The first , 40, terms of the enumerating sequences of walks with given negati\ ve and positive steps that start and end at the x-axis and never go above the x-axis By Shalosh B. Ekhad In this book we will the first, 40, terms of the enumerating sequences for ALL families of walks consisting of at least one negative step, at least on\ e positive steps where the size of each step is between 1 and, 4 and that start and end at the x-axis, but never go above the x-axis We also supply the first, 40, terms of the enumerating sequence Fact Number, 1 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420] ---------------------------------------- This ends Fact No. , that took, 0.061, seconds to generate. Fact Number, 2 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 0, 0, 3, 0, 0, 12, 0, 0, 55, 0, 0, 273, 0, 0, 1428, 0, 0, 7752, 0, 0, 43263, 0, 0, 246675, 0, 0, 1430715, 0, 0, 8414640, 0, 0, 50067108, 0, 0, 300830572, 0] ---------------------------------------- This ends Fact No. , 2 that took, 0.063, seconds to generate. Fact Number, 3 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 22, 0, 0, 0, 140, 0, 0, 0, 969, 0, 0, 0, 7084 , 0, 0, 0, 53820, 0, 0, 0, 420732, 0, 0, 0, 3362260, 0, 0, 0, 27343888] ---------------------------------------- This ends Fact No. , 3 that took, 0.066, seconds to generate. Fact Number, 4 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 0, 1, 0, 0, 0, 0, 5, 0, 0, 0, 0, 35, 0, 0, 0, 0, 285, 0, 0, 0, 0, 2530, 0, 0, 0, 0, 23751, 0, 0, 0, 0, 231880, 0, 0, 0, 0, 2330445] ---------------------------------------- This ends Fact No. , 4 that took, 0.072, seconds to generate. Fact Number, 5 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 5, 8, 21, 42, 96, 222, 495, 1177, 2717, 6435, 15288, 36374, 87516, 210494, 509694, 1237736, 3014882, 7370860, 18059899, 44379535, 109298070, 269766655, 667224480, 1653266565, 4103910930, 10203669285, 25408828065, 63364046190, 158229645720, 395632288590, 990419552730, 2482238709888, 6227850849066, 15641497455612, 39322596749218, 98948326105928] ---------------------------------------- This ends Fact No. , 5 that took, 0.077, seconds to generate. Fact Number, 6 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 0, 11, 0, 46, 0, 207, 0, 979, 0, 4797, 0, 24138, 0, 123998, 0, 647615, 0, 3428493, 0, 18356714, 0, 99229015, 0, 540807165, 0, 2968468275, 0, 16395456762, 0, 91053897066, 0, 508151297602, 0, 2848290555562, 0, 16028132445156] ---------------------------------------- This ends Fact No. , 6 that took, 0.081, seconds to generate. Fact Number, 7 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 1, 5, 7, 14, 36, 47, 165, 198, 715, 975, 3038, 5070, 13056, 26282, 58140, 133361, 271320, 663005, 1321925, 3256156, 6636190, 15954525, 33839325, 78650340, 173501055, 392109696, 889942575, 1980045702, 4561360584, 10114212086, 23385422632, 52117367400, 120140085654, 270136289970, 619450211718, 1405360442161] ---------------------------------------- This ends Fact No. , 7 that took, 0.093, seconds to generate. Fact Number, 8 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 1, 0, 3, 7, 4, 12, 45, 55, 77, 286, 546, 728, 1960, 4760, 7548, 15504 , 39729, 75582, 140448, 336490, 723327, 1366200, 2992990, 6758895, 13522275, 28094040, 63183315, 133231800, 273896532, 600805296, 1305229332, 2720740792, 5843241088, 12797739672, 27206642716, 57941746476, 126405822608] ---------------------------------------- This ends Fact No. , 8 that took, 0.100, seconds to generate. Fact Number, 9 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 0, 1, 3, 0, 8, 12, 5, 55, 55, 78, 364, 308, 840, 2380, 2244, 7752, 15789, 19722, 65835, 109802, 184943, 533830, 822250, 1726725, 4242420, 6667245, 15705066, 33763743, 57500040, 139107664, 273687964, 513312756, 1209003348, 2282162960, 4639186984, 10411233160, 19607898141] ---------------------------------------- This ends Fact No. , 9 that took, 0.110, seconds to generate. Fact Number, 10 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 1, 1, 0, 0, 4, 9, 5, 0, 22, 78, 91, 35, 140, 680, 1224, 969, 1254, 5985, 14630, 17710, 17710, 55660, 164450, 269100, 299520, 593775, 1805076, 3681405, 4951692, 7594752, 20173560, 47303520, 76404460, 110676324, 239784864, 589602585, 1106339923] ---------------------------------------- This ends Fact No. , 10 that took, 0.118, seconds to generate. Fact Number, 11 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 3, 5, 14, 28, 74, 168, 432, 1045, 2684, 6721, 17355, 44408, 115502, 299812, 785570, 2060094, 5434475, 14362841, 38114760, 101360402, 270373303, 722696570, 1936398635, 5198249550, 13982513625, 37674988080, 101685303765, 274867141845, 744093631842, 2017066320624, 5474900965050, 14878450339822, 40479971557162, 110253945275970, 300605644859552, 820399033872096, 2241084167717824] ---------------------------------------- This ends Fact No. , 11 that took, 0.126, seconds to generate. Fact Number, 12 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 6, 8, 28, 50, 132, 317, 715, 1903, 4368, 11349, 27881, 69974, 179112, 448154, 1156986, 2939585, 7571462, 19517522, 50314110, 130672775, 338599560, 882260680, 2299665810, 6007600185, 15730319490, 41221684716, 108277980108, 284671947798, 749663283856, 1976656376706, 5217762017738, 13791262142388, 36487025005964, 96641647221652, 256211125739260, 679892663951293] ---------------------------------------- This ends Fact No. , 12 that took, 0.137, seconds to generate. Fact Number, 13 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 1, 11, 7, 46, 45, 212, 275, 1045, 1651, 5434, 9863, 29458, 58956, 165002, 353685, 948290, 2132655, 5561547, 12934119, 33146588, 78914495, 200119090, 484339635, 1220942985, 2989617540, 7513839651, 18553008216, 46577756370, 115714828152, 290519299114, 725066526924, 1821729163342, 4562705514102, 11476617891706, 28825208705787, 72599069116587] ---------------------------------------- This ends Fact No. , 13 that took, 0.148, seconds to generate. Fact Number, 14 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 1, 1, 3, 7, 12, 21, 50, 110, 209, 442, 1001, 2128, 4480, 9860, 21828, 47481, 103968, 231192, 513513, 1136982, 2533289, 5672260, 12695540, 28448355, 63972675, 144134640, 324977016, 733931913, 1660780236, 3762635044, 8533415880, 19379626068, 44067267808, 100303245980, 228530941928, 521230949853, 1189938422855] ---------------------------------------- This ends Fact No. , 14 that took, 0.162, seconds to generate. Fact Number, 15 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-1, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 3, 6, 14, 35, 82, 213, 527, 1375, 3542, 9308, 24544, 65191, 174302, 467908, 1263134, 3421539, 9306884, 25395191, 69515965, 190821202, 525163025, 1448785065, 4005590760, 11097459750, 30803796285, 85656062715, 238578660846, 665549820519, 1859357891730, 5201651013628, 14570714678278, 40864877919842, 114741473444302, 322526012437664, 907525112122184, 2556109053671163, 7206193195047795] ---------------------------------------- This ends Fact No. , 15 that took, 0.173, seconds to generate. Fact Number, 16 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 0, 0, 3, 0, 0, 12, 0, 0, 55, 0, 0, 273, 0, 0, 1428, 0, 0, 7752, 0, 0, 43263, 0, 0, 246675, 0, 0, 1430715, 0, 0, 8414640, 0, 0, 50067108, 0, 0, 300830572, 0] ---------------------------------------- This ends Fact No. , 16 that took, 0.176, seconds to generate. Fact Number, 17 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 0, 2, 0, 0, 0, 0, 23, 0, 0, 0, 0, 377, 0, 0, 0, 0, 7229, 0, 0, 0, 0, 151491, 0, 0, 0, 0, 3361598, 0, 0, 0, 0, 77635093, 0, 0, 0, 0, 1846620581] ---------------------------------------- This ends Fact No. , 17 that took, 0.186, seconds to generate. Fact Number, 18 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 7, 8, 38, 58, 199, 452, 1149, 3277, 7650, 22696, 55726, 157502, 416967, 1128026, 3122336, 8365304, 23402737, 63505268, 176860650, 487957967, 1353427722, 3774616133, 10483218667, 29371164344, 81965145468, 230030965231, 645265199252, 1813615497166, 5107394107927, 14386545035342, 40621735594210, 114720169872202, 324560293765296, 918870098708832, 2604241833793991, 7388579097551618] ---------------------------------------- This ends Fact No. , 18 that took, 0.201, seconds to generate. Fact Number, 19 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 3, 2, 3, 23, 59, 74, 178, 753, 1859, 3299, 8937, 29884, 73955, 160368 , 445889, 1334825, 3371535, 8167687, 22732271, 64550448, 166944853, 429281385, 1189787311, 3299504856, 8708248080, 23118437489, 63845014804, 175463878127, 470269479575, 1270311652558, 3501884445317, 9604857045847, 26027895342456, 71002490056153, 195692892371919, 537321155970160, 1467430337299719] ---------------------------------------- This ends Fact No. , 19 that took, 0.249, seconds to generate. Fact Number, 20 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 0, 0, 17, 0, 0, 204, 0, 0, 2848, 0, 0, 43335, 0, 0, 697194, 0, 0, 11663971, 0, 0, 200866092, 0, 0, 3537092364, 0, 0, 63397398732, 0, 0, 1152780381018, 0, 0, 21213118756674, 0, 0, 394302904331090, 0] ---------------------------------------- This ends Fact No. , 20 that took, 0.266, seconds to generate. Fact Number, 21 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 2, 5, 17, 14, 103, 65, 544, 515, 2671, 4333, 12920, 32888, 66569, 225063, 389929, 1426875, 2581052, 8652846, 18130991, 51937472, 127733905, 318505753, 879213643, 2034543521, 5892047281, 13539791786, 38764350879, 92547902870, 253609842517, 638716733669, 1670011621498, 4398787899731, 11151727980457, 30093346643625, 75616292374270, 204712934528781] ---------------------------------------- This ends Fact No. , 21 that took, 0.290, seconds to generate. Fact Number, 22 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 0, 2, 3, 0, 23, 12, 23, 207, 55, 560, 1697, 650, 8694, 13252, 15654, 109435, 107696, 324043, 1216953, 1117762, 5441300, 12614367, 16242055, 78430366 , 130185358, 272242662, 1016036652, 1464897950, 4443146662, 12308432648, 19158617589, 67181508815, 145724647279, 282813865564, 943204291083, 1773587239499, 4347509699662] ---------------------------------------- This ends Fact No. , 22 that took, 0.322, seconds to generate. Fact Number, 23 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 9, 31, 78, 248, 705, 2196, 6632, 20780, 64709, 204902, 650000, 2080483, 6683564, 21593311, 70024903, 228022074, 744976876, 2441850778, 8026618762, 26455041139, 87405982153, 289438774174, 960462359139, 3193366842536 , 10636635056279, 35489063311272, 118596791583351, 396914141297320, 1330230442462987, 4464042344334714, 14999217181926990, 50456596364848778, 169921812232536963, 572844723715864685, 1933116776188266041, 6529668152176835624] ---------------------------------------- This ends Fact No. , 23 that took, 0.349, seconds to generate. Fact Number, 24 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 2, 12, 22, 59, 212, 475, 1594, 4750, 13196, 43366, 127252, 391668, 1242498, 3776182, 11978944, 37660098, 118023500, 376764859, 1192384620, 3800996683, 12174327988, 38911906522, 125118761512, 402609044674, 1297297674737 , 4193738729864, 13564531538068, 43963910619655, 142729843336414, 463830423116109, 1509860070733024, 4920351768127197, 16052692656100348, 52437597284067130, 171454732353827446, 561193883071486554, 1838643773175111072] ---------------------------------------- This ends Fact No. , 24 that took, 0.406, seconds to generate. Fact Number, 25 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 3, 2, 17, 56, 82, 266, 1056, 2410, 6157, 22307, 64019, 167114, 536678 , 1678687, 4717571, 14279619, 45164770, 134485030, 403157361, 1261106436, 3873230891, 11739955681, 36411477173, 113212662921, 347970789239, 1077895267865 , 3364407062887, 10448760154791, 32475170028780, 101539568008979, 317371203029902, 991053166160102, 3104840517480353, 9742039327727277, 30552763936651858, 95956784274092846, 301893368320000304] ---------------------------------------- This ends Fact No. , 25 that took, 0.444, seconds to generate. Fact Number, 26 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 7, 8, 38, 65, 199, 543, 1246, 4035, 9556, 28818, 78122, 214933, 629873, 1709610, 5021439, 14093846, 40526640, 117212832, 334644878, 977139174, 2813214366, 8201080331, 23876445760, 69542023022, 203777286111, 595381763799, 1748718032444, 5133556543817, 15099066239211, 44495128105383, 131147147823020, 387426686901620, 1144903084134301, 3388301146564357, 10037246686293247, 29756939669744309] ---------------------------------------- This ends Fact No. , 26 that took, 0.481, seconds to generate. Fact Number, 27 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 5, 14, 45, 132, 425, 1382, 4514, 15074, 50789, 172559, 591478, 2041556, 7087471, 24743296, 86792677, 305748394, 1081274484, 3837319132, 13661640800, 48780359677, 174641599757, 626785406849, 2254641398394, 8127379935041, 29354406339309, 106215414640325, 384982365317509, 1397609307709572, 5081353804008476, 18500467219189397, 67446577033587605, 246194571085717498, 899722241895396679, 3291716072107537857, 12055767543764840101, 44197992941498268988, 162189883599470455418] ---------------------------------------- This ends Fact No. , 27 that took, 0.547, seconds to generate. Fact Number, 28 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 22, 0, 0, 0, 140, 0, 0, 0, 969, 0, 0, 0, 7084 , 0, 0, 0, 53820, 0, 0, 0, 420732, 0, 0, 0, 3362260, 0, 0, 0, 27343888] ---------------------------------------- This ends Fact No. , 28 that took, 0.550, seconds to generate. Fact Number, 29 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 0, 2, 0, 0, 0, 0, 23, 0, 0, 0, 0, 377, 0, 0, 0, 0, 7229, 0, 0, 0, 0, 151491, 0, 0, 0, 0, 3361598, 0, 0, 0, 0, 77635093, 0, 0, 0, 0, 1846620581] ---------------------------------------- This ends Fact No. , 29 that took, 0.560, seconds to generate. Fact Number, 30 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 227, 0, 0, 0, 0, 0, 0, 15090, 0, 0, 0, 0, 0, 0, 1182187, 0, 0, 0, 0, 0, 0, 101527596, 0, 0, 0, 0, 0] ---------------------------------------- This ends Fact No. , 30 that took, 0.585, seconds to generate. Fact Number, 31 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 1, 2, 17, 17, 43, 220, 322, 877, 3495, 6513, 18246, 63069, 137364, 389520, 1240075, 2986569, 8518188, 25878573, 66493272, 190276431, 563345305, 1509236554, 4329167366, 12645267502, 34810974533, 100065738510, 290410780163, 813932210810, 2344530239608, 6787557305833, 19254309739598, 55576193661986, 160849076903780, 460095808260232, 1330726621028529, 3854609838686679, 11091289883698738] ---------------------------------------- This ends Fact No. , 31 that took, 0.610, seconds to generate. Fact Number, 32 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 0, 16, 0, 100, 0, 655, 0, 4465, 0, 31599, 0, 230390, 0, 1717910, 0 , 13034753, 0, 100308732, 0, 781057488, 0, 6142515700, 0, 48719605150, 0, 389274014325, 0, 3130375135624, 0, 25315962247754, 0, 205765906922296, 0, 1679968849194124, 0, 13771490153093158] ---------------------------------------- This ends Fact No. , 32 that took, 0.638, seconds to generate. Fact Number, 33 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 1, 6, 10, 5, 4, 62, 343, 905, 1257, 1417, 6084, 31434, 100115, 203536 , 327731, 862489, 3632078, 12563173, 31362425, 63158050, 147961944, 501825641, 1732253546, 4863749789, 11364582752, 26888942585, 79260215036, 259851147222, 772393923288, 1993593104771, 4932908109711, 13607109223658, 41964005993095, 126676016918733, 348075079482190, 903382764123312, 2442678607763572] ---------------------------------------- This ends Fact No. , 33 that took, 0.731, seconds to generate. Fact Number, 34 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 2, 5, 22, 14, 164, 65, 1030, 657, 5868, 7463, 31765, 73575, 173849 , 631556, 1053086, 4877803, 7526655, 34948691, 61382672, 239407864, 524309309, 1621763388, 4415274965, 11255289437, 35813332389, 82153463817, 279458110861, 633479334487, 2118070224696, 5075741777630, 15824514104397, 41275863623366, 118428013850973, 334523141763061, 899513350738990, 2678023253678681] ---------------------------------------- This ends Fact No. , 34 that took, 0.776, seconds to generate. Fact Number, 35 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 3, 2, 0, 5, 57, 76, 23, 225, 1712, 3096, 2152, 10301, 63344, 135945, 145447, 506300, 2645684, 6334817, 8863943, 26321836, 119924174, 308996066, 518884034, 1421869726, 5774838471, 15623015827, 29889917272, 78796739552, 291401326703, 813202815623, 1712626557848, 4443708508164, 15256966498903, 43361944116807, 98123257670293, 253748908058282, 822563978400235] ---------------------------------------- This ends Fact No. , 35 that took, 0.841, seconds to generate. Fact Number, 36 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 0, 5, 5, 14, 61, 42, 486, 132, 3197, 656, 18845, 6942, 103469, 82337, 540610, 842389, 2738390, 7457658, 13890377, 59018977, 75015627, 428615420, 464370812, 2911703107, 3361816428, 18787058761, 26847573680, 116858010417, 219643015240, 713315739910, 1760480543434, 4372055494146, 13583447394347, 27625019510098, 100561624744208, 183868132825246] ---------------------------------------- This ends Fact No. , 36 that took, 0.922, seconds to generate. Fact Number, 37 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 3, 16, 33, 115, 390, 1087, 4060, 12555, 42953, 148067, 492739, 1735298, 5944320, 20744252, 