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oeis-sequences-benchmark

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sequence_id
string
sequence_name
string
sequence_first_terms
list
sequence_next_term
string
is_easy
int64
A000101
Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps).
[ "3", "5", "11", "29", "97", "127", "541", "907", "1151", "1361", "9587", "15727", "19661", "31469", "156007", "360749", "370373", "492227", "1349651" ]
1357333
0
A000102
a(n) = number of compositions of n in which the maximum part size is 4.
[ "0", "0", "0", "0", "1", "2", "5", "12", "27", "59", "127", "269", "563", "1167", "2400", "4903", "9960", "20135", "40534" ]
81300
1
A000103
Number of n-node triangulations of sphere in which every node has degree >= 4.
[ "0", "0", "1", "1", "2", "5", "12", "34", "130", "525", "2472", "12400", "65619", "357504", "1992985", "11284042", "64719885", "375126827", "2194439398" ]
12941995397
0
A000104
Number of n-celled free polyominoes without holes.
[ "1", "1", "1", "2", "5", "12", "35", "107", "363", "1248", "4460", "16094", "58937", "217117", "805475", "3001127", "11230003", "42161529", "158781106" ]
599563893
0
A000105
Number of free polyominoes (or square animals) with n cells.
[ "1", "1", "1", "2", "5", "12", "35", "108", "369", "1285", "4655", "17073", "63600", "238591", "901971", "3426576", "13079255", "50107909", "192622052" ]
742624232
0
A000106
2nd power of rooted tree enumerator; number of linear forests of 2 rooted trees.
[ "1", "2", "5", "12", "30", "74", "188", "478", "1235", "3214", "8450", "22370", "59676", "160140", "432237", "1172436", "3194870", "8741442", "24007045" ]
66154654
1
A000107
Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.
[ "0", "1", "2", "5", "13", "35", "95", "262", "727", "2033", "5714", "16136", "45733", "130046", "370803", "1059838", "3035591", "8710736", "25036934" ]
72069134
1
A000108
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).
[ "1", "1", "2", "5", "14", "42", "132", "429", "1430", "4862", "16796", "58786", "208012", "742900", "2674440", "9694845", "35357670", "129644790", "477638700" ]
1767263190
1
A000109
Number of simplicial polyhedra with n vertices; simple planar graphs with n vertices and 3n-6 edges; maximal simple planar graphs with n vertices; planar triangulations with n vertices; triangulations of the sphere with n vertices; 3-connected cubic planar graphs on 2n-4 vertices.
[ "1", "1", "1", "2", "5", "14", "50", "233", "1249", "7595", "49566", "339722", "2406841", "17490241", "129664753", "977526957", "7475907149", "57896349553", "453382272049" ]
3585853662949
0
A000110
Bell or exponential numbers: number of ways to partition a set of n labeled elements.
[ "1", "1", "2", "5", "15", "52", "203", "877", "4140", "21147", "115975", "678570", "4213597", "27644437", "190899322", "1382958545", "10480142147", "82864869804", "682076806159" ]
5832742205057
1
A000111
Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).
[ "1", "1", "1", "2", "5", "16", "61", "272", "1385", "7936", "50521", "353792", "2702765", "22368256", "199360981", "1903757312", "19391512145", "209865342976", "2404879675441" ]
29088885112832
1
A000112
Number of partially ordered sets ("posets") with n unlabeled elements.
[ "1", "1", "2", "5", "16", "63", "318", "2045", "16999", "183231", "2567284", "46749427", "1104891746", "33823827452", "1338193159771", "68275077901156" ]
4483130665195087
0
A000113
Number of transformation groups of order n.
[ "1", "3", "4", "3", "6", "12", "8", "6", "4", "18", "12", "12", "14", "24", "24", "6", "18", "12", "20" ]
18
1
A000114
Number of cusps of principal congruence subgroup GAMMA^{hat}(n).
[ "3", "4", "6", "12", "12", "24", "24", "36", "36", "60", "48", "84", "72", "96", "96", "144", "108", "180", "144" ]
192
1
A000115
Denumerants: Expansion of 1/((1-x)*(1-x^2)*(1-x^5)).
[ "1", "1", "2", "2", "3", "4", "5", "6", "7", "8", "10", "11", "13", "14", "16", "18", "20", "22", "24" ]
26
1
A000116
Number of even sequences with period 2n (bisection of A000013).
