Minimally Complex Sequences  


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15, 0, -105, 0, 945, 0, -10395, 0, 135135, 0, -2027025, 0, 34459425, 0, -654729075, 0, 13749310575, 0, -316234143225, 0, ...

MCS26721 (alias MCS106521) : A0 = 0; A1 = 1; AK+1 = - K AK-1 - AK-1 (score: 4)

15, 0, -105, 0, 945, 0, -10395, 0, 135135, 0, -2027025, 0, 34459425, 0, -654729075, 0, 13749310575, 0, -316234143225, 0, ...

MCS854069 (alias MCS1708141) : A0 = 0; A1 = -1; AK+1 = - K AK-1 + AK-1 (score: 5)

15, 9, -104, -87, 927, 974, -10110, -12615, 130456, 186720, -1944225, -3117976, 32865105, 58067793, -621319019, -1194220965, 12989631606, 26894180249, -297567305973, -658449957582, ...

MCS393649 : A0 = 0; A1 = 0; A2 = 1; AK+1 = - K AK-1 - AK-2 (score: 5)

15, -9, -104, 87, 927, -974, -10110, 12615, 130456, -186720, -1944225, 3117976, 32865105, -58067793, -621319019, 1194220965, 12989631606, -26894180249, -297567305973, 658449957582, ...

MCS393521 : A0 = 0; A1 = 0; A2 = 1; AK+1 = - K AK-1 + AK-2 (score: 5)

15, 24, -105, -192, 945, 1920, -10395, -23040, 135135, 322560, -2027025, -5160960, 34459425, 92897280, -654729075, -1857945600, 13749310575, 40874803200, -316234143225, -980995276800, ...

MCS3419190 : A0 = 1; A1 = 1; AK+1 = - K AK-1 - 2 AK-1 (score: 5)

15, 44, -155, -534, 2163, 8536, -38943, -170730, 856735, 4097508, -22275123, -114730238, 668253675, 3671367600, -22720624967, -132169233618, 863383748727, ...

MCS3432630 : A0 = 1; A1 = 1; AK+1 = - 2 K AK-1 - 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 : AK = 2 K3 - 1 (score: 5)

15, -56, 209, -780, 2911, -10864, 40545, -151316, 564719, -2107560, 7865521, -29354524, 109552575, -408855776, 1525870529, -5694626340, 21252634831, -79315912984, 296011017105, ...

MCS27001 : A0 = 0; A1 = 1; AN = - 4 AN-1 - AN-2 (score: 6)

A125905

Horadam sequence Wn(0, 1; -1, -4), Lucas sequence Un(-4, 1)

15, -62, 245, -984, 3931, -15730, 62913, -251660, 1006631, -4026534, 16106125, -64424512, 257698035, -1030792154, 4123168601, -16492674420, 65970697663, -263882790670, ...

MCS497584 : A0 = 1; AK+1 = - 4 AK - K (score: 5)

15, -63, 58, -143, 177, -346, 495, -871, 1334, -2239, 3537, -5814, 9311, -15167, 24434, -39647, 64033, -103730, 167711, -271495, 439150, -710703, 1149793, -1860558, 3010287, -4870911, 7881130, ...

MCS13851142 : A0 = 1; A1 = 1; AK+1 = - AK + AK-1 - 2 K (score: 5)

15, -76, 455, -3186, 25487, -229384, 2293839, -25232230, 302786759, -3936227868, 55107190151, -826607852266, ...

MCS7897 : A0 = 0; AK+1 = - K AK - AK - 1 (score: 4)

15, -86, 588, -4617, 40966, -405042, 4414497, -52568921, 678981477, -9453171756, 141118594864, ...

MCS13581 : A0 = 0; A1 = 1; AK+1 = - K AK - AK-1 + 1 (score: 5)

15, 105, -75, -1545, -345, 27465, 34365, -569865, -1394625, 13421865, 52471365, -350184585, -2029268265, 9877007625, 82930665165, -292395624585, ...

MCS107525 : A0 = 0; A1 = 1; AK+1 = AK - 2 K AK-1 (score: 5)

15, -105, 945, -10395, 135135, -2027025, 34459425, -654729075, 13749310575, -316234143225, ...

MCS249009 : A0 = -1; AK+1 = - 2 K AK - AK (score: 5)

15, -105, 945, -10395, 135135, -2027025, 34459425, -654729075, 13749310575, -316234143225, ...