72905575, 254998049, 903660769, 3195209422, 11355589072, 40507136044, 144620988953, 518478617875, 1861257943227, 6697455408050, 24152234870325, 87226107921628, 315651869078757, 1143924927595869, 4151936835886485, 15091681888691404, 54925223488389666, 200157938880285184, 730258764785275647, 2667274260621421838, 9752675597285646950, 35695329641773808896, 130773052695581564343] ---------------------------------------- This ends Fact No. , 37 that took, 0.975, seconds to generate. Fact Number, 38 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 4, 8, 27, 88, 255, 816, 2652, 8484, 28010, 93519, 311311, 1047079, 3557270, 12116686, 41440458, 142533861, 492066696, 1703991980, 5920941155, 20635503666, 72103747481, 252568582748, 886756183614, 3119870447538, 10998136966961, 38841818713274, 137410335685786, 486890269898723, 1727802215388888, 6139995444922663, 21848255855472370, 77840947711380018, 277659218074236324, 991518669728587943, 3544452787305998644, 12683338799748237350, 45428910418404359917] ---------------------------------------- This ends Fact No. , 38 that took, 1.070, seconds to generate. Fact Number, 39 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 6, 26, 71, 215, 615, 1879, 5923, 19174, 62515, 203978, 666587, 2187956, 7228831, 24041213, 80384059, 269853218, 908761480, 3069017067, 10393205043, 35292352343, 120152439959, 410025684695, 1402229574135, 4804786159886, 16493558252776, 56714159449203, 195327621458483, 673736857127415 , 2327191353954248, 8049180928555926, 27875013503713839, 96647746918324587, 335470346187629947, 1165674067152210376, 4054500970667408497, 14116050642587817343] ---------------------------------------- This ends Fact No. , 39 that took, 1.161, seconds to generate. Fact Number, 40 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 5, 2, 28, 27, 202, 301, 1602, 3177, 13910, 33016, 128266, 342820, 1239098, 3577198, 12386907, 37605752, 127076336, 398632495, 1329629938, 4261217915, 14127149566, 45919144718, 151935298816, 498568851090, 1650229625236 , 5450869527868, 18070744589130, 59971995488605, 199251965396011, 663615612911163, 2210052989218112, 7381286080871931, 24640359998387002, 82485730945578910, 275977010346735803, 925685842356775708, 3103603047697663752] ---------------------------------------- This ends Fact No. , 40 that took, 1.235, seconds to generate. Fact Number, 41 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 6, 22, 66, 252, 848, 3218, 11725, 44502, 169026, 649237, 2518289, 9812886, 38556241, 152092831, 603423111, 2403322940, 9611508342, 38572610558, 155297193204, 627098136122, 2539022823888, 10305860644106, 41926721236162, 170931650675469, 698245783792833, 2857537567523935, 11714387363887240, 48099819158068437, 197797296952047735, 814536604784177900, 3358755787761435674, 13867263598687916962, 57321499879000098991, 237208898321773127799, 982665100210659188886, 4074906222703825280497, 16913964344146139238827] ---------------------------------------- This ends Fact No. , 41 that took, 1.343, seconds to generate. Fact Number, 42 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 0, 1, 0, 0, 0, 0, 5, 0, 0, 0, 0, 35, 0, 0, 0, 0, 285, 0, 0, 0, 0, 2530, 0, 0, 0, 0, 23751, 0, 0, 0, 0, 231880, 0, 0, 0, 0, 2330445] ---------------------------------------- This ends Fact No. , 42 that took, 1.347, seconds to generate. Fact Number, 43 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 227, 0, 0, 0, 0, 0, 0, 15090, 0, 0, 0, 0, 0, 0, 1182187, 0, 0, 0, 0, 0, 0, 101527596, 0, 0, 0, 0, 0] ---------------------------------------- This ends Fact No. , 43 that took, 1.372, seconds to generate. Fact Number, 44 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 3, 1, 3, 33, 58, 43, 296, 1200, 1818, 3494, 17873, 49287, 83946, 260847, 983319, 2307549, 5082075, 17509957, 53854968, 125275193, 338513207, 1112445860, 3069338189, 7671175586, 22775279046, 69565099251, 185664861112, 502807217041, 1518656669438, 4396497183671, 11879683466176, 33895573100792, 100841006640410, 284721688580645, 791082977546277, 2306752037114424, 6729787675037793] ---------------------------------------- This ends Fact No. , 44 that took, 1.410, seconds to generate. Fact Number, 45 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 0, 1, 17, 5, 23, 218, 141, 479, 3409, 3360, 10222, 59920, 77551, 225011, 1137849, 1786350, 5076452, 22845326, 41403956, 116692083, 478629332, 967676930, 2721775080, 10371270281, 22809972634, 64235360622, 230930262906, 542073072103, 1530933120481, 5257647560709, 12980407935867, 36794121686530, 121918287788365, 313013699354907, 890779421265519, 2870517483600895, 7596948088888540] ---------------------------------------- This ends Fact No. , 45 that took, 1.508, seconds to generate. Fact Number, 46 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 1, 5, 16, 14, 157, 47, 1225, 317, 8349, 4089, 52059, 53480, 306209 , 603957, 1759030, 5957945, 10465777, 52788674, 70145619, 430029068, 557385522, 3284474477, 5040523814, 23975716309, 47705189899, 171065437002, 445644890659, 1226653861390, 4008301648929, 9110184969092, 34526452192001, 71613003642538, 285920884207268, 596901552164051, 2296186192981959, 5184623116789519] ---------------------------------------- This ends Fact No. , 46 that took, 1.641, seconds to generate. Fact Number, 47 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 0, 6, 3, 5, 76, 12, 338, 718, 547, 7698, 6287, 31069, 119744, 104392, 918064, 1572814, 3837810, 18998514, 24320474, 123215460, 323755183, 617719052, 3030388182, 5507346575, 18634628139, 61805367626, 118698802731, 498452214396, 1178980116448, 3182517234928, 11477928525904, 24575988302660, 86121282203504, 242166326375800, 600650481324438, 2130520453125270, 5170687302732134] ---------------------------------------- This ends Fact No. , 47 that took, 1.708, seconds to generate. Fact Number, 48 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 0, 5, 5, 14, 75, 42, 710, 132, 5415, 656, 36330, 8442, 223790, 127096, 1296780, 1599366, 7191825, 17000284, 39029291, 158784108, 216908922, 1342917656, 1339175203, 10504886569, 9895257442, 77204253552, 86005434774, 540143631646, 804196436716, 3646888468726, 7512093516175, 24180968667854, 67709007362894, 161292207078680, 582735668471090, 1114680033125948] ---------------------------------------- This ends Fact No. , 48 that took, 1.832, seconds to generate. Fact Number, 49 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 3, 3, 7, 52, 104, 309, 1512, 4103, 13483, 55946, 176679, 611424, 2366567 , 8076977, 28797150, 108439122, 385616674, 1400682434, 5228118841, 19022963870, 69962427824, 260924098247, 962566138443, 3571509151852, 13351205153535, 49697934407487, 185620330580446, 696245098141498, 2608158788832913, 9791671225078267, 36856810723754976, 138729393617351062, 522992182413374808, 1975081475916088209, 7462553157784059215, 28229843679565303229, 106926333323479296929, 405273860833298537790] ---------------------------------------- This ends Fact No. , 49 that took, 1.934, seconds to generate. Fact Number, 50 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 8, 41, 87, 362, 1028, 3722, 11974, 42353, 143059, 505254, 1759797, 6232410, 22110186, 78864073, 282815530, 1016705079, 3674145774, 13299594962, 48355251113, 176071288163, 643381979793, 2354505174602, 8640415951567, 31755910657689, 116970256096214, 431479020116474, 1594526638445841, 5900640133259486, 21869028592831998, 81153889574490442, 301550544397058064, 1121793388897790260, 4177980311592751250, 15576732629781357097, 58134189364844379047, 217170880969920008713] ---------------------------------------- This ends Fact No. , 50 that took, 2.084, seconds to generate. Fact Number, 51 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 2, 15, 22, 97, 288, 824, 3102, 9204, 32651, 109292, 368392, 1293128, 4407884, 15485107, 54098005, 189918002, 672404562, 2377111088, 8461864724, 30163628101, 107831922872, 386706240333, 1389000550245, 5002604881563, 18048926417904, 65241923293743, 236263608449912, 856888545036236, 3112789267430932, 11323472633987856, 41247386778048645, 150442417056681492, 549360231881172775, 2008361895366503951, 7350049036451878478, 26926575646754690704, 98739950253472417485] ---------------------------------------- This ends Fact No. , 51 that took, 2.235, seconds to generate. Fact Number, 52 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 13, 8, 109, 134, 766, 2222, 5667, 27825, 57823, 291595, 758231, 2888278, 10019673, 30553000, 122532818, 362736708, 1415922277, 4594544294, 16288545726, 58193764683, 194253053346, 720179999943, 2412584922755, 8793964423849, 30562231165838, 108005451494368, 387301565469087, 1349166079379134, 4882357369086739, 17113374293107050, 61501403849121275, 218700782342812768, 779036582786730989, 2799184161473324138, 9949047480322149410, 35839494559135870332] ---------------------------------------- This ends Fact No. , 52 that took, 2.377, seconds to generate. Fact Number, 53 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 5, 28, 90, 300, 1340, 4720, 19224, 78820, 309895, 1299231, 5338874, 22148050, 93434582, 391938929, 1662056604, 7074526063, 30168464264, 129408353368, 556198907308, 2398053883570, 10371629134077, 44948387189829, 195299032310816, 850344880548907, 3709508198783748, 16213694839495304, 70985794462036252, 311290410148601782, 1367172855288292117, 6012971476311705743 , 26481376365070704301, 116771849758445934030, 515526878612169852893, 2278532324221682665744, 10081407770093577248123, 44650557420113720402486, 197947643610523846362502] ---------------------------------------- This ends Fact No. , 53 that took, 2.537, seconds to generate. Fact Number, 54 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 5, 8, 21, 42, 96, 222, 495, 1177, 2717, 6435, 15288, 36374, 87516, 210494, 509694, 1237736, 3014882, 7370860, 18059899, 44379535, 109298070, 269766655, 667224480, 1653266565, 4103910930, 10203669285, 25408828065, 63364046190, 158229645720, 395632288590, 990419552730, 2482238709888, 6227850849066, 15641497455612, 39322596749218, 98948326105928] ---------------------------------------- This ends Fact No. , 54 that took, 2.542, seconds to generate. Fact Number, 55 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 7, 8, 38, 58, 199, 452, 1149, 3277, 7650, 22696, 55726, 157502, 416967, 1128026, 3122336, 8365304, 23402737, 63505268, 176860650, 487957967, 1353427722, 3774616133, 10483218667, 29371164344, 81965145468, 230030965231, 645265199252, 1813615497166, 5107394107927, 14386545035342, 40621735594210, 114720169872202, 324560293765296, 918870098708832, 2604241833793991, 7388579097551618] ---------------------------------------- This ends Fact No. , 55 that took, 2.556, seconds to generate. Fact Number, 56 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 1, 2, 17, 17, 43, 220, 322, 877, 3495, 6513, 18246, 63069, 137364, 389520, 1240075, 2986569, 8518188, 25878573, 66493272, 190276431, 563345305, 1509236554, 4329167366, 12645267502, 34810974533, 100065738510, 290410780163, 813932210810, 2344530239608, 6787557305833, 19254309739598, 55576193661986, 160849076903780, 460095808260232, 1330726621028529, 3854609838686679, 11091289883698738] ---------------------------------------- This ends Fact No. , 56 that took, 2.581, seconds to generate. Fact Number, 57 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 3, 1, 3, 33, 58, 43, 296, 1200, 1818, 3494, 17873, 49287, 83946, 260847, 983319, 2307549, 5082075, 17509957, 53854968, 125275193, 338513207, 1112445860, 3069338189, 7671175586, 22775279046, 69565099251, 185664861112, 502807217041, 1518656669438, 4396497183671, 11879683466176, 33895573100792, 100841006640410, 284721688580645, 791082977546277, 2306752037114424, 6729787675037793] ---------------------------------------- This ends Fact No. , 57 that took, 2.619, seconds to generate. Fact Number, 58 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 11, 24, 93, 272, 971, 3194, 11293, 39148, 139687, 497756, 1798002, 6517194, 23807731, 87336870, 322082967, 1192381270, 4431889344, 16527495396, 61831374003, 231973133544, 872598922407, 3290312724374, 12434632908623, 47089829065940, 178672856753641, 679155439400068, 2585880086336653, 9861191391746256, 37660870323158835, 144029959800495438, 551546279543420059, 2114684919809270434, 8117356580480783638, 31193334574672753772, 119994768635233629431, 462054434301743595662, 1780873197452044558004] ---------------------------------------- This ends Fact No. , 58 that took, 2.635, seconds to generate. Fact Number, 59 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 6, 21, 66, 206, 694, 2343, 8006, 27865, 97842, 346560, 1238017, 4451859, 16104105, 58569206, 214013423, 785324563, 2892811352, 10692822131, 39649034086, 147443120646, 549749019862, 2054764213960, 7697272862049, 28894655660026, 108677590661657, 409493420065062, 1545562470596778, 5842680517890137, 22119801344728755, 83860065166879578, 318345575635570632, 1209984597470883971, 4604353717435642583, 17540344670612420506, 66890292116966006476, 255340921774236374052, 975637911204231539435] ---------------------------------------- This ends Fact No. , 59 that took, 2.686, seconds to generate. Fact Number, 60 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 5, 19, 45, 168, 519, 1718, 5923, 19772, 69314, 239913, 843440, 2987539, 10607907, 37999955, 136469506, 492661835, 1785189586, 6489630289, 23673676337, 86591259225, 317602030338, 1167735170362, 4303093334571, 15890533775030, 58793965087761, 217931442312430, 809172627869999, 3009199409999055, 11207473083719999, 41799473690206514, 156101189392112522, 583687975336595193, 2185067701649255408, 8189009346853774636, 30722376511573875180, 115375093192928177806, 433692106341616988780] ---------------------------------------- This ends Fact No. , 60 that took, 2.729, seconds to generate. Fact Number, 61 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 3, 23, 48, 155, 612, 1609, 6255, 20608, 67954, 250621, 837858, 2997773, 10682234, 37447731, 135767710, 484626014, 1747304695, 6345838687, 22949010094, 83737139716, 305552771525, 1117272519230, 4101926037674, 15064915295407, 55488468018578, 204729501895013, 756350275118646, 2800027056148971, 10377918836812794, 38521413439022433, 143184883556986540, 532814894354798905, 1985211915360494824, 7404609038051674655, 27647122999848204744, 103334287176052280814, 386579072024085298844] ---------------------------------------- This ends Fact No. , 61 that took, 2.757, seconds to generate. Fact Number, 62 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 5, 13, 45, 126, 409, 1330, 4334, 14379, 48547, 164243, 561757, 1935335, 6702697, 23343984, 81711201, 287179404, 1013288990, 3587966035, 12744788675, 45402974551, 162181082707, 580738971260, 2084245423378, 7496032481347, 27012370625607, 97518043503485, 352651739855174, 1277315657327468, 4633389008133849, 16830948220061852, 61219815349778619, 222954622656362950, 812928026349030113, 2967367732392875777, 10842988100678903747, 39660807051648167870, 145206957905349494122] ---------------------------------------- This ends Fact No. , 62 that took, 2.824, seconds to generate. Fact Number, 63 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 3, 4, 3, 42, 117, 171, 949, 3537, 7632, 28765, 115189, 315534, 1040806, 4055775, 12773335, 41226403, 153201604, 518969067, 1705182586, 6118352563, 21406751812, 72154670394, 254356245660, 900150854840, 3099873232059, 10871104611942, 38581306372934, 134854878746180, 473650031509210, 1681877247755816, 5934007868931549, 20928312762436191, 74366048412855518, 263879579937685840, 934890115024132352, 3326678645979956643, 11847241110192142045, 42142027265956747320] ---------------------------------------- This ends Fact No. , 63 that took, 2.864, seconds to generate. Fact Number, 64 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 15, 54, 197, 778, 3046, 12378, 50688, 210821, 885836, 3755794, 16053550, 69077136, 299051044, 1301497997, 5691174700, 24991961429, 110168793923, 487328507125, 2162490768266, 9623634039899, 42941087514502, 192072611056724, 861064794485586, 3868232998947027, 17411324425937991, 78511976428487851, 354628109413966644, 1604341160570722942, 7268823842760461184 , 32978945219886339519, 149823281943148384308, 681490024904553949414, 3103471368624092952988, 14148708406910701942775, 64571602595986047193440, 294984276632934745080426, 1348861148942366519449640] ---------------------------------------- This ends Fact No. , 64 that took, 2.898, seconds to generate. Fact Number, 65 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 14, 43, 164, 617, 2375, 9465, 37776, 154315, 633321, 2632378, 11010055, 46404863, 196718589, 838474916, 3590934328, 15444460191, 66683955076, 288921124823, 1255802647438, 5474219850958, 23926618975537, 104834748880803, 460376934797848, 2025973157117432, 8933078015857335, 39460076600107456, 174603055385215294, 773812426883313328, 3434525009893030608, 15265318066197588429, 67938771200586087448, 302739905433425969561, 1350616317896028768618, 6032220951129635074688, 26969950798997812341575, 120702816205128656679356, 540713729414372654541714] ---------------------------------------- This ends Fact No. , 65 that took, 2.974, seconds to generate. Fact Number, 66 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 9, 35, 125, 453, 1728, 6635, 26019, 103175, 413415, 1673562, 6824013, 28020830, 115766799, 480795028, 2006376164, 8408428486, 35373640393, 149333199475, 632422174306, 