[ "1", "2", "4", "8", "20", "56", "180", "596", "2068", "7316", "26272", "95420", "349716", "1290872", "4794088", "17896832", "67110932", "252648992", "954444608" ]
3616828364
1
A000117
Number of even sequences with period 2n (bisection of A000011).
[ "1", "2", "4", "8", "18", "44", "122", "362", "1162", "3914", "13648", "48734", "176906", "649532", "2405236", "8964800", "33588234", "126390032", "477353376" ]
1808676326
1
A000118
Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.
[ "1", "8", "24", "32", "24", "48", "96", "64", "24", "104", "144", "96", "96", "112", "192", "192", "24", "144", "312" ]
160
1
A000119
Number of representations of n as a sum of distinct Fibonacci numbers.
[ "1", "1", "1", "2", "1", "2", "2", "1", "3", "2", "2", "3", "1", "3", "3", "2", "4", "2", "3" ]
3
0
A000120
1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).
[ "0", "1", "1", "2", "1", "2", "2", "3", "1", "2", "2", "3", "2", "3", "3", "4", "1", "2", "2" ]
3
1
A000121
Number of representations of n as a sum of Fibonacci numbers (1 is allowed twice as a part).
[ "1", "2", "2", "3", "3", "3", "4", "3", "4", "5", "4", "5", "4", "4", "6", "5", "6", "6", "5" ]
6
0
A000122
Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n).
[ "1", "2", "0", "0", "2", "0", "0", "0", "0", "2", "0", "0", "0", "0", "0", "0", "2", "0", "0" ]
0
1
A000123
Number of binary partitions: number of partitions of 2n into powers of 2.
[ "1", "2", "4", "6", "10", "14", "20", "26", "36", "46", "60", "74", "94", "114", "140", "166", "202", "238", "284" ]
330
1
A000124
Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.
[ "1", "2", "4", "7", "11", "16", "22", "29", "37", "46", "56", "67", "79", "92", "106", "121", "137", "154", "172" ]
191
1
A000125
Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3) + n + 1.
[ "1", "2", "4", "8", "15", "26", "42", "64", "93", "130", "176", "232", "299", "378", "470", "576", "697", "834", "988" ]
1160
1
A000126
A nonlinear binomial sum.
[ "1", "2", "4", "8", "15", "27", "47", "80", "134", "222", "365", "597", "973", "1582", "2568", "4164", "6747", "10927", "17691" ]
28636
1
A000127
Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.
[ "1", "2", "4", "8", "16", "31", "57", "99", "163", "256", "386", "562", "794", "1093", "1471", "1941", "2517", "3214", "4048" ]
5036
1
A000128
A nonlinear binomial sum.
[ "1", "2", "4", "8", "16", "31", "58", "105", "185", "319", "541", "906", "1503", "2476", "4058", "6626", "10790", "17537", "28464" ]
46155
1
A000129
Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
[ "0", "1", "2", "5", "12", "29", "70", "169", "408", "985", "2378", "5741", "13860", "33461", "80782", "195025", "470832", "1136689", "2744210" ]
6625109
1
A000130
One-half the number of permutations of length n with exactly 1 rising or falling successions.
[ "0", "0", "1", "2", "5", "20", "115", "790", "6217", "55160", "545135", "5938490", "70686805", "912660508", "12702694075", "189579135710", "3019908731105", "51139445487680", "917345570926087" ]
17376071107513090
0
A000131
Number of asymmetrical dissections of n-gon.
[ "2", "5", "21", "61", "214", "669", "2240", "7330", "24695", "83257", "284928", "981079", "3410990", "11937328", "42075242", "149171958", "531866972", "1905842605", "6861162880" ]
24805692978
0
A000132
Number of ways of writing n as a sum of 5 squares.
[ "1", "10", "40", "80", "90", "112", "240", "320", "200", "250", "560", "560", "400", "560", "800", "960", "730", "480", "1240" ]
1520
1
A000133
Number of Boolean functions of n variables.
[ "2", "5", "30", "2288", "67172352", "144115192303714304", "1329227995784915891206435945914040320" ]
226156424291633194186662080095093570364871077725232774230036394136943198208
1
A000134
Positive zeros of Bessel function of order 0 rounded to nearest integer.
[ "2", "6", "9", "12", "15", "18", "21", "24", "27", "31", "34", "37", "40", "43", "46", "49", "53", "56", "59" ]
62
0
A000135
Number of partitions into non-integral powers.
