 (score: 5)

5, 21, -45, -231, 585, 3465, -9945, -65835, 208845, 1514205, -5221125, -40883535, 151412625, 1267389585, -4996616625, -44358635475, 184874815125, ...

MCS3416246 : A₀ = 1; A₁ = 1; A_K+1 = - 2 K A_K-1 + A_K-1 (score: 5)

5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, 1398101, 5592405, 22369621, 89478485, 357913941, 1431655765, 5726623061, 22906492245, 91625968981, 366503875925, ...

MCS3915 (alias MCS250592) : A₀ = 0; A_N = 4 A_N-1 + 1 (score: 5)

(4^N-1)/3 (A2450)

5, -21, 107, -647, 4550, -36507, 329210, -3296650, 36299657, -435925094, 5670322872, -79420819865, ...

MCS197425 : A₀ = 0; A₁ = 1; A₂ = 1; A_K+1 = - K A_K - A_K-2 (score: 5)

5, -22, 107, -646, 4517, -36142, 325271, -3252718, 35779889, -429358678, 5581662803, -78143279254, ...

MCS497368 : A₀ = 1; A_K+1 = - K A_K - 2 A_K - K (score: 5)

5, -22, 107, -646, 4517, -36142, 325271, -3252718, 35779889, -429358678, 5581662803, -78143279254, ...

MCS31341 : A₀ = 0; A_K+1 = - K A_K - A_K - K + 1 (score: 5)

5, 23, 59, 119, 209, 335, 503, 719, 989, 1319, 1715, 2183, 2729, 3359, 4079, 4895, 5813, 6839, 7979, 9239, 10625, 12143, 13799, 15599, 17549, 19655, 21923, 24359, 26969, 29759, 32735, 35903, ...

MCS14188 : A_K = K³ - K - 1 (score: 5)

5, 23, 119, 719, 5039, 40319, 362879, 3628799, 39916799, 479001599, 6227020799, 87178291199, ...

MCS3876 : A₀ = 0; A_K+1 = K A_K + A_K + K (score: 4)

5, 24, 115, 551, 2640, 12649, 60605, 290376, 1391275, 6665999, 31938720, 153027601, 733199285, 3512968824, 16831644835, 80645255351, 386394631920, ...

MCS13553 : A₀ = 0; A₁ = 1; A_N = 5 A_N-1 - A_N-2 (score: 7)

Horadam sequence W_n(0, 1; -1, 5), Lucas sequence U_n(5, 1)

5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 762939453125, ...

MCS62400 : A₀ = 1; A_N = 5 A_N-1 (score: 5)

Powers of 5 (A0351)

5, 26, 77, 190, 429, 922, 1925, 3950, 8021, 16186, 32541, 65278, 130781, 261818, 523925, 1048174, 2096709, 4193818, 8388077, 16776638, 33553805, 67108186, 134216997, 268434670, 536870069, ...

MCS124177 : A₀ = -1; A_K+1 = 2 A_K + K² (score: 5)

5, 26, 135, 701, 3640, 18901, 98145, 509626, 2646275, 13741001, 71351280, 370497401, 1923838285, 9989688826, 51872282415, 269351100901, ...

MCS6776 : A₀ = 0; A₁ = 1; A_N = 5 A_N-1 + A_N-2 (score: 7)

Horadam sequence W_n(0, 1; 1, 5), Lucas sequence U_n(5, -1)

5, -26, 181, -1630, 17929, -233078, 3496169, -59434874, 1129262605, -23714514706, 545433838237, ...

MCS505432 : A₀ = 1; A_K+1 = - 2 K A_K - A_K - 1 (score: 5)

5, 28, 200, 1805, 19861, 258200, 3873008, 65841145, 1250981765, 26270617076, 604224192760, ...

MCS15570 : A₀ = 0; A_K+1 = 2 K A_K - A_K + K (score: 5)

5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, 38613965, 225058681, 1311738121, 7645370045, 44560482149, 259717522849, ...

MCS13882118 : A₀ = 1; A₁ = 1; A_N = 6 A_N-1 - A_N-2 (score: 8)

2X²-1 is a square (A1653)

5, -29, 233, -2329, 27949, -391285, 6260561, -112690097, 2253801941, -49583642701, ...

MCS7813 (alias MCS125016, MCS250033) : A₀ = 0; A_K+1 = - 2 K A_K + 1 (score: 4)

5, -31, 253, -2561, 30985, -436351, 7012601, -126663169, 2540275981, -56012734751, ...

MCS3417478 : A₀ = 1; A₁ = 1; A_K+1 = - 2 K A_K + A_K-1 (score: 5)

5, 33, 13, -119, -171, 305, 989, -231, -4187, -3263, 13485, 26537, -27403, -133551, -23939, 510265, 606021, -1435039, -3859123, 1881033, 17317525, 9793393, -59476707, -98650279, 139256549, ...