2686049488173, 11438629470361, 48831003190507, 208928783583116, 895797735746866, 3848277802797746, 16561892402742964, 71398420127824528, 308287098220305259, 1333114870664775287, 5772799599221969986 , 25030912431116987558, 108669028818295142400, 472328430829679812265, 2055248165770684461994, 8952434515586478345212, 39034757378159103550152, 170362396512970263333300] ---------------------------------------- This ends Fact No. , 66 that took, 3.022, seconds to generate. Fact Number, 67 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 6, 29, 107, 310, 1285, 4755, 17072, 68097, 261074, 1008449, 4019538, 15856730, 63120997, 253955943, 1021134596, 4130964757, 16792455962, 68395002809 , 279669185840, 1147006764628, 4714858037720, 19433651768421, 80280672686204, 332305871983609, 1378320168716942, 5727177767108796, 23837465768727622, 99375775609570540, 414899721471863616, 1734653866111787864, 7262031828592394561 , 30439797146760801971, 127742603344567668902, 536677521075129555005, 2257085847121359584834, 9502047618152094682026, 40040473246167731114949] ---------------------------------------- This ends Fact No. , 67 that took, 3.069, seconds to generate. Fact Number, 68 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-2, -1, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 18, 73, 290, 1223, 5259, 23049, 102703, 463508, 2114586, 9737060, 45192097, 211199028, 992997848, 4693803931, 22293043296, 106332602178, 509135524570, 2446319933288, 11791527427505, 57001393740128, 276284001879588, 1342423134398429, 6537407560700816, 31903039573287447, 155992783588117250, 764126173919465124, 3749399159675148078, 18426685432205212254, 90694091814132640031, 447010928136072932856, 2206120331479291546459, 10901335981633850369590, 53931185491617579479953, 267105334498062695387913, 1324287541095964236040935, 6572277200614525363625511, 32648388198061017486845493] ---------------------------------------- This ends Fact No. , 68 that took, 3.151, seconds to generate. Fact Number, 69 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 0, 11, 0, 46, 0, 207, 0, 979, 0, 4797, 0, 24138, 0, 123998, 0, 647615, 0, 3428493, 0, 18356714, 0, 99229015, 0, 540807165, 0, 2968468275, 0, 16395456762, 0, 91053897066, 0, 508151297602, 0, 2848290555562, 0, 16028132445156] ---------------------------------------- This ends Fact No. , 69 that took, 3.155, seconds to generate. Fact Number, 70 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 3, 2, 3, 23, 59, 74, 178, 753, 1859, 3299, 8937, 29884, 73955, 160368 , 445889, 1334825, 3371535, 8167687, 22732271, 64550448, 166944853, 429281385, 1189787311, 3299504856, 8708248080, 23118437489, 63845014804, 175463878127, 470269479575, 1270311652558, 3501884445317, 9604857045847, 26027895342456, 71002490056153, 195692892371919, 537321155970160, 1467430337299719] ---------------------------------------- This ends Fact No. , 70 that took, 3.180, seconds to generate. Fact Number, 71 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 0, 16, 0, 100, 0, 655, 0, 4465, 0, 31599, 0, 230390, 0, 1717910, 0 , 13034753, 0, 100308732, 0, 781057488, 0, 6142515700, 0, 48719605150, 0, 389274014325, 0, 3130375135624, 0, 25315962247754, 0, 205765906922296, 0, 1679968849194124, 0, 13771490153093158] ---------------------------------------- This ends Fact No. , 71 that took, 3.208, seconds to generate. Fact Number, 72 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 0, 1, 17, 5, 23, 218, 141, 479, 3409, 3360, 10222, 59920, 77551, 225011, 1137849, 1786350, 5076452, 22845326, 41403956, 116692083, 478629332, 967676930, 2721775080, 10371270281, 22809972634, 64235360622, 230930262906, 542073072103, 1530933120481, 5257647560709, 12980407935867, 36794121686530, 121918287788365, 313013699354907, 890779421265519, 2870517483600895, 7596948088888540] ---------------------------------------- This ends Fact No. , 72 that took, 3.301, seconds to generate. Fact Number, 73 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 6, 21, 66, 206, 694, 2343, 8006, 27865, 97842, 346560, 1238017, 4451859, 16104105, 58569206, 214013423, 785324563, 2892811352, 10692822131, 39649034086, 147443120646, 549749019862, 2054764213960, 7697272862049, 28894655660026, 108677590661657, 409493420065062, 1545562470596778, 5842680517890137, 22119801344728755, 83860065166879578, 318345575635570632, 1209984597470883971, 4604353717435642583, 17540344670612420506, 66890292116966006476, 255340921774236374052, 975637911204231539435] ---------------------------------------- This ends Fact No. , 73 that took, 3.331, seconds to generate. Fact Number, 74 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 0, 13, 0, 120, 0, 1288, 0, 15046, 0, 185658, 0, 2380720, 0, 31411376, 0, 423660504, 0, 5814905977, 0, 80956085304, 0, 1140478875656, 0, 16227516683124, 0, 232870988052180, 0, 3366482778363616, 0, 48981220255732960, 0, 716707681487535144, 0, 10539913681632290532, 0, 155697664218428455520, 0, 2309297999296926348448] ---------------------------------------- This ends Fact No. , 74 that took, 3.362, seconds to generate. Fact Number, 75 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 3, 21, 38, 156, 552, 1512, 6317, 19891, 70514, 262600, 886366, 3324275, 11917581, 43075004, 160465736, 582553382, 2165827825, 8035397386, 29765494555, 111532020794, 416139099821, 1561328302808, 5873799646865, 22096414953525, 83473571039613, 315506419432058, 1194838387405440, 4534322636772920, 17221949385346138, 65535146413192692, 249671343922622483, 952305694778871837, 3637275047915282859, 13905699591780752658, 53223860882997743324, 203918412648437136447, 781994588942192034001] ---------------------------------------- This ends Fact No. , 75 that took, 3.474, seconds to generate. Fact Number, 76 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 6, 15, 52, 156, 528, 1765, 6114, 21236, 74776, 265291, 949595, 3422399, 12410903, 45247919, 165752707, 609808036, 2252176796, 8347161607, 31035445149, 115729542652, 432703786186, 1621835578983, 6092706372588, 22936590083350, 86516207497332, 326932457744595, 1237536547815069, 4691914691117079, 17815222622885223, 67739562218872488, 257909787594052190, 983184309422800372, 3752440379470237568, 14337614322431343300, 54840178875567735030, 209969537892201797328, 804687981679200752808] ---------------------------------------- This ends Fact No. , 76 that took, 3.529, seconds to generate. Fact Number, 77 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 3, 6, 3, 42, 193, 300, 1045, 6099, 16734, 43217, 210817, 782925, 2204773 , 8485820, 34912398, 114022747, 394138850, 1586046526, 5759557242, 19841733394, 75718209771, 288068491153, 1028137670356, 3799702261969, 14538125472039, 53684595930388, 197344794259924, 748242277946288, 2817171295843109, 10447324282076692, 39320074452507852, 149043614017066577, 558940601895391705, 2101140641931237724, 7969855121731563589, 30131317547692018637, 113620558686097683264, 430852951323620245061] ---------------------------------------- This ends Fact No. , 77 that took, 3.637, seconds to generate. Fact Number, 78 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 3, 15, 33, 114, 374, 1148, 4014, 13218, 45251, 156691, 540459, 1898433, 6670470, 23602581, 83999017, 299828723, 1075578570, 3870451106, 13973579863, 50606121977, 183742900600, 668874849424, 2440432319190, 8923053466466, 32691099204201, 119988702131927, 441167062238030, 1624669451725207, 5992148068258060, 22131837671897606, 81852496903185977, 303105431236230035, 1123756293191475308, 4170990262745090702, 15497773816780295343, 57641948516878681768, 214597818608804245249] ---------------------------------------- This ends Fact No. , 78 that took, 3.717, seconds to generate. Fact Number, 79 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 16, 48, 208, 778, 3305, 13499, 57999, 247426, 1080038, 4725641, 20929207, 93118686, 417432294, 1879871543, 8510737402, 38686261748, 176564376942, 808602162394, 3715180084791, 17119059401564, 79095591109170, 366346002995878, 1700682157965819, 7911704506752742, 36878195675123781, 172211459271608956, 805555550951269942, 3774174295168643551, 17709186843741843257, 83212029736653279793, 391515337546259034843, 1844394597724020439367, 8699040178460691523419, 41074630787462631404420, 194149125687057441136325, 918614252117154387538888, 4350560194974760150572558] ---------------------------------------- This ends Fact No. , 79 that took, 3.801, seconds to generate. Fact Number, 80 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 5, 9, 42, 182, 593, 2581, 10957, 43420, 188471, 813065, 3463970, 15218878, 66912366, 294387495, 1310788491, 5851890115, 26203798664, 118014897338, 532997318579, 2414210327413, 10973073735408, 49997307201615, 228355683827178, 1045524124363610, 4796707597515697, 22049637042067194, 101548378550822286, 468464987749445297, 2164590099847430866, 10016842234891063409, 46418949804128268189, 215395376850006865160, 1000742784297131319331, 4655016518949470669326, 21677341163691048135998, 101053555648429442926525, 471556590217779488901190, 2202577602437903342889461] ---------------------------------------- This ends Fact No. , 80 that took, 3.930, seconds to generate. Fact Number, 81 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 13, 35, 147, 549, 2160, 8867, 35765, 150772, 628464, 2683757, 11456093, 49473738, 214539108, 936161664, 4104468256, 18071975575, 79898601922, 354443312181, 1577645717021, 7042409083701, 31523012983176, 141450890769500, 636189953631244, 2867417631833429, 12949532955572112, 58589145884064847, 265537708473571351, 1205407567101080169, 5480195896811295332, 24950170906420844504, 113744350209745417424, 519196937329810060808, 2372739294180022605528, 10855647444031754020668, 49719138352905825910456, 227944931325178226874747, 1046051034104785013658717] ---------------------------------------- This ends Fact No. , 81 that took, 4.024, seconds to generate. Fact Number, 82 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 9, 30, 114, 435, 1632, 6579, 26242, 106833, 439395, 1825652, 7640859, 32214100, 136676305, 582912766, 2498452213, 10754837780, 46476863187, 201558528689, 876928763837, 3826500214994, 16742047926682, 73433163298618, 322826846258108, 1422223345190754, 6277996996195399, 27763285161878786, 122988389682909472, 545698946551522530, 2424903800447054625, 10790682240448018921, 48081758717108570141, 214514159797536743989, 958176738963435645716, 4284711813074472760815, 19180433623946052630684, 85947346068967519992406, 385497851671482105467914] ---------------------------------------- This ends Fact No. , 82 that took, 4.147, seconds to generate. Fact Number, 83 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -1, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 19, 83, 347, 1558, 7163, 33363, 158849, 764668, 3723905, 18311102, 90753476, 453017546, 2275288314, 11490033498, 58306220057, 297160459897, 1520424304907, 7806833844973, 40214457567004, 207761908832432, 1076267071688253 , 5589222169763409, 29092253152085330, 151748567111547764, 793099243676858407, 4152669418745126818, 21780664402751249922, 114422382678727944803, 602009826602592042005, 3171826794792340438352, 16733726510866175251444, 88393492534719713377817, 467479081409996431509100, 2475087790725635009123409, 13118374671923072912752337, 69599620069376221637273227, 369614759969301977577341515] ---------------------------------------- This ends Fact No. , 83 that took, 4.281, seconds to generate. Fact Number, 84 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 1, 5, 7, 14, 36, 47, 165, 198, 715, 975, 3038, 5070, 13056, 26282, 58140, 133361, 271320, 663005, 1321925, 3256156, 6636190, 15954525, 33839325, 78650340, 173501055, 392109696, 889942575, 1980045702, 4561360584, 10114212086, 23385422632, 52117367400, 120140085654, 270136289970, 619450211718, 1405360442161] ---------------------------------------- This ends Fact No. , 84 that took, 4.291, seconds to generate. Fact Number, 85 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 0, 0, 17, 0, 0, 204, 0, 0, 2848, 0, 0, 43335, 0, 0, 697194, 0, 0, 11663971, 0, 0, 200866092, 0, 0, 3537092364, 0, 0, 63397398732, 0, 0, 1152780381018, 0, 0, 21213118756674, 0, 0, 394302904331090, 0] ---------------------------------------- This ends Fact No. , 85 that took, 4.309, seconds to generate. Fact Number, 86 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 1, 6, 10, 5, 4, 62, 343, 905, 1257, 1417, 6084, 31434, 100115, 203536 , 327731, 862489, 3632078, 12563173, 31362425, 63158050, 147961944, 501825641, 1732253546, 4863749789, 11364582752, 26888942585, 79260215036, 259851147222, 772393923288, 1993593104771, 4932908109711, 13607109223658, 41964005993095, 126676016918733, 348075079482190, 903382764123312, 2442678607763572] ---------------------------------------- This ends Fact No. , 86 that took, 4.382, seconds to generate. Fact Number, 87 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 1, 5, 16, 14, 157, 47, 1225, 317, 8349, 4089, 52059, 53480, 306209 , 603957, 1759030, 5957945, 10465777, 52788674, 70145619, 430029068, 557385522, 3284474477, 5040523814, 23975716309, 47705189899, 171065437002, 445644890659, 1226653861390, 4008301648929, 9110184969092, 34526452192001, 71613003642538, 285920884207268, 596901552164051, 2296186192981959, 5184623116789519] ---------------------------------------- This ends Fact No. , 87 that took, 4.527, seconds to generate. Fact Number, 88 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 5, 19, 45, 168, 519, 1718, 5923, 19772, 69314, 239913, 843440, 2987539, 10607907, 37999955, 136469506, 492661835, 1785189586, 6489630289, 23673676337, 86591259225, 317602030338, 1167735170362, 4303093334571, 15890533775030, 58793965087761, 217931442312430, 809172627869999, 3009199409999055, 11207473083719999, 41799473690206514, 156101189392112522, 583687975336595193, 2185067701649255408, 8189009346853774636, 30722376511573875180, 115375093192928177806, 433692106341616988780] ---------------------------------------- This ends Fact No. , 88 that took, 4.594, seconds to generate. Fact Number, 89 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 3, 21, 38, 156, 552, 1512, 6317, 19891, 70514, 262600, 886366, 3324275, 11917581, 43075004, 160465736, 582553382, 2165827825, 8035397386, 29765494555, 111532020794, 416139099821, 1561328302808, 5873799646865, 22096414953525, 83473571039613, 315506419432058, 1194838387405440, 4534322636772920, 17221949385346138, 65535146413192692, 249671343922622483, 952305694778871837, 3637275047915282859, 13905699591780752658, 53223860882997743324, 203918412648437136447, 781994588942192034001] ---------------------------------------- This ends Fact No. , 89 that took, 4.675, seconds to generate. Fact Number, 90 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 0, 11, 2, 86, 46, 805, 778, 8413, 11878, 94852, 173984, 1131252, 2504528 , 14091392, 35821680, 181672096, 511820780, 2407668034, 7325806982, 32627492623 , 105194203456, 450246051074, 1516507955452, 6306326174143, 21955982861752, 89421401069958, 319264578186104, 1281020032435482, 4662416952523702, 18510274085957975, 68370859229450878, 269431128270139919, 1006578376046689460, 3946457992968826824, 14874688246776367900, 58120022046397289843, 220584658819812831730, 860011965567157025094] ---------------------------------------- This ends Fact No. , 90 that took, 4.825, seconds to generate. Fact Number, 91 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 7, 12, 27, 145, 542, 1484, 4717, 18912, 69837, 232014, 812826, 3071902, 11366874, 40707093, 148601720, 556981881, 2078644528, 7688270359, 28655504504, 107884665273, 406245118858, 1527302121578, 5763019307157, 21842298191740, 82883070965849, 314669019951097, 1197199488323510, 4565609438963327, 17433298689456342, 66632408011178909, 255048489784936380, 977706227455184425, 3752109923111391023, 14413431556469767587, 55428845426448705134, 213390292338576409122, 822277537506099282221] ---------------------------------------- This ends Fact No. , 91 that took, 4.928, seconds to generate. Fact Number, 92 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 2, 21, 22, 158, 426, 1227, 5903, 14069, 66827, 205889, 745643, 2913306, 9383495, 38221380, 130637623, 493823616, 1847361711, 6633770656, 25638537003, 92974985748, 353715796924, 1324858454636, 4948780352629, 18875738540175, 70545212099359, 269014986585439, 1017064321986165, 3859590813198634, 14725866957485841, 55888525299099115, 213810548972298479, 815183648914943154, 3118041148131277489, 11943290635288962106, 45717056577344079917, 175557684502312773670, 673562621994925044285] ---------------------------------------- This ends Fact No. , 92 that took, 5.087, seconds to generate. Fact Number, 93 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 7, 36, 117, 400, 1245, 4027, 13396, 46752, 167365, 605010, 2185763 , 7876802, 28403728, 102834605, 374646145, 1373362305, 5059661823, 18705574433, 69327171291, 257475183672, 958229408041, 3574005895189, 13360143857095, 50049035006696, 187857489700137, 706351338258899, 2660106240405986, 10032556743745659, 37889968121931664, 143288138970975747, 542555156099692131, 2056826017314191699, 7806248457632168464, 29658574681836452594, 112796507367897893963, 429393885857106695958] ---------------------------------------- This ends Fact No. , 93 that took, 5.237, seconds to generate. Fact Number, 94 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 12, 45, 178, 721, 2980, 12618, 