MCS247985 : A0 = -1; AK+1 = - 2 K AK + AK (score: 5)

15, -116, 1165, -13974, 195643, -3130280, 56345049, -1126900970, 24791821351, -595003712412, ...

MCS7717 (alias MCS123480, MCS246961) : A0 = 0; AK+1 = - 2 K AK + K (score: 4)

15, -306, 1799, -14904, 133407, -1335070, 14684439, -176214996, 2290792751, -32071101258, 481066515495, ...

MCS7705 (alias MCS30822) : A0 = 0; AK+1 = - K AK - K3 (score: 5)

16, 0, -64, 0, 256, 0, -1024, 0, 4096, 0, -16384, 0, 65536, 0, -262144, 0, 1048576, 0, -4194304, 0, 16777216, 0, -67108864, 0, 268435456, 0, -1073741824, 0, 4294967296, 0, -17179869184, 0, ...

MCS6679 : A0 = 0; A1 = 1; AN = - 4 AN-2 (score: 5)

(Not in OEIS)

Horadam sequence Wn(0, 1; -4, 0), Lucas sequence Un(0, 4)

16, 13, -35, -74, 31, 253, 160, -599, -1079, 718, 3955, 1801, -10064, -15467, 14725, 61126, 16951, -166427, -217280, 282001, 933841, 87838, -2713685, -2977199, 5163856, 14095453, -1396115, ...

MCS13455 (alias MCS6889222) : A0 = 0; A1 = 1; AN = AN-1 - 3 AN-2 (score: 5)

A106852

Horadam sequence Wn(0, 1; -3, 1), Lucas sequence Un(1, 3) ; Horadam sequence Wn(1, 1; -3, 1)

16, 16, -64, -64, 256, 256, -1024, -1024, 4096, 4096, -16384, -16384, 65536, 65536, -262144, -262144, 1048576, 1048576, -4194304, -4194304, 16777216, 16777216, -67108864, -67108864, 268435456, ...

MCS3419910 : A0 = 1; A1 = 1; AN = - 4 AN-2 (score: 5)

Horadam sequence Wn(1, 1; -4, 0)

16, -10, -108, 56, 848, -522, -8500, 5720, 101976, -74386, -1427692, 1115760, 22843040, -18967954, -411174756, 360391088, 8223495080, -7568212890, -180916891804, 174068896424, ...

MCS107201 : A0 = 0; A1 = 1; AK+1 = - K AK-1 - 2 K (score: 5)

16, -64, 256, -1024, 4096, -16384, 65536, -262144, 1048576, -4194304, 16777216, -67108864, 268435456, -1073741824, 4294967296, -17179869184, 68719476736, -274877906944, ...

MCS124384 : A0 = 1; AN = - 4 AN-1 (score: 4)

16, -90, 528, -3710, 29664, -266994, 2669920, -29369142, 352429680, -4581585866, 64142202096, -962133031470, ...

MCS7725 (alias MCS123608, MCS247217) : A0 = 0; AK+1 = - K AK - 2 K (score: 4)

17, 39, -171, -469, 2393, 7503, -43075, -150061, 947649, 3601463, -24638875, -100840965, 739166249, 3226910879, -25131652467, -116168791645, 955002793745, ...

MCS3506358 : A0 = 1; A1 = 1; AK+1 = - 2 K AK-1 - 1 (score: 5)

17, -66, 267, -1064, 4261, -17038, 68159, -272628, 1090521, -4362074, 17448307, -69793216, 279172877, -1116691494, 4466765991, -17867063948, 71468255809, -285873023218, ...

MCS497568 : A0 = 1; AK+1 = - 4 AK + K (score: 5)

17, -72, 305, -1292, 5473, -23184, 98209, -416020, 1762289, -7465176, 31622993, -133957148, 567451585, -2403763488, 10182505537, -43133785636, 182717648081, -774004377960, ...

MCS13500 : A0 = 0; A1 = 1; AN = - 4 AN-1 + AN-2 (score: 6)

(Not in OEIS)

Horadam sequence Wn(0, 1; 1, -4), Lucas sequence Un(-4, -1)

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

MCS124632 : A0 = 1; AK+1 = - K AK - AK - K (score: 4)

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

MCS7821 (alias MCS125144, MCS250289) : A0 = 0; AK+1 = - K AK - K + 1 (score: 4)

18, 28, -152, -261, 1510, 2860, -18132, -37193, 253834, 557880, -4061360, -9483977, 73104462, 180195544, -1462089260, -3784106445, 32165963698, 87034448212, -771983128776, ...