MCS26903 : A₀ = 0; A₁ = 1; A_N = A_N-1 - 4 A_N-2 (score: 6)

A106853

Horadam sequence W_n(0, 1; -4, 1), Lucas sequence U_n(1, 4)

5, -34, 33, -114, 165, -410, 721, -1562, 2981, -6130, 12065, -24354, 48453, -97194, 194065, -388490, 776581, -1553602, 3106721, -6213970, 12427365, -24855354, 49710033, -99420794, 198840805, ...

MCS62641 : A₀ = 0; A_K+1 = - 2 A_K - K² + 1 (score: 5)

5, 35, -1, -253, -253, 2015, 4535, -17641, -72073, 157247, 1166255, -1192465, -19852561, 419327, 357765407, 349798175, -6805509985, -14151271681, 135569947967, 461049196607, ...

MCS107713 : A₀ = 0; A₁ = 1; A_K+1 = A_K - K A_K-1 - K (score: 5)

5, -36, -84, 224, 675, -2340, -7546, 27936, 97929, -391300, -1469160, 6260544, 24975431, -112690116, -474533550, 2253801920, 9965204109, -49583642724, -229199695036, ...

MCS213197 : A₀ = 0; A₁ = 0; A_K+1 = - K A_K-1 - K² (score: 5)

5, 39, -51, -469, 713, 7503, -12835, -150061, 282369, 3601463, -7341595, -100840965, 220247849, 3226910879, -7488426867, -116168791645, 284560220945, ...

MCS109573 : A₀ = 0; A₁ = 1; A_K+1 = - 2 K A_K-1 - 1 (score: 5)

5, -49, 259, -1861, 14825, -133505, 1334951, -14684581, 176214829, -2290792945, 32071101035, -481066515749, ...

MCS15629 (alias MCS125036) : A₀ = 0; A_K+1 = - K A_K - K² + 1 (score: 5)

6, 0, 36, 0, 216, 0, 1296, 0, 7776, 0, 46656, 0, 279936, 0, 1679616, 0, 10077696, 0, 60466176, 0, 362797056, 0, 2176782336, 0, 13060694016, 0, 78364164096, 0, 470184984576, 0, ...

MCS6686 : A₀ = 0; A₁ = 1; A_N = 6 A_N-2 (score: 7)

Horadam sequence W_n(0, 1; 6, 0), Lucas sequence U_n(0, -6)

6, 0, 72, 0, 1296, 0, 31104, 0, 933120, 0, 33592320, 0, 1410877440, 0, 67722117120, 0, ...

MCS13315 : A₀ = 0; A₁ = 1; A_K+1 = 3 K A_K-1 (score: 5)

6, -2, 31, -57, 220, -596, 1869, -5507, 16642, -49782, 149515, -448349, 1345272, -4035560, 12106969, -36320583, 108962110, -326885930, 980658231, -2941974209, 8825923156, -26477768892, ...

MCS15608 : A₀ = 0; A_K+1 = - 3 A_K + K² (score: 5)

6, -3, -28, 45, 150, -511, -840, 5949, 3290, -74679, 31908, 1013597, -1492218, -14725335, 40093040, 224962989, -986730750, -3512529031, 24233874780, 53041763901, -610420883842, -662581449783, ...

MCS109953 : A₀ = 0; A₁ = 1; A_K+1 = - A_K - K A_K-1 - 1 (score: 5)

6, -4, -4, 16, -24, 16, 16, -64, 96, -64, -64, 256, -384, 256, 256, -1024, 1536, -1024, -1024, 4096, -6144, 4096, 4096, -16384, 24576, -16384, -16384, 65536, -98304, 65536, 65536, -262144, 393216, ...

MCS13814534 : A₀ = 1; A₁ = 1; A_N = - 2 A_N-1 - 2 A_N-2 (score: 5)

Horadam sequence W_n(1, 1; -2, -2)

6, -4, 26, -20, 90, -108, 314, -500, 1162, -2124, 4490, -8692, 17722, -35052, 70554, -140596, 281770, -562892, 1126506, -2252212, 4505306, -9009644, 18020346, -36039540, 72080330, -144159308, ...

MCS248596 : A₀ = 1; A_K+1 = - 2 A_K + 2 K² (score: 5)

6, 4, 35, 30, 252, 248, 2277, 2490, 25058, 29892, 325767, 418502, 4886520, 6696048, 83070857, 120528882, 1578346302, 2410577660, 33145272363, 53032708542, 762341264372, ...

MCS856101 : A₀ = 0; A₁ = -1; A_K+1 = K A_K-1 + K (score: 5)

6, 4, -40, 12, 308, -416, -2664, 7240, 24728, -118848, -227344, 2010064, 1627440, -35798528, 6504608, 673667424, -803759584, -13343256320, 31025967168, 275868928192, ...

MCS108129 : A₀ = 0; A₁ = 1; A_K+1 = - A_K - K A_K-1 - A_K-1 (score: 5)

6, 4, -40, 12, 308, -416, -2664, 7240, 24728, -118848, -227344, 2010064, 1627440, -35798528, 6504608, 673667424, -803759584, -13343256320, 31025967168, 275868928192, ...

MCS432153 : A₀ = -1; A₁ = 0; A_K+1 = - A_K - K A_K-1 (score: 5)

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