54266, 236383, 1041368, 4630533, 20756934, 93698601, 425560318, 1943310451, 8916943760, 41092648342, 190108239860, 882605979597, 4110776878895, 19202243069826, 89938540569976, 422291998164797, 1987334469375444, 9372321716452823, 44287004439716584, 209652387433138625, 994179994698277927, 4721981838734316695, 22461280717792572650, 106993245383170929687, 510333442194048183737, 2437220121666623121530, 11653288409046276771048, 55781171290260584231067, 267292461323901388822788, 1282101001674019236995727, 6155629536869738188221870] ---------------------------------------- This ends Fact No. , 94 that took, 5.366, seconds to generate. Fact Number, 95 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 14, 40, 159, 697, 2459, 12042, 44672, 215443, 868306, 4040563, 17379348, 79161088, 354070173, 1604712407, 7323617933, 33342897873, 153683705026, 705219259336, 3268607211799, 15116316348041, 70354769218781, 327494817808456, 1530075812280634, 7159125931674967, 33570907598135401, 157721481404908286, 742146999526013695, 3498542432662058412, 16513846215268476941, 78073866083378497353, 369558905952506118153, 1751670492286817641149, 8312100884655159944328, 39488886489216574895314, 187799603980096788738853, 894040135980110448412790, 4260276202829358230528217] ---------------------------------------- This ends Fact No. , 95 that took, 5.557, seconds to generate. Fact Number, 96 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 12, 36, 140, 585, 2220, 9788, 39408, 173456, 735429, 3238167, 14168618, 62836717, 279920245, 1253536916, 5647232095, 25524125879, 115930468614, 528086721376, 2414452731458, 11069079478914, 50891683497612, 234555033508350, 1083567461637042, 5016467518268546, 23270207462071840, 108146135175780795, 503464333942663879, 2347638058349136824, 10963519657363825862, 51272927124942285046, 240108477142472990681, 1125837383731639275390, 5285204358150588598638, 24839217524698244407563, 116863329477560536219173, 550375288891860404813649, 2594525559976878381839036] ---------------------------------------- This ends Fact No. , 96 that took, 5.738, seconds to generate. Fact Number, 97 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 9, 33, 119, 465, 1842, 7593, 31547, 132973, 565997, 2433703, 10551396 , 46073446, 202424887, 894219578, 3969529256, 17698183707, 79218036743, 355845761188, 1603622644830, 7248091381618, 32848838026095, 149244205098392, 679630890563661, 3101509622852940, 14181810462297919, 64966529546499114, 298121930030616458, 1370239229393886772, 6307445860137827026, 29075401730765256677, 134207522346745066047, 620260192169601243801, 2870031058236734986342, 13294961730655108760282, 61652367586963523840353, 286187678985212386480646, 1329746075157892562412721] ---------------------------------------- This ends Fact No. , 97 that took, 5.917, seconds to generate. Fact Number, 98 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -1, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 21, 80, 379, 1706, 8225, 39664, 196297, 980574, 4963069, 25340871, 130498727, 676723549, 3531411785, 18529267945, 97700050628, 517401200142, 2750869368345, 14677721462643, 78569435214289, 421825341903492, 2270852774406485, 12255428487665875, 66293054280114936, 359363186334713677, 1951906934072249979, 10621487830462605860, 57897393105933378657, 316105389396670180818, 1728464126122428825586, 9464653657999162443391, 51895389267009079068629, 284904571409905552493816, 1565980383960187297304960, 8617109094860609735400486, 47467849365367837580730225, 261743859011672427519003223, 1444676125021442286928312139] ---------------------------------------- This ends Fact No. , 98 that took, 6.123, seconds to generate. Fact Number, 99 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 1, 0, 3, 7, 4, 12, 45, 55, 77, 286, 546, 728, 1960, 4760, 7548, 15504 , 39729, 75582, 140448, 336490, 723327, 1366200, 2992990, 6758895, 13522275, 28094040, 63183315, 133231800, 273896532, 600805296, 1305229332, 2720740792, 5843241088, 12797739672, 27206642716, 57941746476, 126405822608] ---------------------------------------- This ends Fact No. , 99 that took, 6.130, seconds to generate. Fact Number, 100 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 2, 5, 17, 14, 103, 65, 544, 515, 2671, 4333, 12920, 32888, 66569, 225063, 389929, 1426875, 2581052, 8652846, 18130991, 51937472, 127733905, 318505753, 879213643, 2034543521, 5892047281, 13539791786, 38764350879, 92547902870, 253609842517, 638716733669, 1670011621498, 4398787899731, 11151727980457, 30093346643625, 75616292374270, 204712934528781] ---------------------------------------- This ends Fact No. , 100 that took, 6.155, seconds to generate. Fact Number, 101 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 2, 5, 22, 14, 164, 65, 1030, 657, 5868, 7463, 31765, 73575, 173849 , 631556, 1053086, 4877803, 7526655, 34948691, 61382672, 239407864, 524309309, 1621763388, 4415274965, 11255289437, 35813332389, 82153463817, 279458110861, 633479334487, 2118070224696, 5075741777630, 15824514104397, 41275863623366, 118428013850973, 334523141763061, 899513350738990, 2678023253678681] ---------------------------------------- This ends Fact No. , 101 that took, 6.238, seconds to generate. Fact Number, 102 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 0, 6, 3, 5, 76, 12, 338, 718, 547, 7698, 6287, 31069, 119744, 104392, 918064, 1572814, 3837810, 18998514, 24320474, 123215460, 323755183, 617719052, 3030388182, 5507346575, 18634628139, 61805367626, 118698802731, 498452214396, 1178980116448, 3182517234928, 11477928525904, 24575988302660, 86121282203504, 242166326375800, 600650481324438, 2130520453125270, 5170687302732134] ---------------------------------------- This ends Fact No. , 102 that took, 6.309, seconds to generate. Fact Number, 103 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 3, 23, 48, 155, 612, 1609, 6255, 20608, 67954, 250621, 837858, 2997773, 10682234, 37447731, 135767710, 484626014, 1747304695, 6345838687, 22949010094, 83737139716, 305552771525, 1117272519230, 4101926037674, 15064915295407, 55488468018578, 204729501895013, 756350275118646, 2800027056148971, 10377918836812794, 38521413439022433, 143184883556986540, 532814894354798905, 1985211915360494824, 7404609038051674655, 27647122999848204744, 103334287176052280814, 386579072024085298844] ---------------------------------------- This ends Fact No. , 103 that took, 6.339, seconds to generate. Fact Number, 104 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 6, 15, 52, 156, 528, 1765, 6114, 21236, 74776, 265291, 949595, 3422399, 12410903, 45247919, 165752707, 609808036, 2252176796, 8347161607, 31035445149, 115729542652, 432703786186, 1621835578983, 6092706372588, 22936590083350, 86516207497332, 326932457744595, 1237536547815069, 4691914691117079, 17815222622885223, 67739562218872488, 257909787594052190, 983184309422800372, 3752440379470237568, 14337614322431343300, 54840178875567735030, 209969537892201797328, 804687981679200752808] ---------------------------------------- This ends Fact No. , 104 that took, 6.426, seconds to generate. Fact Number, 105 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 7, 12, 27, 145, 542, 1484, 4717, 18912, 69837, 232014, 812826, 3071902, 11366874, 40707093, 148601720, 556981881, 2078644528, 7688270359, 28655504504, 107884665273, 406245118858, 1527302121578, 5763019307157, 21842298191740, 82883070965849, 314669019951097, 1197199488323510, 4565609438963327, 17433298689456342, 66632408011178909, 255048489784936380, 977706227455184425, 3752109923111391023, 14413431556469767587, 55428845426448705134, 213390292338576409122, 822277537506099282221] ---------------------------------------- This ends Fact No. , 105 that took, 6.545, seconds to generate. Fact Number, 106 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 0, 11, 4, 86, 78, 805, 1192, 8452, 16990, 96270, 236808, 1164643, 3284046, 14743066, 45644056, 193172365, 637821100, 2598848478, 8972601330, 35688679563, 127118293564, 498084862421, 1813552391314, 7042085802066, 26046219527470, 100619518600486, 376413085980710, 1450310346991644, 5471244629599182, 21059012766071621, 79947499571174084, 307715177621912926, 1173886029775191684, 4520932319331403266, 17312819399699279000, 66739472841683131101, 256368332246429186536, 989408338157591223177] ---------------------------------------- This ends Fact No. , 106 that took, 6.613, seconds to generate. Fact Number, 107 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 13, 41, 142, 423, 1565, 5031, 18317, 62800, 226120, 805869, 2908590, 10574603, 38508998, 141519613, 520445593, 1926634784, 7144979470, 26609773651, 99344634768, 371959565008, 1396126526467, 5251913565336, 19800366728401, 74790976340113, 283041863568142, 1072947093583434, 4073933208022754, 15491633629534717, 58992488747586041, 224943679315165279, 858806351946684037, 3282708526533806537, 12561906699087323909, 48121690306976634685, 184528536718044245160, 708275860717755281083] ---------------------------------------- This ends Fact No. , 107 that took, 6.729, seconds to generate. Fact Number, 108 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 21, 8, 197, 233, 1408, 5049, 10488, 71079, 130001, 782233, 2201941 , 8026957, 34227885, 94397860, 457903743, 1347084031, 5635328466, 20320964097, 70631100651, 292032321663, 962115513981, 3966803759358, 13937561312596, 53051843744805, 202863349866624, 728732895941242, 2884574137311771, 10387479666546096, 40343107500630748, 150740730513027441, 566466520267270869, 2181535928801844651, 8090036239456081140, 31306403343772671433, 117260445061898684128, 448372729922843768927] ---------------------------------------- This ends Fact No. , 108 that took, 6.810, seconds to generate. Fact Number, 109 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 15, 52, 196, 848, 3285, 14647, 60702, 270321, 1175419, 5255914, 23491263, 106039951, 481526629, 2196502989, 10079226967, 46405794075, 214644323424, 995842152796, 4636334700470, 21645379432275, 101333050920532, 475554054349281, 2236867470931726, 10543910516031653, 49798007373922185, 235624174511043642, 1116777689178292522, 5301626970881411615, 25205850251590771049, 120006708503224545684, 572117331555550458916, 2730911696937623440681, 13050977933834611429924, 62440110843797303986797, 299050203169880371941497, 1433708348090695408846840, 6880058216300684749855770] ---------------------------------------- This ends Fact No. , 109 that took, 6.910, seconds to generate. Fact Number, 110 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 12, 40, 173, 680, 2738, 11592, 49560, 213043, 930426, 4104324, 18224691, 81515943, 366985964, 1660844289, 7552464664, 34495078833, 158166160773, 727771087540, 3359490043697, 15553431607032, 72201388024457, 336000477077523, 1567210120565075, 7325442897415681, 34307937043387379, 160972191966314062, 756571165393072453, 3561587852889538685, 16791490580230611025, 79276999748006450145, 374784514473292472980, 1774024972741778277300, 8407200078952753702115, 39886776045811315915460, 189437910930095337570179, 900620301134491911240989, 4285797716605589717246128] ---------------------------------------- This ends Fact No. , 110 that took, 7.011, seconds to generate. Fact Number, 111 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 12, 40, 142, 563, 2289, 9727, 40469, 172805, 743270, 3243883, 14239798, 62935619, 279857285, 1251361707, 5623320148, 25375881395, 114960382053, 522629400640, 2383651108802, 10903365260933, 50008581050214, 229932888607024, 1059617777704406, 4893461077487489, 22643110549619545, 104966491249405277, 487424239965507547, 2267033308971383675, 10559899348010542261, 49257626409056359429, 230071679802184434665, 1075959829408813703189, 5037813762224184057949, 23614224431340528916377, 110806524896816430366480, 520467417284075888657318, 2447014667977586175683040] ---------------------------------------- This ends Fact No. , 111 that took, 7.138, seconds to generate. Fact Number, 112 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 1, 14, 28, 145, 516, 1979, 8908, 32441, 154161, 585672, 2740838, 11061624, 50413130, 213687612, 957236549, 4188565357, 18653392705, 83110071258, 370756924711, 1668000143121, 7479757197768, 33836186180325, 152641293855293, 693103339062975, 3143907958532025, 14320921951988689, 65258577234059766, 298145590895329589, 1363718159921737018, 6248066678501022899, 28667371686851204749, 131692952209625592984, 605824896356913007642, 2789854967643889563545, 12863630295346101943215, 59368850862446003976810, 274302757655275776867112, 1268501890425313783407797] ---------------------------------------- This ends Fact No. , 112 that took, 7.270, seconds to generate. Fact Number, 113 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 24, 82, 410, 1785, 8725, 41430, 205221, 1016199, 5125126, 25993548, 133175275, 686393117, 3561569210, 18575651599, 97368087820, 512549531921, 2708750703907, 14365708065579, 76433925018425, 407866200085910, 2182328511230655, 11705718596847398, 62931821554624832, 339049323572699808, 1830250984719160443, 9898165151387946036, 53621902634723942488, 290954704851412757057, 1581105635733268646594, 8604176533948514498281, 46885087489221705470263, 255802197823545269079079, 1397294018291880210521059, 7641130449027926071567603, 41830031736621289983307661, 229221830789605978076342582, 1257300713220926794579300018] ---------------------------------------- This ends Fact No. , 113 that took, 7.417, seconds to generate. Fact Number, 114 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 0, 1, 3, 0, 8, 12, 5, 55, 55, 78, 364, 308, 840, 2380, 2244, 7752, 15789, 19722, 65835, 109802, 184943, 533830, 822250, 1726725, 4242420, 6667245, 15705066, 33763743, 57500040, 139107664, 273687964, 513312756, 1209003348, 2282162960, 4639186984, 10411233160, 19607898141] ---------------------------------------- This ends Fact No. , 114 that took, 7.426, seconds to generate. Fact Number, 115 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 3, 2, 0, 5, 57, 76, 23, 225, 1712, 3096, 2152, 10301, 63344, 135945, 145447, 506300, 2645684, 6334817, 8863943, 26321836, 119924174, 308996066, 518884034, 1421869726, 5774838471, 15623015827, 29889917272, 78796739552, 291401326703, 813202815623, 1712626557848, 4443708508164, 15256966498903, 43361944116807, 98123257670293, 253748908058282, 822563978400235] ---------------------------------------- This ends Fact No. , 115 that took, 7.494, seconds to generate. Fact Number, 116 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 5, 13, 45, 126, 409, 1330, 4334, 14379, 48547, 164243, 561757, 1935335, 6702697, 23343984, 81711201, 287179404, 1013288990, 3587966035, 12744788675, 45402974551, 162181082707, 580738971260, 2084245423378, 7496032481347, 27012370625607, 97518043503485, 352651739855174, 1277315657327468, 4633389008133849, 16830948220061852, 61219815349778619, 222954622656362950, 812928026349030113, 2967367732392875777, 10842988100678903747, 39660807051648167870, 145206957905349494122] ---------------------------------------- This ends Fact No. , 116 that took, 7.570, seconds to generate. Fact Number, 117 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 3, 6, 3, 42, 193, 300, 1045, 6099, 16734, 43217, 210817, 782925, 2204773 , 8485820, 34912398, 114022747, 394138850, 1586046526, 5759557242, 19841733394, 75718209771, 288068491153, 1028137670356, 3799702261969, 14538125472039, 53684595930388, 197344794259924, 748242277946288, 2817171295843109, 10447324282076692, 39320074452507852, 149043614017066577, 558940601895391705, 2101140641931237724, 7969855121731563589, 30131317547692018637, 113620558686097683264, 430852951323620245061] ---------------------------------------- This ends Fact No. , 117 that took, 7.649, seconds to generate. Fact Number, 118 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 2, 21, 22, 158, 426, 1227, 5903, 14069, 66827, 205889, 745643, 2913306, 9383495, 38221380, 130637623, 493823616, 1847361711, 6633770656, 25638537003, 92974985748, 353715796924, 1324858454636, 4948780352629, 18875738540175, 70545212099359, 269014986585439, 1017064321986165, 3859590813198634, 14725866957485841, 55888525299099115, 213810548972298479, 815183648914943154, 3118041148131277489, 11943290635288962106, 45717056577344079917, 175557684502312773670, 673562621994925044285] ---------------------------------------- This ends Fact No. , 118 that took, 7.801, seconds to generate. Fact Number, 119 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 13, 41, 142, 423, 1565, 5031, 18317, 62800, 226120, 805869, 2908590, 10574603, 38508998, 141519613, 520445593, 1926634784, 7144979470, 26609773651, 99344634768, 371959565008, 1396126526467, 5251913565336, 19800366728401, 74790976340113, 283041863568142, 1072947093583434, 4073933208022754, 15491633629534717, 58992488747586041, 224943679315165279, 858806351946684037, 3282708526533806537, 12561906699087323909, 48121690306976634685, 184528536718044245160, 708275860717755281083] ---------------------------------------- This ends Fact No. , 119 that took, 7.917, seconds to generate. Fact Number, 120 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 9, 41, 108, 384, 1305, 4292, 15718, 52573, 195642, 673385, 2511848 , 8886337, 33092680, 119706321, 445640960, 1637747863, 6108825653, 22696378874, 84942997383, 318015659342, 1194923337216, 4498783400806, 16971785054563, 64176317350006, 243017737585660, 922234453787969, 3504086271401526, 13338654333027299, 