MCS858165 : A0 = 0; A1 = -1; AK+1 = - K AK-1 - K (score: 5)

18, -80, 356, -1584, 7048, -31360, 139536, -620864, 2762528, -12291840, 54692416, -243353344, 1082798208, -4817899520, 21437194496, -95384577024, 424412697088, ...

MCS54002 : A0 = 0; A1 = 1; AN = - 4 AN-1 + 2 AN-2 (score: 7)

Horadam sequence Wn(0, 1; 2, -4), Lucas sequence Un(-4, -2)

19, 29, -151, -260, 1511, 2861, -18131, -37192, 253835, 557881, -4061359, -9483976, 73104463, 180195545, -1462089259, -3784106444, 32165963699, 87034448213, -771983128775, ...

MCS869133 : A0 = 0; A1 = 0; AK+1 = - K AK-1 - AK-1 + 1 (score: 5)

19, 29, -151, -260, 1511, 2861, -18131, -37192, 253835, 557881, -4061359, -9483976, 73104463, 180195545, -1462089259, -3784106444, 32165963699, 87034448213, -771983128775, ...

MCS3473622 : A0 = 1; A1 = 0; AK+1 = - K AK-1 + 1 (score: 5)

19, -11, -179, 311, 2195, -7171, -32339, 175759, 535699, -4753915, -9174259, 142283879, 132943891, -4686028019, 165935725, 168531072959, -174836630509, ...

MCS3457206 : A0 = 1; A1 = 1; AK+1 = - AK - 2 K AK-1 (score: 5)

19, -77, 307, -1229, 4915, -19661, 78643, -314573, 1258291, -5033165, 20132659, -80530637, 322122547, -1288490189, 5153960755, -20615843021, 82463372083, -329853488333, ...

MCS505312 : A0 = 1; AN = - 4 AN-1 - 1 (score: 5)

19, -111, 641, -4523, 36135, -325279, 3252709, -35779899, 429358667, -5581662815, 78143279241, ...

MCS124524 : A0 = 1; AK+1 = - K AK - AK - K2 (score: 5)

20, 20, -120, -280, 800, 3600, -5200, -48400, 19200, 696800, 408800, -10740000, -17689600, 175630400, 511732800, -3000875200, -13747264000, 52271990400, 368459062400, -886068707200, ...

MCS26881 : A0 = 0; A1 = 1; AK+1 = AK - K AK-1 (score: 4)

20, 20, -120, -280, 800, 3600, -5200, -48400, 19200, 696800, 408800, -10740000, -17689600, 175630400, 511732800, -3000875200, -13747264000, 52271990400, 368459062400, -886068707200, ...

MCS3443766 : A0 = 1; A1 = 1; AK+1 = AK - K AK-1 - AK-1 (score: 5)

20, 35, -234, -483, 3752, 8703, -75030, -191455, 1800732, 4977843, -50420482, -149335275, 1613455440, 5077399367, -58084395822, -192941175927, ...

MCS107013 : A0 = 0; A1 = 1; AK+1 = - 2 K AK-1 + K (score: 5)

20, -20, -120, 280, 800, -3600, -5200, 48400, 19200, -696800, 408800, 10740000, -17689600, -175630400, 511732800, 3000875200, -13747264000, -52271990400, 368459062400, 886068707200, ...

MCS864309 : A0 = 0; A1 = -1; AK+1 = - AK - K AK-1 (score: 5)

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

MCS877153 : A0 = 0; A1 = 0; AK+1 = - K AK + AK-1 - 1 (score: 5)

21, 0, -231, 0, 3465, 0, -65835, 0, 1514205, 0, -40883535, 0, 1267389585, 0, -44358635475, 0, ...

MCS106757 : A0 = 0; A1 = 1; AK+1 = - 2 K AK-1 + AK-1 (score: 5)

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

MCS3420342 : A0 = 1; A1 = 1; AK+1 = - 2 K AK-1 - AK-1 (score: 5)

21, -11, -153, 81, 1369, -819, -15069, 9817, 195885, -137451, -2938289, 2199201, 49950897, -39585635, -949067061, 791712681, 19930408261, -17417679003, -458399390025, 418024296049, ...

MCS107361 : A0 = 0; A1 = 1; AK+1 = - K AK-1 - AK-1 - K (score: 5)

24, 0, -192, 0, 1920, 0, -23040, 0, 322560, 0, -5160960, 0, 92897280, 0, -1857945600, 0, 40874803200, 0, -980995276800, 0, ...