50833483280809771, 194026270629674966, 741398242537467382, 2836676419875402233, 10864845110928346626, 41660512356834417121, 159900937587870759671, 614334359906086506478] ---------------------------------------- This ends Fact No. , 120 that took, 8.073, seconds to generate. Fact Number, 121 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 11, 47, 172, 712, 2989, 12512, 54358, 237130, 1047429, 4675704, 21014115, 95178810, 433636944, 1986368758, 9143750434, 42270618717, 196179650060, 913688439575, 4269064226697, 20005114057167, 93997212012314, 442754534013583, 2090272020160383, 9889198848821871, 46878407628310594, 222627782711405355, 1059077986510979275, 5046275318560217266, 24080480126793825889, 115072421968276796337, 550621834398102612726, 2638024846409493080153, 12653700415989916849919, 60763361406450695591690, 292096361529844718077536, 1405552608376005310630932, 6769910241991793935533704] ---------------------------------------- This ends Fact No. , 121 that took, 8.197, seconds to generate. Fact Number, 122 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 14, 45, 174, 705, 2800, 12071, 50511, 222316, 967499, 4316149, 19260799, 86978299, 394472309, 1800797616, 8259869037, 38055053055, 176051000021, 817301397283, 3807104188102, 17785888949091, 83323447600135, 391327548812882, 1842171612604341, 8690605539109116, 41080783917266559, 194551027555493522, 922956714765385134, 4385642207332457900, 20871074681098995915, 99466366947203937622, 474668873014738798567, 2268062564105608268668, 10850215210041176856053, 51965205417225750889842, 249144882910747349428617, 1195728100841626240263498, 5744231311905114416199054] ---------------------------------------- This ends Fact No. , 122 that took, 8.376, seconds to generate. Fact Number, 123 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 8, 37, 159, 525, 2438, 10042, 41525, 185548, 801462, 3529370, 15839885, 70692828, 319336202, 1451168239, 6608519719, 30288406388, 139296525886, 642637894372, 2976038764761, 13819096891959, 64342031682708, 300361760409456, 1405257565517345, 6588774191205851, 30954308294418044, 145689560892329823, 686892645081487527, 3243760841098195210, 15341313867232060292, 72659797444569360871, 344592030585316531227, 1636299576905358897298, 7779242608860177776633, 37025432233530294285494, 176411070882997673332990, 841377250137556015049995, 4016731604260519937116224] ---------------------------------------- This ends Fact No. , 123 that took, 8.553, seconds to generate. Fact Number, 124 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 14, 32, 159, 524, 2338, 9082, 39516, 165400, 722149, 3135027, 13841240, 61327446, 273934220, 1229936066, 5549881920, 25160782183, 114494874135, 523035436496, 2396690608896, 11017034973520, 50775851725898, 234629462228955, 1086656911086627, 5043771756404914, 23457103429023694, 109297225320797700, 510143458543390700, 2384974385809157044, 11166922697405924006, 52360652310059059353, 245843911662281125334, 1155752014079370429492, 5439877402788341601314, 25633338178551940672130, 120916776552465814662832, 570965072595477380660308, 2698682919563333007403026] ---------------------------------------- This ends Fact No. , 124 that took, 8.742, seconds to generate. Fact Number, 125 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 20, 93, 399, 1941, 9268, 46056, 231133, 1178069, 6071454, 31587133, 165722945, 875572969, 4655258827, 24886810879, 133698537851, 721411301508, 3907970421062, 21245559580445, 115875807979820, 633874170787929, 3476896181543665, 19118978975571799, 105375235434670288, 582021783378845451, 3221073023191972811, 17859250614304540906, 99191356883751539068, 551803105586217829029, 3074329772707969871664, 17152721061127098082779, 95828707447007095535495, 536049799444002397243984, 3002141641526611590329954, 16832392294545062898919940, 94476420276619204644852131, 530811355404767546964415312, 2985205413525769848330080375] ---------------------------------------- This ends Fact No. , 125 that took, 8.952, seconds to generate. Fact Number, 126 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 0, 1, 1, 0, 0, 4, 9, 5, 0, 22, 78, 91, 35, 140, 680, 1224, 969, 1254, 5985, 14630, 17710, 17710, 55660, 164450, 269100, 299520, 593775, 1805076, 3681405, 4951692, 7594752, 20173560, 47303520, 76404460, 110676324, 239784864, 589602585, 1106339923] ---------------------------------------- This ends Fact No. , 126 that took, 8.961, seconds to generate. Fact Number, 127 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 0, 2, 3, 0, 23, 12, 23, 207, 55, 560, 1697, 650, 8694, 13252, 15654, 109435, 107696, 324043, 1216953, 1117762, 5441300, 12614367, 16242055, 78430366 , 130185358, 272242662, 1016036652, 1464897950, 4443146662, 12308432648, 19158617589, 67181508815, 145724647279, 282813865564, 943204291083, 1773587239499, 4347509699662] ---------------------------------------- This ends Fact No. , 127 that took, 8.993, seconds to generate. Fact Number, 128 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 0, 5, 5, 14, 61, 42, 486, 132, 3197, 656, 18845, 6942, 103469, 82337, 540610, 842389, 2738390, 7457658, 13890377, 59018977, 75015627, 428615420, 464370812, 2911703107, 3361816428, 18787058761, 26847573680, 116858010417, 219643015240, 713315739910, 1760480543434, 4372055494146, 13583447394347, 27625019510098, 100561624744208, 183868132825246] ---------------------------------------- This ends Fact No. , 128 that took, 9.080, seconds to generate. Fact Number, 129 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 2, 0, 5, 5, 14, 75, 42, 710, 132, 5415, 656, 36330, 8442, 223790, 127096, 1296780, 1599366, 7191825, 17000284, 39029291, 158784108, 216908922, 1342917656, 1339175203, 10504886569, 9895257442, 77204253552, 86005434774, 540143631646, 804196436716, 3646888468726, 7512093516175, 24180968667854, 67709007362894, 161292207078680, 582735668471090, 1114680033125948] ---------------------------------------- This ends Fact No. , 129 that took, 9.211, seconds to generate. Fact Number, 130 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 3, 4, 3, 42, 117, 171, 949, 3537, 7632, 28765, 115189, 315534, 1040806, 4055775, 12773335, 41226403, 153201604, 518969067, 1705182586, 6118352563, 21406751812, 72154670394, 254356245660, 900150854840, 3099873232059, 10871104611942, 38581306372934, 134854878746180, 473650031509210, 1681877247755816, 5934007868931549, 20928312762436191, 74366048412855518, 263879579937685840, 934890115024132352, 3326678645979956643, 11847241110192142045, 42142027265956747320] ---------------------------------------- This ends Fact No. , 130 that took, 9.251, seconds to generate. Fact Number, 131 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 3, 15, 33, 114, 374, 1148, 4014, 13218, 45251, 156691, 540459, 1898433, 6670470, 23602581, 83999017, 299828723, 1075578570, 3870451106, 13973579863, 50606121977, 183742900600, 668874849424, 2440432319190, 8923053466466, 32691099204201, 119988702131927, 441167062238030, 1624669451725207, 5992148068258060, 22131837671897606, 81852496903185977, 303105431236230035, 1123756293191475308, 4170990262745090702, 15497773816780295343, 57641948516878681768, 214597818608804245249] ---------------------------------------- This ends Fact No. , 131 that took, 9.354, seconds to generate. Fact Number, 132 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 7, 36, 117, 400, 1245, 4027, 13396, 46752, 167365, 605010, 2185763 , 7876802, 28403728, 102834605, 374646145, 1373362305, 5059661823, 18705574433, 69327171291, 257475183672, 958229408041, 3574005895189, 13360143857095, 50049035006696, 187857489700137, 706351338258899, 2660106240405986, 10032556743745659, 37889968121931664, 143288138970975747, 542555156099692131, 2056826017314191699, 7806248457632168464, 29658574681836452594, 112796507367897893963, 429393885857106695958] ---------------------------------------- This ends Fact No. , 132 that took, 9.509, seconds to generate. Fact Number, 133 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 21, 8, 197, 233, 1408, 5049, 10488, 71079, 130001, 782233, 2201941 , 8026957, 34227885, 94397860, 457903743, 1347084031, 5635328466, 20320964097, 70631100651, 292032321663, 962115513981, 3966803759358, 13937561312596, 53051843744805, 202863349866624, 728732895941242, 2884574137311771, 10387479666546096, 40343107500630748, 150740730513027441, 566466520267270869, 2181535928801844651, 8090036239456081140, 31306403343772671433, 117260445061898684128, 448372729922843768927] ---------------------------------------- This ends Fact No. , 133 that took, 9.588, seconds to generate. Fact Number, 134 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 9, 41, 108, 384, 1305, 4292, 15718, 52573, 195642, 673385, 2511848 , 8886337, 33092680, 119706321, 445640960, 1637747863, 6108825653, 22696378874, 84942997383, 318015659342, 1194923337216, 4498783400806, 16971785054563, 64176317350006, 243017737585660, 922234453787969, 3504086271401526, 13338654333027299, 50833483280809771, 194026270629674966, 741398242537467382, 2836676419875402233, 10864845110928346626, 41660512356834417121, 159900937587870759671, 614334359906086506478] ---------------------------------------- This ends Fact No. , 134 that took, 9.742, seconds to generate. Fact Number, 135 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 0, 11, 0, 86, 10, 805, 272, 8402, 5234, 94306, 88376, 1115323, 1402720, 13726778, 21554764, 174384395, 325382606, 2273715203, 4863679958, 30297332697, 72317018462, 411214089798, 1072551174738, 5669702018354, 15894336501786, 79235197530024, 235606235603822, 1120304174577041, 3495865129155220, 16000519492326746, 51944591955436124, 230534966698267644, 773153714164626994, 3346995722417436604, 11529341706911930830, 48918509055790444073, 172264678066957981484, 719178618177413109270] ---------------------------------------- This ends Fact No. , 135 that took, 9.887, seconds to generate. Fact Number, 136 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 5, 6, 50, 174, 565, 3080, 11191, 49034, 227206, 933171, 4311955, 19310558, 85635238, 396319116, 1794446568, 8232499503, 38150174439, 175738911268, 817972348985, 3813047007426, 17794767253652, 83489899681079, 392030365100993, 1845586602721322, 8711855196774592, 41175251027563057, 195046091158926423, 925468390984776918, 4397547470606916133, 20931121000048214049, 99757237528309643876, 476074405425889160011, 2274940628235451294906, 10883243913150570303722, 52124659814241345416820, 249913426306982128794065, 1199409165236299556318623, 5761904154571992418133017] ---------------------------------------- This ends Fact No. , 136 that took, 10.007, seconds to generate. Fact Number, 137 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 9, 42, 144, 611, 2503, 10595, 45887, 199596, 882898, 3932658, 17674871, 80002082, 364292134, 1668204472, 7675750332, 35473327847, 164582488869, 766313377208, 3579595307820, 16770309736453, 78781386331826, 371011065979814, 1751250014490917, 8283879399943508, 39262482483753727, 186432284840407400, 886769304811076784, 4224723769284715645, 20157681118747248154, 96315860743576406076, 460822767706964854318, 2207577205532196340887, 10587988359085863296649, 50839206707946649283969, 244369049447877429792939, 1175796314499567428137012, 5662843491312921829787856] ---------------------------------------- This ends Fact No. , 137 that took, 10.185, seconds to generate. Fact Number, 138 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 12, 35, 145, 591, 2382, 10432, 43664, 194085, 843258, 3780514, 16859418, 76309679, 346163000, 1582070419, 7260873642, 33476318422, 154970500673, 719854368101, 3355281579644, 15683637412269, 73517632827284, 345459740596778, 1627145915782916, 7680266521836452, 36324189829942464, 172114540320427285, 816941996946739829, 3883890762356052732, 18492736536820842926, 88176741397595277570, 421006472639548012484, 2012669347834850035371, 9633280506436145623545, 46160048518629558843426, 221422941434925750000323, 1063211162864711626762878, 5110164649379414414466014] ---------------------------------------- This ends Fact No. , 138 that took, 10.360, seconds to generate. Fact Number, 139 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 1, 14, 28, 145, 576, 1993, 10820, 34209, 199219, 669788, 3712506, 13897436, 71203702, 294233863, 1414210779, 6269756485, 29036471404, 134120314010, 612540959851, 2884031274762, 13183216941756, 62458531170436, 287763886503395, 1363940212039958, 6344531495418125, 30041986726274074, 140937702147274056, 667101525149799718, 3150025373968215933, 14921520605296739009, 70782943107581828669, 335871561508315295364, 1598337752997159551148, 7601251857372211031392, 36255878028082081970773, 172833025265715998744285, 825872089720081621587041, 3945888969760363419264442] ---------------------------------------- This ends Fact No. , 139 that took, 10.534, seconds to generate. Fact Number, 140 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 18, 103, 375, 2106, 9569, 49152, 251313, 1280581, 6778689, 35417695, 189284212, 1011771708, 5453066888, 29566502151, 160884413244, 880388306191, 4832512200881, 26632927096920, 147231798404710, 816341656082818, 4538865746430343, 25297532772313916, 141329266342997491, 791226069742202571, 4438490524168795481, 24944263019403452860, 140427823076010800536, 791839345602523829828, 4471733695015749171105, 25289019778362724107676, 143208197125076098225042, 811990412603012918033033, 4609466041363415319663577, 26196242758840065370481710, 149035761201708748836313867, 848752794766974317905637717, 4838261097069496793835514902] ---------------------------------------- This ends Fact No. , 140 that took, 10.728, seconds to generate. Fact Number, 141 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 3, 5, 14, 28, 74, 168, 432, 1045, 2684, 6721, 17355, 44408, 115502, 299812, 785570, 2060094, 5434475, 14362841, 38114760, 101360402, 270373303, 722696570, 1936398635, 5198249550, 13982513625, 37674988080, 101685303765, 274867141845, 744093631842, 2017066320624, 5474900965050, 14878450339822, 40479971557162, 110253945275970, 300605644859552, 820399033872096, 2241084167717824] ---------------------------------------- This ends Fact No. , 141 that took, 10.736, seconds to generate. Fact Number, 142 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 9, 31, 78, 248, 705, 2196, 6632, 20780, 64709, 204902, 650000, 2080483, 6683564, 21593311, 70024903, 228022074, 744976876, 2441850778, 8026618762, 26455041139, 87405982153, 289438774174, 960462359139, 3193366842536 , 10636635056279, 35489063311272, 118596791583351, 396914141297320, 1330230442462987, 4464042344334714, 14999217181926990, 50456596364848778, 169921812232536963, 572844723715864685, 1933116776188266041, 6529668152176835624] ---------------------------------------- This ends Fact No. , 142 that took, 10.764, seconds to generate. Fact Number, 143 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 3, 16, 33, 115, 390, 1087, 4060, 12555, 42953, 148067, 492739, 1735298, 5944320, 20744252, 72905575, 254998049, 903660769, 3195209422, 11355589072, 40507136044, 144620988953, 518478617875, 1861257943227, 6697455408050, 24152234870325, 87226107921628, 315651869078757, 1143924927595869, 4151936835886485, 15091681888691404, 54925223488389666, 200157938880285184, 730258764785275647, 2667274260621421838, 9752675597285646950, 35695329641773808896, 130773052695581564343] ---------------------------------------- This ends Fact No. , 143 that took, 10.843, seconds to generate. Fact Number, 144 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 3, 3, 7, 52, 104, 309, 1512, 4103, 13483, 55946, 176679, 611424, 2366567 , 8076977, 28797150, 108439122, 385616674, 1400682434, 5228118841, 19022963870, 69962427824, 260924098247, 962566138443, 3571509151852, 13351205153535, 49697934407487, 185620330580446, 696245098141498, 2608158788832913, 9791671225078267, 36856810723754976, 138729393617351062, 522992182413374808, 1975081475916088209, 7462553157784059215, 28229843679565303229, 106926333323479296929, 405273860833298537790] ---------------------------------------- This ends Fact No. , 144 that took, 10.923, seconds to generate. Fact Number, 145 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 15, 54, 197, 778, 3046, 12378, 50688, 210821, 885836, 3755794, 16053550, 69077136, 299051044, 1301497997, 5691174700, 24991961429, 110168793923, 487328507125, 2162490768266, 9623634039899, 42941087514502, 192072611056724, 861064794485586, 3868232998947027, 17411324425937991, 78511976428487851, 354628109413966644, 1604341160570722942, 7268823842760461184 , 32978945219886339519, 149823281943148384308, 681490024904553949414, 3103471368624092952988, 14148708406910701942775, 64571602595986047193440, 294984276632934745080426, 1348861148942366519449640] ---------------------------------------- This ends Fact No. , 145 that took, 10.986, seconds to generate. Fact Number, 146 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 16, 48, 208, 778, 3305, 13499, 57999, 247426, 1080038, 4725641, 20929207, 93118686, 417432294, 1879871543, 8510737402, 38686261748, 176564376942, 808602162394, 3715180084791, 17119059401564, 79095591109170, 366346002995878, 1700682157965819, 7911704506752742, 36878195675123781, 172211459271608956, 805555550951269942, 3774174295168643551, 17709186843741843257, 83212029736653279793, 391515337546259034843, 1844394597724020439367, 8699040178460691523419, 41074630787462631404420, 194149125687057441136325, 