MCS106849 : A0 = 0; A1 = 1; AK+1 = - K AK-1 - 2 AK-1 (score: 5)

24, -93, 131, -328, 578, -1247, 2389, -4898, 9660, -19473, 38775, -77740, 155270, -310771, 621289, -1242854, 2485408, -4971141, 9941931, -19884240, 39768074, -79536583, 159072701, -318145898, ...

MCS1731905 : A0 = 0; A1 = 0; AK+1 = - AK + 2 AK-1 - K (score: 5)

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

MCS27005 : A0 = 0; A1 = 1; AN = - 5 AN-1 - AN-2 (score: 7)

Horadam sequence Wn(0, 1; -1, -5), Lucas sequence Un(-5, 1)

24, -135, 792, -5565, 44496, -400491, 4004880, -44053713, 528644520, -6872378799, 96213303144, ...

MCS15485 (alias MCS247768) : A0 = 0; AK+1 = - K AK - 3 K (score: 5)

24, -212, 2337, -30375, 455632, -7745736, 147168993, -3090548843, 71082623400, ...

MCS31141 : A0 = 0; AK+1 = - 2 K AK - AK + K (score: 5)

25, 0, -125, 0, 625, 0, -3125, 0, 15625, 0, -78125, 0, 390625, 0, -1953125, 0, 9765625, 0, -48828125, 0, 244140625, 0, -1220703125, 0, 6103515625, 0, -30517578125, 0, 152587890625, 0, ...

MCS13359 : A0 = 0; A1 = 1; AN = - 5 AN-2 (score: 6)

(Not in OEIS)

Horadam sequence Wn(0, 1; -5, 0), Lucas sequence Un(0, 5)

25, 51, -299, -713, 4785, 12835, -95699, -282369, 2296777, 7341595, -64309755, -220247849, 2057912161, 7488426867, -74084837795, -284560220945, ...

MCS108549 : A0 = 0; A1 = 1; AK+1 = - 2 K AK-1 + 1 (score: 5)

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

MCS124400 : A0 = 1; AN = - 5 AN-1 (score: 5)

25, -134, 793, -5564, 44497, -400490, 4004881, -44053712, 528644521, -6872378798, 96213303145, ...

MCS31277 (alias MCS500440) : A0 = 0; AK+1 = - K AK - 2 K + 1 (score: 5)

25, -137, 841, -6031, 49081, -447769, 4526761, -50242151, 607432561, -7946865457, 111863548945, ...

MCS3434886 : A0 = 1; A1 = 1; AK+1 = - K AK + AK-1 - K (score: 5)

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

MCS13502 : A0 = 0; A1 = 1; AN = - 5 AN-1 + AN-2 (score: 7)

Horadam sequence Wn(0, 1; 1, -5), Lucas sequence Un(-5, -1)

27, 72, -405, -1296, 8505, 31104, -229635, -933120, 7577955, 33592320, -295540245, -1410877440, 13299311025, 67722117120, -678264862275, ...

MCS1704438 : A0 = 1; A1 = 1; AK+1 = - 3 K AK-1 (score: 5)

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

MCS7877 (alias MCS126040, MCS252081) : A0 = 0; AK+1 = - 2 K AK - 1 (score: 4)

29, -260, 2861, -37192, 557881, -9483976, 180195545, -3784106444, 87034448213, ...

MCS31301 : A0 = 0; AK+1 = - 2 K AK + AK + 1 (score: 5)

29, -260, 2861, -37192, 557881, -9483976, 180195545, -3784106444, 87034448213, ...

MCS501336 : A0 = 1; AK+1 = - 2 K AK - AK + 1 (score: 5)

31, -191, 1331, -10655, 95887, -958879, 10547659, -126571919, 1645434935, -23036089103, 345541336531, ...

MCS7789 : A0 = 0; AK+1 = - K AK - AK - K (score: 4)

32, 0, -384, 0, 6144, 0, -122880, 0, 2949120, 0, -82575360, 0, 2642411520, 0, -95126814720, 0, ...

MCS26629 : A0 = 0; A1 = 1; AK+1 = - 2 K AK-1 (score: 4)

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, ...

MCS13373 : A0 = 0; A1 = 1; AN = - 6 AN-2 (score: 7)

Horadam sequence Wn(0, 1; -6, 0), Lucas sequence Un(0, 6)

39, -271, 1847, -14841, 133487, -1334971, 14684559, -176214853, 2290792919, -32071101063, 481066515719, ...