918614252117154387538888, 4350560194974760150572558] ---------------------------------------- This ends Fact No. , 146 that took, 11.050, seconds to generate. Fact Number, 147 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 12, 45, 178, 721, 2980, 12618, 54266, 236383, 1041368, 4630533, 20756934, 93698601, 425560318, 1943310451, 8916943760, 41092648342, 190108239860, 882605979597, 4110776878895, 19202243069826, 89938540569976, 422291998164797, 1987334469375444, 9372321716452823, 44287004439716584, 209652387433138625, 994179994698277927, 4721981838734316695, 22461280717792572650, 106993245383170929687, 510333442194048183737, 2437220121666623121530, 11653288409046276771048, 55781171290260584231067, 267292461323901388822788, 1282101001674019236995727, 6155629536869738188221870] ---------------------------------------- This ends Fact No. , 147 that took, 11.178, seconds to generate. Fact Number, 148 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 15, 52, 196, 848, 3285, 14647, 60702, 270321, 1175419, 5255914, 23491263, 106039951, 481526629, 2196502989, 10079226967, 46405794075, 214644323424, 995842152796, 4636334700470, 21645379432275, 101333050920532, 475554054349281, 2236867470931726, 10543910516031653, 49798007373922185, 235624174511043642, 1116777689178292522, 5301626970881411615, 25205850251590771049, 120006708503224545684, 572117331555550458916, 2730911696937623440681, 13050977933834611429924, 62440110843797303986797, 299050203169880371941497, 1433708348090695408846840, 6880058216300684749855770] ---------------------------------------- This ends Fact No. , 148 that took, 11.242, seconds to generate. Fact Number, 149 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 11, 47, 172, 712, 2989, 12512, 54358, 237130, 1047429, 4675704, 21014115, 95178810, 433636944, 1986368758, 9143750434, 42270618717, 196179650060, 913688439575, 4269064226697, 20005114057167, 93997212012314, 442754534013583, 2090272020160383, 9889198848821871, 46878407628310594, 222627782711405355, 1059077986510979275, 5046275318560217266, 24080480126793825889, 115072421968276796337, 550621834398102612726, 2638024846409493080153, 12653700415989916849919, 60763361406450695591690, 292096361529844718077536, 1405552608376005310630932, 6769910241991793935533704] ---------------------------------------- This ends Fact No. , 149 that took, 11.369, seconds to generate. Fact Number, 150 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 5, 6, 50, 174, 565, 3080, 11191, 49034, 227206, 933171, 4311955, 19310558, 85635238, 396319116, 1794446568, 8232499503, 38150174439, 175738911268, 817972348985, 3813047007426, 17794767253652, 83489899681079, 392030365100993, 1845586602721322, 8711855196774592, 41175251027563057, 195046091158926423, 925468390984776918, 4397547470606916133, 20931121000048214049, 99757237528309643876, 476074405425889160011, 2274940628235451294906, 10883243913150570303722, 52124659814241345416820, 249913426306982128794065, 1199409165236299556318623, 5761904154571992418133017] ---------------------------------------- This ends Fact No. , 150 that took, 11.499, seconds to generate. Fact Number, 151 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 6, 35, 138, 689, 3272, 16522, 83792, 434749, 2278888, 12093271, 64741330 , 349470487, 1899418046, 10387322922, 57111322368, 315523027610, 1750681516380, 9751416039535, 54507046599094, 305650440453943, 1718956630038438, 9693209009913658, 54794959143735984, 310457570693809237, 1762705520682665544, 10027857107877385345, 57151686288411033894, 326279642607630891545, 1865707051569388661592, 10684308742180810054918, 61271675090362129009048, 351842023188483741065950, 2022914958249315372328220, 11644465032923181133034754 , 67103614092340792585084972, 387107295639502088469299294, 2235385554839773350530747032, 12920742642773813918875894076] ---------------------------------------- This ends Fact No. , 151 that took, 11.572, seconds to generate. Fact Number, 152 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 7, 27, 125, 585, 2833, 14118, 71723, 370251, 1937070, 10246782, 54715075 , 294531484, 1596604091, 8708189187, 47754368496, 263144086228, 1456299382751, 8090922913342, 45110329043546, 252316864526423, 1415430767035524, 7961551391236514, 44893214712013256, 253720462310171143, 1436974290244317735, 8154473631126499566, 46359392025340641107, 264010777537791807344, 1505913565771035357805, 8602599161172668341840, 49212029952098180231478, 281896038877909747571814, 1616776467354651001994102, 9283766947905138168796190, 53368388042787537986433852, 307117158348290806219764243, 1769133889313374235807138284, 10200752071072611764253509601] ---------------------------------------- This ends Fact No. , 152 that took, 11.732, seconds to generate. Fact Number, 153 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 6, 25, 115, 527, 2543, 12630, 63569, 326545, 1697119, 8921094, 47344056, 253258115, 1364381346, 7395544297, 40305245926, 220725694389, 1214008928461, 6703220103017, 37143019250523, 206473636449997, 1151132503139380, 6435077295543729, 36062602355754011, 202559653694417233, 1140166430013004419, 6430394451160630714, 36333095756335782202, 205641053733279226445, 1165768067295268456197, 6618600224908285415116, 37629848041907997718718, 214227498423128204181501, 1221129116351598192575845, 6968855841340925863336699, 39815047311035224760216802, 227715905500311549871746638, 1303696016855881180091743410, 7470925055954882600844480878] ---------------------------------------- This ends Fact No. , 153 that took, 11.875, seconds to generate. Fact Number, 154 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 6, 21, 117, 449, 2387, 11234, 56355, 289252, 1471858, 7724107, 40427754, 214442115, 1144284334, 6139037391, 33152981513, 179749651008, 979272133586, 5354970609236, 29386618328805, 161793108491311, 893349780901737, 4946162401427134, 27452595269075619, 152719470064311219, 851385478194669127, 4755670197695159219, 26613040311658985001, 149183191076710677717, 837608540735970579807, 4709929978217260226714, 26521651814763104950884, 149542154142854387322557, 844249221319698387870556, 4771903174533925414114749, 27002176072591155534082429, 152956025362129647474539518, 867304385711510843270881168, 4922569102038997258451514869] ---------------------------------------- This ends Fact No. , 154 that took, 11.979, seconds to generate. Fact Number, 155 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-3, -2, -1, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 9, 47, 236, 1255, 6958, 39302, 227439, 1335100, 7948216, 47832942, 290608143, 1779883158, 10977966112, 68127590542, 425090391580, 2665251082146, 16783155395439, 106096597710740, 673071844607004, 4283661267869412, 27342741560011682, 174999464650230043, 1122809783075305501, 7220515474260568103 , 46531773058036815488, 300458862623285346399, 1943639973423848468549, 12594740239019599708079, 81744301452907705557981, 531345960488654354893658, 3458673487385365659862484, 22543334242180428959049014, 147119527691524356842839582, 961249191362960963631903186, 6287637353055041210486027120, 41171750323955206472628950356, 269865887991244159127005781623, 1770564273916956121948841979311] ---------------------------------------- This ends Fact No. , 155 that took, 12.129, seconds to generate. Fact Number, 156 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 6, 8, 28, 50, 132, 317, 715, 1903, 4368, 11349, 27881, 69974, 179112, 448154, 1156986, 2939585, 7571462, 19517522, 50314110, 130672775, 338599560, 882260680, 2299665810, 6007600185, 15730319490, 41221684716, 108277980108, 284671947798, 749663283856, 1976656376706, 5217762017738, 13791262142388, 36487025005964, 96641647221652, 256211125739260, 679892663951293] ---------------------------------------- This ends Fact No. , 156 that took, 12.140, seconds to generate. Fact Number, 157 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 2, 12, 22, 59, 212, 475, 1594, 4750, 13196, 43366, 127252, 391668, 1242498, 3776182, 11978944, 37660098, 118023500, 376764859, 1192384620, 3800996683, 12174327988, 38911906522, 125118761512, 402609044674, 1297297674737 , 4193738729864, 13564531538068, 43963910619655, 142729843336414, 463830423116109, 1509860070733024, 4920351768127197, 16052692656100348, 52437597284067130, 171454732353827446, 561193883071486554, 1838643773175111072] ---------------------------------------- This ends Fact No. , 157 that took, 12.179, seconds to generate. Fact Number, 158 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 4, 8, 27, 88, 255, 816, 2652, 8484, 28010, 93519, 311311, 1047079, 3557270, 12116686, 41440458, 142533861, 492066696, 1703991980, 5920941155, 20635503666, 72103747481, 252568582748, 886756183614, 3119870447538, 10998136966961, 38841818713274, 137410335685786, 486890269898723, 1727802215388888, 6139995444922663, 21848255855472370, 77840947711380018, 277659218074236324, 991518669728587943, 3544452787305998644, 12683338799748237350, 45428910418404359917] ---------------------------------------- This ends Fact No. , 158 that took, 12.286, seconds to generate. Fact Number, 159 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 5, 8, 41, 87, 362, 1028, 3722, 11974, 42353, 143059, 505254, 1759797, 6232410, 22110186, 78864073, 282815530, 1016705079, 3674145774, 13299594962, 48355251113, 176071288163, 643381979793, 2354505174602, 8640415951567, 31755910657689, 116970256096214, 431479020116474, 1594526638445841, 5900640133259486, 21869028592831998, 81153889574490442, 301550544397058064, 1121793388897790260, 4177980311592751250, 15576732629781357097, 58134189364844379047, 217170880969920008713] ---------------------------------------- This ends Fact No. , 159 that took, 12.444, seconds to generate. Fact Number, 160 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 14, 43, 164, 617, 2375, 9465, 37776, 154315, 633321, 2632378, 11010055, 46404863, 196718589, 838474916, 3590934328, 15444460191, 66683955076, 288921124823, 1255802647438, 5474219850958, 23926618975537, 104834748880803, 460376934797848, 2025973157117432, 8933078015857335, 39460076600107456, 174603055385215294, 773812426883313328, 3434525009893030608, 15265318066197588429, 67938771200586087448, 302739905433425969561, 1350616317896028768618, 6032220951129635074688, 26969950798997812341575, 120702816205128656679356, 540713729414372654541714] ---------------------------------------- This ends Fact No. , 160 that took, 12.493, seconds to generate. Fact Number, 161 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 5, 9, 42, 182, 593, 2581, 10957, 43420, 188471, 813065, 3463970, 15218878, 66912366, 294387495, 1310788491, 5851890115, 26203798664, 118014897338, 532997318579, 2414210327413, 10973073735408, 49997307201615, 228355683827178, 1045524124363610, 4796707597515697, 22049637042067194, 101548378550822286, 468464987749445297, 2164590099847430866, 10016842234891063409, 46418949804128268189, 215395376850006865160, 1000742784297131319331, 4655016518949470669326, 21677341163691048135998, 101053555648429442926525, 471556590217779488901190, 2202577602437903342889461] ---------------------------------------- This ends Fact No. , 161 that took, 12.617, seconds to generate. Fact Number, 162 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 14, 40, 159, 697, 2459, 12042, 44672, 215443, 868306, 4040563, 17379348, 79161088, 354070173, 1604712407, 7323617933, 33342897873, 153683705026, 705219259336, 3268607211799, 15116316348041, 70354769218781, 327494817808456, 1530075812280634, 7159125931674967, 33570907598135401, 157721481404908286, 742146999526013695, 3498542432662058412, 16513846215268476941, 78073866083378497353, 369558905952506118153, 1751670492286817641149, 8312100884655159944328, 39488886489216574895314, 187799603980096788738853, 894040135980110448412790, 4260276202829358230528217] ---------------------------------------- This ends Fact No. , 162 that took, 12.797, seconds to generate. Fact Number, 163 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 12, 40, 173, 680, 2738, 11592, 49560, 213043, 930426, 4104324, 18224691, 81515943, 366985964, 1660844289, 7552464664, 34495078833, 158166160773, 727771087540, 3359490043697, 15553431607032, 72201388024457, 336000477077523, 1567210120565075, 7325442897415681, 34307937043387379, 160972191966314062, 756571165393072453, 3561587852889538685, 16791490580230611025, 79276999748006450145, 374784514473292472980, 1774024972741778277300, 8407200078952753702115, 39886776045811315915460, 189437910930095337570179, 900620301134491911240989, 4285797716605589717246128] ---------------------------------------- This ends Fact No. , 163 that took, 12.929, seconds to generate. Fact Number, 164 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 3, 14, 45, 174, 705, 2800, 12071, 50511, 222316, 967499, 4316149, 19260799, 86978299, 394472309, 1800797616, 8259869037, 38055053055, 176051000021, 817301397283, 3807104188102, 17785888949091, 83323447600135, 391327548812882, 1842171612604341, 8690605539109116, 41080783917266559, 194551027555493522, 922956714765385134, 4385642207332457900, 20871074681098995915, 99466366947203937622, 474668873014738798567, 2268062564105608268668, 10850215210041176856053, 51965205417225750889842, 249144882910747349428617, 1195728100841626240263498, 5744231311905114416199054] ---------------------------------------- This ends Fact No. , 164 that took, 13.113, seconds to generate. Fact Number, 165 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 9, 42, 144, 611, 2503, 10595, 45887, 199596, 882898, 3932658, 17674871, 80002082, 364292134, 1668204472, 7675750332, 35473327847, 164582488869, 766313377208, 3579595307820, 16770309736453, 78781386331826, 371011065979814, 1751250014490917, 8283879399943508, 39262482483753727, 186432284840407400, 886769304811076784, 4224723769284715645, 20157681118747248154, 96315860743576406076, 460822767706964854318, 2207577205532196340887, 10587988359085863296649, 50839206707946649283969, 244369049447877429792939, 1175796314499567428137012, 5662843491312921829787856] ---------------------------------------- This ends Fact No. , 165 that took, 13.299, seconds to generate. Fact Number, 166 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 7, 27, 125, 585, 2833, 14118, 71723, 370251, 1937070, 10246782, 54715075 , 294531484, 1596604091, 8708189187, 47754368496, 263144086228, 1456299382751, 8090922913342, 45110329043546, 252316864526423, 1415430767035524, 7961551391236514, 44893214712013256, 253720462310171143, 1436974290244317735, 8154473631126499566, 46359392025340641107, 264010777537791807344, 1505913565771035357805, 8602599161172668341840, 49212029952098180231478, 281896038877909747571814, 1616776467354651001994102, 9283766947905138168796190, 53368388042787537986433852, 307117158348290806219764243, 1769133889313374235807138284, 10200752071072611764253509601] ---------------------------------------- This ends Fact No. , 166 that took, 13.438, seconds to generate. Fact Number, 167 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 4, 33, 102, 598, 2544, 13676, 66376, 352074, 1814324, 9695689, 51552276, 278766701, 1510571560, 8261267102, 45351526196, 250443975175, 1388272169900, 7728694037714, 43171753065550, 241966997205773, 1360096129902940, 7666224264755313, 43318044771874766, 245337979440231404, 1392468090987410006, 7918988208879512369, 45118567235896080050, 257507888918292710659, 1472061425165776335468, 8427880496340570639502, 48320079616930497360570, 277407172494432691200086, 1594612562636791665849852, 9177191374543356046459577, 52875380728953935356776980, 304972458585557505879551905, 1760789125449417641580244900, 10175873838683738101570829127] ---------------------------------------- This ends Fact No. , 167 that took, 13.637, seconds to generate. Fact Number, 168 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 6, 21, 116, 474, 2441, 11964, 60376, 314230, 1632196, 8658074, 46174396, 248498500, 1347490282, 7347493544, 40303415382, 222108850452, 1229464846249, 6832478437045, 38104355448182, 213199540197523, 1196400038438485, 6732018628924680, 37974930405337963, 214707865395659647, 1216536492028395828, 6906560392414260168, 39282500772963974855, 223812146845658449201, 1277224510471476285971, 7299721511751094009619, 41779292791433478687128, 239439312796025940370186, 1373967052294041202263832, 7893567695726185429303811, 45400291607566749187145885, 261400584701848078745200891, 1506584076695419808919164010, 8691548745202259798394442097] ---------------------------------------- This ends Fact No. , 168 that took, 13.835, seconds to generate. Fact Number, 169 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 24, 99, 467, 2244, 11088, 55990, 287175, 1494246, 7859987, 41751210, 223584684, 1205933290, 6544890962, 35716714784, 195867367871, 1078828297122, 5965573113464, 33105420479596, 184311598004294, 1029178415527183, 5762437400106894, 32344855509348933, 181971931852627460, 1025958463310283994, 5795824741994932413, 32802092122144099425, 185966765450609575638, 1056011000640987887517, 6005610877816069880621, 34202816939108641130626, 195050154725945233764750, 1113723432012470632340513, 6366851255646933338257723, 36438496786114207646090687, 208765589362151609308583440, 1197281801336040615847143089, 6873072314552569018362060742] ---------------------------------------- This ends Fact No. , 169 that took, 14.041, seconds to generate. Fact Number, 170 