MCS15757 (alias MCS126060) : A0 = 0; AK+1 = - K AK - K2 - 1 (score: 5)

39, -464, 6965, -125364, 2632651, -63183616, 1705957641, -51178729220, ...

MCS15439 (alias MCS247032) : A0 = 0; AK+1 = - 3 K AK + K (score: 5)

40, -270, 1848, -14840, 133488, -1334970, 14684560, -176214852, 2290792920, -32071101062, 481066515720, ...

MCS15437 (alias MCS123500) : A0 = 0; AK+1 = - K AK - K2 + K (score: 5)

41, 31, -461, -895, 6481, 22591, -107029, -604031, 1964665, 17669471, -37341149, -567425279, 627491489, 19919950975, -2669742629, -759627879679, -652838174519, ...

MCS3440822 : A0 = 1; A1 = 1; AK+1 = AK - 2 K AK-1 (score: 5)

43, -266, 1853, -14834, 133495, -1334962, 14684569, -176214842, 2290792931, -32071101050, 481066515733, ...

MCS15533 : A0 = 0; AK+1 = - K AK + 2 AK - K (score: 5)

43, -266, 1853, -14834, 133495, -1334962, 14684569, -176214842, 2290792931, -32071101050, 481066515733, ...

MCS31565 (alias MCS505048) : A0 = 0; AK+1 = - K AK + AK - K - 1 (score: 5)

43, -266, 1853, -14834, 133495, -1334962, 14684569, -176214842, 2290792931, -32071101050, 481066515733, ...

MCS31437 (alias MCS503000) : A0 = 0; AK+1 = - K AK - K - 2 (score: 5)

45, 0, -585, 0, 9945, 0, -208845, 0, 5221125, 0, -151412625, 0, 4996616625, 0, -184874815125, 0, ...

MCS106885 : A0 = 0; A1 = 1; AK+1 = - 2 K AK-1 - AK-1 (score: 5)

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

MCS868397 : A0 = 0; A1 = 0; AK+1 = - 2 K AK-1 + 1 (score: 5)

52, -473, 5197, -67568, 1013512, -17229713, 327364537, -6874655288, 158117071612, ...

MCS31029 : A0 = 0; AK+1 = - 2 K AK + AK - K (score: 5)

56, -585, 6984, -97825, 1565136, -28172529, 563450480, -12395910681, 297501856200, ...

MCS15405 (alias MCS123244) : A0 = 0; AK+1 = - 2 K AK - K2 (score: 5)

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

MCS53263 : A0 = 0; A1 = 1; AK+1 = - 3 K AK-1 (score: 5)

Statistics:

Complexity level 1 : 1 sequence and 0 aliases.

Complexity level 2 : 12 sequences and 5 aliases.

Complexity level 3 : 90 sequences and 44 aliases.

Complexity level 4 : 545 sequences and 183 aliases.

Complexity level 5 : 2595 sequences and 725 aliases.

Complexity level 6 : 22 sequences and 5 aliases.

Complexity level 7 : 27 sequences and 8 aliases.

Higher complexity levels: 16 sequences.

There are 3308 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

AN = JAN-1 + KAN-2 + LAN-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.

In addition to those terms, I also allow terms like:

... + J + KN + LN2 + ...

and

... + KNAN-1 + LNAN-2 + ...


Quick Index:
    Sequences Beginning 2,0,...
    Sequences Beginning 2,1,...
    Sequences Beginning 2,2,0,... ; 2,2,1,... ; etc.
    Sequences Beginning 2,2,4,... ; 2,2,5,... ; etc.
    Sequences Beginning 2,3,0,... ; 2,3,1,... ; etc.
    Sequences Beginning 2,3,4,... ; 2,3,5,... ; etc.
    Sequences Beginning 2,4,...
    Sequences Beginning 2,5,... ; 2,6,... and 2,7,...
    Sequences Beginning 2,8,... ; 2,9,... ; etc.
    Sequences Beginning 3,0,... ; 3,1,... ; etc.
    Sequences Beginning 3,4,... ; 3,5,... ; etc.
    Sequences Beginning 3,8,... ; 3,9,... ; etc.
    Sequences Beginning 4,0,... ; 4,1,... ; etc.
    Sequences Beginning 4,6,... ; 4,7,... ; etc.
    Sequences Beginning 5,...
    Sequences Beginning 6,... ; 7,... ; etc.



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