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -2, -1, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 8, 46, 224, 1226, 6831, 39180, 229513, 1365735, 8241531, 50284073, 309775925, 1923969783, 12034822349, 75748984620, 479399092456, 3048847393883, 19474677151785, 124885427781600, 803707503490482, 5189062935157686, 33601806280718427, 218178727534017686, 1420183204095809574, 9265640254001917065 , 60580340822209660764, 396869047279229152722, 2604730640762604297904, 17124788854648694776491, 112768118963450266205826, 743708153538948382746211, 4911726720925964577667997, 32482177702359776513065069, 215081129689466972529194665, 1425855117056824213475406656, 9463168029887154809683760910, 62872410179319386099496237645, 418140643926299625657053340368, 2783565611195766239903843790392] ---------------------------------------- This ends Fact No. , 170 that took, 14.268, seconds to generate. Fact Number, 171 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 1, 11, 7, 46, 45, 212, 275, 1045, 1651, 5434, 9863, 29458, 58956, 165002, 353685, 948290, 2132655, 5561547, 12934119, 33146588, 78914495, 200119090, 484339635, 1220942985, 2989617540, 7513839651, 18553008216, 46577756370, 115714828152, 290519299114, 725066526924, 1821729163342, 4562705514102, 11476617891706, 28825208705787, 72599069116587] ---------------------------------------- This ends Fact No. , 171 that took, 14.279, seconds to generate. Fact Number, 172 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 2, 3, 2, 17, 56, 82, 266, 1056, 2410, 6157, 22307, 64019, 167114, 536678 , 1678687, 4717571, 14279619, 45164770, 134485030, 403157361, 1261106436, 3873230891, 11739955681, 36411477173, 113212662921, 347970789239, 1077895267865 , 3364407062887, 10448760154791, 32475170028780, 101539568008979, 317371203029902, 991053166160102, 3104840517480353, 9742039327727277, 30552763936651858, 95956784274092846, 301893368320000304] ---------------------------------------- This ends Fact No. , 172 that took, 14.348, seconds to generate. Fact Number, 173 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 3, 6, 26, 71, 215, 615, 1879, 5923, 19174, 62515, 203978, 666587, 2187956, 7228831, 24041213, 80384059, 269853218, 908761480, 3069017067, 10393205043, 35292352343, 120152439959, 410025684695, 1402229574135, 4804786159886, 16493558252776, 56714159449203, 195327621458483, 673736857127415 , 2327191353954248, 8049180928555926, 27875013503713839, 96647746918324587, 335470346187629947, 1165674067152210376, 4054500970667408497, 14116050642587817343] ---------------------------------------- This ends Fact No. , 173 that took, 14.426, seconds to generate. Fact Number, 174 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 2, 15, 22, 97, 288, 824, 3102, 9204, 32651, 109292, 368392, 1293128, 4407884, 15485107, 54098005, 189918002, 672404562, 2377111088, 8461864724, 30163628101, 107831922872, 386706240333, 1389000550245, 5002604881563, 18048926417904, 65241923293743, 236263608449912, 856888545036236, 3112789267430932, 11323472633987856, 41247386778048645, 150442417056681492, 549360231881172775, 2008361895366503951, 7350049036451878478, 26926575646754690704, 98739950253472417485] ---------------------------------------- This ends Fact No. , 174 that took, 14.578, seconds to generate. Fact Number, 175 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 9, 35, 125, 453, 1728, 6635, 26019, 103175, 413415, 1673562, 6824013, 28020830, 115766799, 480795028, 2006376164, 8408428486, 35373640393, 149333199475, 632422174306, 2686049488173, 11438629470361, 48831003190507, 208928783583116, 895797735746866, 3848277802797746, 16561892402742964, 71398420127824528, 308287098220305259, 1333114870664775287, 5772799599221969986 , 25030912431116987558, 108669028818295142400, 472328430829679812265, 2055248165770684461994, 8952434515586478345212, 39034757378159103550152, 170362396512970263333300] ---------------------------------------- This ends Fact No. , 175 that took, 14.661, seconds to generate. Fact Number, 176 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 13, 35, 147, 549, 2160, 8867, 35765, 150772, 628464, 2683757, 11456093, 49473738, 214539108, 936161664, 4104468256, 18071975575, 79898601922, 354443312181, 1577645717021, 7042409083701, 31523012983176, 141450890769500, 636189953631244, 2867417631833429, 12949532955572112, 58589145884064847, 265537708473571351, 1205407567101080169, 5480195896811295332, 24950170906420844504, 113744350209745417424, 519196937329810060808, 2372739294180022605528, 10855647444031754020668, 49719138352905825910456, 227944931325178226874747, 1046051034104785013658717] ---------------------------------------- This ends Fact No. , 176 that took, 14.755, seconds to generate. Fact Number, 177 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 12, 36, 140, 585, 2220, 9788, 39408, 173456, 735429, 3238167, 14168618, 62836717, 279920245, 1253536916, 5647232095, 25524125879, 115930468614, 528086721376, 2414452731458, 11069079478914, 50891683497612, 234555033508350, 1083567461637042, 5016467518268546, 23270207462071840, 108146135175780795, 503464333942663879, 2347638058349136824, 10963519657363825862, 51272927124942285046, 240108477142472990681, 1125837383731639275390, 5285204358150588598638, 24839217524698244407563, 116863329477560536219173, 550375288891860404813649, 2594525559976878381839036] ---------------------------------------- This ends Fact No. , 177 that took, 14.932, seconds to generate. Fact Number, 178 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 12, 40, 142, 563, 2289, 9727, 40469, 172805, 743270, 3243883, 14239798, 62935619, 279857285, 1251361707, 5623320148, 25375881395, 114960382053, 522629400640, 2383651108802, 10903365260933, 50008581050214, 229932888607024, 1059617777704406, 4893461077487489, 22643110549619545, 104966491249405277, 487424239965507547, 2267033308971383675, 10559899348010542261, 49257626409056359429, 230071679802184434665, 1075959829408813703189, 5037813762224184057949, 23614224431340528916377, 110806524896816430366480, 520467417284075888657318, 2447014667977586175683040] ---------------------------------------- This ends Fact No. , 178 that took, 15.068, seconds to generate. Fact Number, 179 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 8, 37, 159, 525, 2438, 10042, 41525, 185548, 801462, 3529370, 15839885, 70692828, 319336202, 1451168239, 6608519719, 30288406388, 139296525886, 642637894372, 2976038764761, 13819096891959, 64342031682708, 300361760409456, 1405257565517345, 6588774191205851, 30954308294418044, 145689560892329823, 686892645081487527, 3243760841098195210, 15341313867232060292, 72659797444569360871, 344592030585316531227, 1636299576905358897298, 7779242608860177776633, 37025432233530294285494, 176411070882997673332990, 841377250137556015049995, 4016731604260519937116224] ---------------------------------------- This ends Fact No. , 179 that took, 15.251, seconds to generate. Fact Number, 180 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 12, 35, 145, 591, 2382, 10432, 43664, 194085, 843258, 3780514, 16859418, 76309679, 346163000, 1582070419, 7260873642, 33476318422, 154970500673, 719854368101, 3355281579644, 15683637412269, 73517632827284, 345459740596778, 1627145915782916, 7680266521836452, 36324189829942464, 172114540320427285, 816941996946739829, 3883890762356052732, 18492736536820842926, 88176741397595277570, 421006472639548012484, 2012669347834850035371, 9633280506436145623545, 46160048518629558843426, 221422941434925750000323, 1063211162864711626762878, 5110164649379414414466014] ---------------------------------------- This ends Fact No. , 180 that took, 15.437, seconds to generate. Fact Number, 181 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 6, 25, 115, 527, 2543, 12630, 63569, 326545, 1697119, 8921094, 47344056, 253258115, 1364381346, 7395544297, 40305245926, 220725694389, 1214008928461, 6703220103017, 37143019250523, 206473636449997, 1151132503139380, 6435077295543729, 36062602355754011, 202559653694417233, 1140166430013004419, 6430394451160630714, 36333095756335782202, 205641053733279226445, 1165768067295268456197, 6618600224908285415116, 37629848041907997718718, 214227498423128204181501, 1221129116351598192575845, 6968855841340925863336699, 39815047311035224760216802, 227715905500311549871746638, 1303696016855881180091743410, 7470925055954882600844480878] ---------------------------------------- This ends Fact No. , 181 that took, 15.577, seconds to generate. Fact Number, 182 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 6, 21, 116, 474, 2441, 11964, 60376, 314230, 1632196, 8658074, 46174396, 248498500, 1347490282, 7347493544, 40303415382, 222108850452, 1229464846249, 6832478437045, 38104355448182, 213199540197523, 1196400038438485, 6732018628924680, 37974930405337963, 214707865395659647, 1216536492028395828, 6906560392414260168, 39282500772963974855, 223812146845658449201, 1277224510471476285971, 7299721511751094009619, 41779292791433478687128, 239439312796025940370186, 1373967052294041202263832, 7893567695726185429303811, 45400291607566749187145885, 261400584701848078745200891, 1506584076695419808919164010, 8691548745202259798394442097] ---------------------------------------- This ends Fact No. , 182 that took, 15.780, seconds to generate. Fact Number, 183 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 4, 29, 98, 516, 2346, 11954, 59870, 310530, 1620212, 8583056, 45839028, 247042027, 1340584492, 7321844976, 40209683152, 221920587012, 1230198545078, 6846674544463, 38242139930990, 214300025781975, 1204468235474154, 6788187359629484, 38353267837797488, 217198590390067809, 1232659037451487146, 7009605766655301002, 39934579732010299064, 227906006347127010720, 1302761735199632124434, 7458178871381320544536, 42758177771253650183380, 245463953759134198090647, 1410928758183268942139722, 8119713990109618760484580, 46780688250054693430536144, 269809273437771146226448526, 1557713847355137855878910282, 9001957850273498566051449430] ---------------------------------------- This ends Fact No. , 183 that took, 15.975, seconds to generate. Fact Number, 184 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 24, 89, 470, 2129, 10980, 54658, 284849, 1481762, 7858805, 41900508, 225750402, 1223494672, 6676540873, 36622040063, 201896517476, 1117826366106, 6213674868016, 34662396151415, 193990236249636, 1088885812690092, 6128628034177263, 34580124294809255, 195564515211411558, 1108356048154963106, 6294025238935490316, 35807850776069646642, 204068217177747803286, 1164854379026284217317, 6659220841993341285409, 38123346860616832542411, 218543683488824652763864, 1254388433572313034263366, 7208452344005099399652685, 41470603209459772556500247, 238836843909290269618638510, 1376896319383477781164589806, 7945443960404671590646519185] ---------------------------------------- This ends Fact No. , 184 that took, 16.179, seconds to generate. Fact Number, 185 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -1, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 8, 44, 224, 1205, 6840, 39318, 232444, 1392592, 8467716, 52051503, 323080389, 2021815791, 12742735306, 80814679057, 515351618433, 3302478877283, 21255700235421, 137347432168055, 890661479850665, 5794439070450058, 37808945270000250, 247374683788793626, 1622554809760500329, 10667034127346822067, 70277154363763220662, 463921753526840225510, 3068146790853596703244, 20326144981085652653100, 134875544349445658645282, 896328157188699012358513, 5965087294111445544705924, 39750839074671464464803274 , 265229867257603037200868949, 1771800585635190145069233788, 11849380684930749354236212046, 79330314329811673494572228530, 531644954798530241023209551816, 3566327991735747678817417834095] ---------------------------------------- This ends Fact No. , 185 that took, 16.384, seconds to generate. Fact Number, 186 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 0, 1, 1, 1, 3, 7, 12, 21, 50, 110, 209, 442, 1001, 2128, 4480, 9860, 21828, 47481, 103968, 231192, 513513, 1136982, 2533289, 5672260, 12695540, 28448355, 63972675, 144134640, 324977016, 733931913, 1660780236, 3762635044, 8533415880, 19379626068, 44067267808, 100303245980, 228530941928, 521230949853, 1189938422855] ---------------------------------------- This ends Fact No. , 186 that took, 16.394, seconds to generate. Fact Number, 187 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 7, 8, 38, 65, 199, 543, 1246, 4035, 9556, 28818, 78122, 214933, 629873, 1709610, 5021439, 14093846, 40526640, 117212832, 334644878, 977139174, 2813214366, 8201080331, 23876445760, 69542023022, 203777286111, 595381763799, 1748718032444, 5133556543817, 15099066239211, 44495128105383, 131147147823020, 387426686901620, 1144903084134301, 3388301146564357, 10037246686293247, 29756939669744309] ---------------------------------------- This ends Fact No. , 187 that took, 16.470, seconds to generate. Fact Number, 188 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 0, 5, 2, 28, 27, 202, 301, 1602, 3177, 13910, 33016, 128266, 342820, 1239098, 3577198, 12386907, 37605752, 127076336, 398632495, 1329629938, 4261217915, 14127149566, 45919144718, 151935298816, 498568851090, 1650229625236 , 5450869527868, 18070744589130, 59971995488605, 199251965396011, 663615612911163, 2210052989218112, 7381286080871931, 24640359998387002, 82485730945578910, 275977010346735803, 925685842356775708, 3103603047697663752] ---------------------------------------- This ends Fact No. , 188 that took, 16.543, seconds to generate. Fact Number, 189 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 2, 13, 8, 109, 134, 766, 2222, 5667, 27825, 57823, 291595, 758231, 2888278, 10019673, 30553000, 122532818, 362736708, 1415922277, 4594544294, 16288545726, 58193764683, 194253053346, 720179999943, 2412584922755, 8793964423849, 30562231165838, 108005451494368, 387301565469087, 1349166079379134, 4882357369086739, 17113374293107050, 61501403849121275, 218700782342812768, 779036582786730989, 2799184161473324138, 9949047480322149410, 35839494559135870332] ---------------------------------------- This ends Fact No. , 189 that took, 16.683, seconds to generate. Fact Number, 190 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 4, 6, 29, 107, 310, 1285, 4755, 17072, 68097, 261074, 1008449, 4019538, 15856730, 63120997, 253955943, 1021134596, 4130964757, 16792455962, 68395002809 , 279669185840, 1147006764628, 4714858037720, 19433651768421, 80280672686204, 332305871983609, 1378320168716942, 5727177767108796, 23837465768727622, 99375775609570540, 414899721471863616, 1734653866111787864, 7262031828592394561 , 30439797146760801971, 127742603344567668902, 536677521075129555005, 2257085847121359584834, 9502047618152094682026, 40040473246167731114949] ---------------------------------------- This ends Fact No. , 190 that took, 16.759, seconds to generate. Fact Number, 191 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 9, 30, 114, 435, 1632, 6579, 26242, 106833, 439395, 1825652, 7640859, 32214100, 136676305, 582912766, 2498452213, 10754837780, 46476863187, 201558528689, 876928763837, 3826500214994, 16742047926682, 73433163298618, 322826846258108, 1422223345190754, 6277996996195399, 27763285161878786, 122988389682909472, 545698946551522530, 2424903800447054625, 10790682240448018921, 48081758717108570141, 214514159797536743989, 958176738963435645716, 4284711813074472760815, 19180433623946052630684, 85947346068967519992406, 385497851671482105467914] ---------------------------------------- This ends Fact No. , 191 that took, 16.854, seconds to generate. Fact Number, 192 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 9, 33, 119, 465, 1842, 7593, 31547, 132973, 565997, 2433703, 10551396 , 46073446, 202424887, 894219578, 3969529256, 17698183707, 79218036743, 355845761188, 1603622644830, 7248091381618, 32848838026095, 149244205098392, 679630890563661, 3101509622852940, 14181810462297919, 64966529546499114, 298121930030616458, 1370239229393886772, 6307445860137827026, 29075401730765256677, 134207522346745066047, 620260192169601243801, 2870031058236734986342, 13294961730655108760282, 61652367586963523840353, 286187678985212386480646, 1329746075157892562412721] ---------------------------------------- This ends Fact No. , 192 that took, 17.061, seconds to generate. Fact Number, 193 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 1, 14, 28, 145, 516, 1979, 8908, 32441, 154161, 585672, 2740838, 11061624, 50413130, 213687612, 957236549, 4188565357, 18653392705, 83110071258, 370756924711, 1668000143121, 7479757197768, 33836186180325, 152641293855293, 693103339062975, 3143907958532025, 14320921951988689, 65258577234059766, 298145590895329589, 1363718159921737018, 6248066678501022899, 28667371686851204749, 131692952209625592984, 605824896356913007642, 2789854967643889563545, 12863630295346101943215, 59368850862446003976810, 274302757655275776867112, 1268501890425313783407797] ---------------------------------------- This ends Fact No. , 193 that took, 17.182, seconds to generate. Fact Number, 194 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 2, 14, 32, 159, 524, 2338, 9082, 39516, 165400, 722149, 3135027, 13841240, 61327446, 273934220, 1229936066, 5549881920, 25160782183, 114494874135, 523035436496, 2396690608896, 11017034973520, 50775851725898, 234629462228955, 1086656911086627, 5043771756404914, 23457103429023694, 109297225320797700, 510143458543390700, 2384974385809157044, 11166922697405924006, 52360652310059059353, 245843911662281125334, 1155752014079370429492, 5439877402788341601314, 25633338178551940672130, 120916776552465814662832, 570965072595477380660308, 2698682919563333007403026] ---------------------------------------- This ends Fact No. , 194 that took, 17.359, seconds to generate. Fact Number, 195 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 1, 14, 28, 145, 576, 1993, 10820, 34209, 199219, 669788, 3712506, 13897436, 71203702, 294233863, 1414210779, 6269756485, 29036471404, 134120314010, 612540959851, 2884031274762, 13183216941756, 62458531170436, 287763886503395, 1363940212039958, 6344531495418125, 30041986726274074, 140937702147274056, 667101525149799718, 3150025373968215933, 14921520605296739009, 70782943107581828669, 335871561508315295364, 1598337752997159551148, 7601251857372211031392, 36255878028082081970773, 172833025265715998744285, 825872089720081621587041, 3945888969760363419264442] ---------------------------------------- This ends Fact No. , 195 that took, 17.536, seconds to generate. Fact Number, 196 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 6, 21, 117, 449, 2387, 11234, 56355, 289252, 1471858, 7724107, 40427754, 214442115, 1144284334, 6139037391, 33152981513, 179749651008, 979272133586, 5354970609236, 29386618328805, 161793108491311, 893349780901737, 4946162401427134, 27452595269075619, 152719470064311219, 851385478194669127, 4755670197695159219, 26613040311658985001, 149183191076710677717, 837608540735970579807, 4709929978217260226714, 26521651814763104950884, 149542154142854387322557, 844249221319698387870556, 4771903174533925414114749, 27002176072591155534082429, 152956025362129647474539518, 867304385711510843270881168, 4922569102038997258451514869] ---------------------------------------- This ends Fact No. , 196 that took, 17.669, seconds to generate. Fact Number, 197 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 24, 99, 467, 2244, 11088, 55990, 287175, 1494246, 7859987, 41751210, 223584684, 1205933290, 6544890962, 35716714784, 195867367871, 1078828297122, 5965573113464, 33105420479596, 184311598004294, 1029178415527183, 5762437400106894, 32344855509348933, 181971931852627460, 1025958463310283994, 5795824741994932413, 32802092122144099425, 185966765450609575638, 1056011000640987887517, 6005610877816069880621, 34202816939108641130626, 195050154725945233764750, 1113723432012470632340513, 6366851255646933338257723, 36438496786114207646090687, 208765589362151609308583440, 1197281801336040615847143089, 6873072314552569018362060742] ---------------------------------------- This ends Fact No. , 197 that took, 17.867, seconds to generate. Fact Number, 198 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 24, 89, 470, 2129, 10980, 54658, 284849, 1481762, 7858805, 41900508, 225750402, 1223494672, 6676540873, 36622040063, 201896517476, 1117826366106, 6213674868016, 34662396151415, 193990236249636, 1088885812690092, 6128628034177263, 34580124294809255, 195564515211411558, 1108356048154963106, 6294025238935490316, 35807850776069646642, 204068217177747803286, 1164854379026284217317, 6659220841993341285409, 38123346860616832542411, 218543683488824652763864, 1254388433572313034263366, 7208452344005099399652685, 41470603209459772556500247, 238836843909290269618638510, 1376896319383477781164589806, 7945443960404671590646519185] ---------------------------------------- This ends Fact No. , 198 that took, 18.062, seconds to generate. Fact Number, 199 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 2, 33, 66, 567, 1856, 12279, 51472, 304486, 1451832, 8203376, 41915700, 233072192, 1237979182, 6864234453, 37310614660, 207423272288, 1144188631542, 6390369615330, 35609962951332, 199889290801917, 1122231114895000, 6329956908308426, 35745816167751470, 202515271155474359, 1149057092821556004, 6535551318528771722, 37230027691876546624, 212494302831417357939, 1214616595740886649306, 6953978490550821738644, 39867127514816369305072, 228875533886845877148935, 1315580854871015499446334, 7571182786717452519888998, 43620714963706975978344234, 251588466612272646199119942, 1452531511600394185756421184, 8394231042401234597547800818] ---------------------------------------- This ends Fact No. , 199 that took, 18.253, seconds to generate. Fact Number, 200 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 7, 46, 214, 1211, 6797, 39292, 233928, 1402067, 8583291, 52872607, 329844518, 2071001475, 13108529709, 83448520958, 534282941239, 3437152982092, 22209755550285, 144075707613003, 937963029135836, 6126139218663923, 40130195298575245, 263592695992953473, 1735713593106120119, 11455732995839046111, 75769321196965311238, 502138844806826923379, 3333916920849457267940, 22173437294581749515021, 147710159535520283790436, 985469474682831050341732, 6584029316677878702480727, 44047363781240060345607118 , 295049295550039657442549400, 1978724835239361998189146389, 13285090351631288172840267058, 89290699873293723260330446863, 600740406992292157873483266398, 4045613934505507202258855526102] ---------------------------------------- This ends Fact No. , 200 that took, 18.458, seconds to generate. Fact Number, 201 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 1}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 1, 3, 6, 14, 35, 82, 213, 527, 1375, 3542, 9308, 24544, 65191, 174302, 467908, 1263134, 3421539, 9306884, 25395191, 69515965, 190821202, 525163025, 1448785065, 4005590760, 11097459750, 30803796285, 85656062715, 238578660846, 665549820519, 1859357891730, 5201651013628, 14570714678278, 40864877919842, 114741473444302, 322526012437664, 907525112122184, 2556109053671163, 7206193195047795] ---------------------------------------- This ends Fact No. , 201 that took, 18.470, seconds to generate. Fact Number, 202 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 5, 14, 45, 132, 425, 1382, 4514, 15074, 50789, 172559, 591478, 2041556, 7087471, 24743296, 86792677, 305748394, 1081274484, 3837319132, 13661640800, 48780359677, 174641599757, 626785406849, 2254641398394, 8127379935041, 29354406339309, 106215414640325, 384982365317509, 1397609307709572, 5081353804008476, 18500467219189397, 67446577033587605, 246194571085717498, 899722241895396679, 3291716072107537857, 12055767543764840101, 44197992941498268988, 162189883599470455418] ---------------------------------------- This ends Fact No. , 202 that took, 18.546, seconds to generate. Fact Number, 203 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 2, 6, 22, 66, 252, 848, 3218, 11725, 44502, 169026, 649237, 2518289, 9812886, 38556241, 152092831, 603423111, 2403322940, 9611508342, 38572610558, 155297193204, 627098136122, 2539022823888, 10305860644106, 41926721236162, 170931650675469, 698245783792833, 2857537567523935, 11714387363887240, 48099819158068437, 197797296952047735, 814536604784177900, 3358755787761435674, 13867263598687916962, 57321499879000098991, 237208898321773127799, 982665100210659188886, 4074906222703825280497, 16913964344146139238827] ---------------------------------------- This ends Fact No. , 203 that took, 18.632, seconds to generate. Fact Number, 204 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 1, 3, 5, 28, 90, 300, 1340, 4720, 19224, 78820, 309895, 1299231, 5338874, 22148050, 93434582, 391938929, 1662056604, 7074526063, 30168464264, 129408353368, 556198907308, 2398053883570, 10371629134077, 44948387189829, 195299032310816, 850344880548907, 3709508198783748, 16213694839495304, 70985794462036252, 311290410148601782, 1367172855288292117, 6012971476311705743 , 26481376365070704301, 116771849758445934030, 515526878612169852893, 2278532324221682665744, 10081407770093577248123, 44650557420113720402486, 197947643610523846362502] ---------------------------------------- This ends Fact No. , 204 that took, 18.810, seconds to generate. Fact Number, 205 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 1, 2}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 18, 73, 290, 1223, 5259, 23049, 102703, 463508, 2114586, 9737060, 45192097, 211199028, 992997848, 4693803931, 22293043296, 106332602178, 509135524570, 2446319933288, 11791527427505, 57001393740128, 276284001879588, 1342423134398429, 6537407560700816, 31903039573287447, 155992783588117250, 764126173919465124, 3749399159675148078, 18426685432205212254, 90694091814132640031, 447010928136072932856, 2206120331479291546459, 10901335981633850369590, 53931185491617579479953, 267105334498062695387913, 1324287541095964236040935, 6572277200614525363625511, 32648388198061017486845493] ---------------------------------------- This ends Fact No. , 205 that took, 18.891, seconds to generate. Fact Number, 206 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 1, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 19, 83, 347, 1558, 7163, 33363, 158849, 764668, 3723905, 18311102, 90753476, 453017546, 2275288314, 11490033498, 58306220057, 297160459897, 1520424304907, 7806833844973, 40214457567004, 207761908832432, 1076267071688253 , 5589222169763409, 29092253152085330, 151748567111547764, 793099243676858407, 4152669418745126818, 21780664402751249922, 114422382678727944803, 602009826602592042005, 3171826794792340438352, 16733726510866175251444, 88393492534719713377817, 467479081409996431509100, 2475087790725635009123409, 13118374671923072912752337, 69599620069376221637273227, 369614759969301977577341515] ---------------------------------------- This ends Fact No. , 206 that took, 18.993, seconds to generate. Fact Number, 207 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 1, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 21, 80, 379, 1706, 8225, 39664, 196297, 980574, 4963069, 25340871, 130498727, 676723549, 3531411785, 18529267945, 97700050628, 517401200142, 2750869368345, 14677721462643, 78569435214289, 421825341903492, 2270852774406485, 12255428487665875, 66293054280114936, 359363186334713677, 1951906934072249979, 10621487830462605860, 57897393105933378657, 316105389396670180818, 1728464126122428825586, 9464653657999162443391, 51895389267009079068629, 284904571409905552493816, 1565980383960187297304960, 8617109094860609735400486, 47467849365367837580730225, 261743859011672427519003223, 1444676125021442286928312139] ---------------------------------------- This ends Fact No. , 207 that took, 19.220, seconds to generate. Fact Number, 208 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 4, 24, 82, 410, 1785, 8725, 41430, 205221, 1016199, 5125126, 25993548, 133175275, 686393117, 3561569210, 18575651599, 97368087820, 512549531921, 2708750703907, 14365708065579, 76433925018425, 407866200085910, 2182328511230655, 11705718596847398, 62931821554624832, 339049323572699808, 1830250984719160443, 9898165151387946036, 53621902634723942488, 290954704851412757057, 1581105635733268646594, 8604176533948514498281, 46885087489221705470263, 255802197823545269079079, 1397294018291880210521059, 7641130449027926071567603, 41830031736621289983307661, 229221830789605978076342582, 1257300713220926794579300018] ---------------------------------------- This ends Fact No. , 208 that took, 19.322, seconds to generate. Fact Number, 209 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 20, 93, 399, 1941, 9268, 46056, 231133, 1178069, 6071454, 31587133, 165722945, 875572969, 4655258827, 24886810879, 133698537851, 721411301508, 3907970421062, 21245559580445, 115875807979820, 633874170787929, 3476896181543665, 19118978975571799, 105375235434670288, 582021783378845451, 3221073023191972811, 17859250614304540906, 99191356883751539068, 551803105586217829029, 3074329772707969871664, 17152721061127098082779, 95828707447007095535495, 536049799444002397243984, 3002141641526611590329954, 16832392294545062898919940, 94476420276619204644852131, 530811355404767546964415312, 2985205413525769848330080375] ---------------------------------------- This ends Fact No. , 209 that took, 19.591, seconds to generate. Fact Number, 210 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 2, 5, 18, 103, 375, 2106, 9569, 49152, 251313, 1280581, 6778689, 35417695, 189284212, 1011771708, 5453066888, 29566502151, 160884413244, 880388306191, 4832512200881, 26632927096920, 147231798404710, 816341656082818, 4538865746430343, 25297532772313916, 141329266342997491, 791226069742202571, 4438490524168795481, 24944263019403452860, 140427823076010800536, 791839345602523829828, 4471733695015749171105, 25289019778362724107676, 143208197125076098225042, 811990412603012918033033, 4609466041363415319663577, 26196242758840065370481710, 149035761201708748836313867, 848752794766974317905637717, 4838261097069496793835514902] ---------------------------------------- This ends Fact No. , 210 that took, 19.786, seconds to generate. Fact Number, 211 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 1, 2, 3}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 9, 47, 236, 1255, 6958, 39302, 227439, 1335100, 7948216, 47832942, 290608143, 1779883158, 10977966112, 68127590542, 425090391580, 2665251082146, 16783155395439, 106096597710740, 673071844607004, 4283661267869412, 27342741560011682, 174999464650230043, 1122809783075305501, 7220515474260568103 , 46531773058036815488, 300458862623285346399, 1943639973423848468549, 12594740239019599708079, 81744301452907705557981, 531345960488654354893658, 3458673487385365659862484, 22543334242180428959049014, 147119527691524356842839582, 961249191362960963631903186, 6287637353055041210486027120, 41171750323955206472628950356, 269865887991244159127005781623, 1770564273916956121948841979311] ---------------------------------------- This ends Fact No. , 211 that took, 19.929, seconds to generate. Fact Number, 212 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 1, 2, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 8, 46, 224, 1226, 6831, 39180, 229513, 1365735, 8241531, 50284073, 309775925, 1923969783, 12034822349, 75748984620, 479399092456, 3048847393883, 19474677151785, 124885427781600, 803707503490482, 5189062935157686, 33601806280718427, 218178727534017686, 1420183204095809574, 9265640254001917065 , 60580340822209660764, 396869047279229152722, 2604730640762604297904, 17124788854648694776491, 112768118963450266205826, 743708153538948382746211, 4911726720925964577667997, 32482177702359776513065069, 215081129689466972529194665, 1425855117056824213475406656, 9463168029887154809683760910, 62872410179319386099496237645, 418140643926299625657053340368, 2783565611195766239903843790392] ---------------------------------------- This ends Fact No. , 212 that took, 20.134, seconds to generate. Fact Number, 213 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 1, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 8, 44, 224, 1205, 6840, 39318, 232444, 1392592, 8467716, 52051503, 323080389, 2021815791, 12742735306, 80814679057, 515351618433, 3302478877283, 21255700235421, 137347432168055, 890661479850665, 5794439070450058, 37808945270000250, 247374683788793626, 1622554809760500329, 10667034127346822067, 70277154363763220662, 463921753526840225510, 3068146790853596703244, 20326144981085652653100, 134875544349445658645282, 896328157188699012358513, 5965087294111445544705924, 39750839074671464464803274 , 265229867257603037200868949, 1771800585635190145069233788, 11849380684930749354236212046, 79330314329811673494572228530, 531644954798530241023209551816, 3566327991735747678817417834095] ---------------------------------------- This ends Fact No. , 213 that took, 20.374, seconds to generate. Fact Number, 214 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 3, 7, 46, 214, 1211, 6797, 39292, 233928, 1402067, 8583291, 52872607, 329844518, 2071001475, 13108529709, 83448520958, 534282941239, 3437152982092, 22209755550285, 144075707613003, 937963029135836, 6126139218663923, 40130195298575245, 263592695992953473, 1735713593106120119, 11455732995839046111, 75769321196965311238, 502138844806826923379, 3333916920849457267940, 22173437294581749515021, 147710159535520283790436, 985469474682831050341732, 6584029316677878702480727, 44047363781240060345607118 , 295049295550039657442549400, 1978724835239361998189146389, 13285090351631288172840267058, 89290699873293723260330446863, 600740406992292157873483266398, 4045613934505507202258855526102] ---------------------------------------- This ends Fact No. , 214 that took, 20.582, seconds to generate. Fact Number, 215 Theorem: Let a(n) be the number of sequences, of length n ,in the alphabet, {-4, -3, -2, -1, 1, 2, 3, 4}, that add-up to zero and such that their partial sums are NEVER positive. Then the first, 40, terms are [0, 4, 12, 82, 454, 2912, 18652, 124299, 841400, 5800725, 40506816, 286137616, 2040430976, 14670243774, 106225269954, 773958961125, 5670067999156, 41742291894425, 308645064367896, 2291123920091484, 17067970534656790, 127561390515473136, 956178202877392924, 7186775021944900011, 54151565796975489136, 408965523168941116374, 3095189509742107222596, 23471817329902172794262, 178322431645587215705708, 1357099829315140907449281, 10344656683501531412175872, 78972383829301271624255725, 603739597784276570651376116, 4621715223961674977454921412, 35424421248282763919182980330, 271842971557901397835976125542, 2088436977509949669000724559036, 16061529918152158484629286236917, 123649123324846199302225344244184, 952822187230646758734825649542187] ---------------------------------------- This ends Fact No. , 215 that took, 20.793, seconds to generate. -------------------------------------