[ "1", "2", "6", "13", "24", "42", "73", "125", "204", "324", "511", "801", "1228", "1856", "2780", "4135", "6084", "8873", "12847" ]
18481
0
A000136
Number of ways of folding a strip of n labeled stamps.
[ "1", "2", "6", "16", "50", "144", "462", "1392", "4536", "14060", "46310", "146376", "485914", "1557892", "5202690", "16861984", "56579196", "184940388", "622945970" ]
2050228360
0
A000137
Series-parallel numbers.
[ "1", "2", "6", "18", "58", "186", "614", "2034", "6818", "22970", "77858", "264970", "905294", "3102434", "10661370", "36722642", "126752218", "438294018", "1518032598" ]
5265341314
1
A000138
Expansion of e.g.f. exp(-x^4/4)/(1-x).
[ "1", "1", "2", "6", "18", "90", "540", "3780", "31500", "283500", "2835000", "31185000", "372972600", "4848643800", "67881013200", "1018215198000", "16294848570000", "277012425690000", "4986223662420000" ]
94738249585980000
1
A000139
a(n) = 2*(3*n)! / ((2*n+1)!*(n+1)!).
[ "2", "1", "2", "6", "22", "91", "408", "1938", "9614", "49335", "260130", "1402440", "7702632", "42975796", "243035536", "1390594458", "8038677054", "46892282815", "275750636070" ]
1633292229030
1
A000140
Kendall-Mann numbers: the most common number of inversions in a permutation on n letters is floor(n*(n-1)/4); a(n) is the number of permutations with this many inversions.
[ "1", "1", "2", "6", "22", "101", "573", "3836", "29228", "250749", "2409581", "25598186", "296643390", "3727542188", "50626553988", "738680521142", "11501573822788", "190418421447330", "3344822488498265" ]
62119523114983224
1
A000141
Number of ways of writing n as a sum of 6 squares.
[ "1", "12", "60", "160", "252", "312", "544", "960", "1020", "876", "1560", "2400", "2080", "2040", "3264", "4160", "4092", "3480", "4380" ]
7200
1
A000142
Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).
[ "1", "1", "2", "6", "24", "120", "720", "5040", "40320", "362880", "3628800", "39916800", "479001600", "6227020800", "87178291200", "1307674368000", "20922789888000", "355687428096000", "6402373705728000" ]
121645100408832000
1
A000143
Number of ways of writing n as a sum of 8 squares.
[ "1", "16", "112", "448", "1136", "2016", "3136", "5504", "9328", "12112", "14112", "21312", "31808", "35168", "38528", "56448", "74864", "78624", "84784" ]
109760
1
A000144
Number of ways of writing n as a sum of 10 squares.
[ "1", "20", "180", "960", "3380", "8424", "16320", "28800", "52020", "88660", "129064", "175680", "262080", "386920", "489600", "600960", "840500", "1137960", "1330420" ]
1563840
1
A000145
Number of ways of writing n as a sum of 12 squares.
[ "1", "24", "264", "1760", "7944", "25872", "64416", "133056", "253704", "472760", "825264", "1297056", "1938336", "2963664", "4437312", "6091584", "8118024", "11368368", "15653352" ]
19822176
1
A000146
From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)|2n} 1/p.
[ "1", "1", "1", "1", "1", "1", "2", "-6", "56", "-528", "6193", "-86579", "1425518", "-27298230", "601580875", "-15116315766", "429614643062", "-13711655205087", "488332318973594" ]
-19296579341940067
1
A000147
Number of trees of diameter 5.
[ "0", "0", "0", "0", "0", "1", "2", "7", "14", "32", "58", "110", "187", "322", "519", "839", "1302", "2015", "3032" ]
4542
0
A000148
Number of partitions into non-integral powers.
[ "1", "2", "7", "15", "28", "45", "70", "100", "138", "183", "242", "310", "388", "481", "583", "701", "838", "984", "1152" ]
1337
0
A000149
a(n) = floor(e^n).
[ "1", "2", "7", "20", "54", "148", "403", "1096", "2980", "8103", "22026", "59874", "162754", "442413", "1202604", "3269017", "8886110", "24154952", "65659969" ]
178482300
1
A000150
Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.
[ "0", "0", "1", "2", "7", "20", "66", "212", "715", "2424", "8398", "29372", "104006", "371384", "1337220", "4847208", "17678835", "64821680", "238819350" ]
883629164
1
A000151
Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.
[ "1", "2", "7", "26", "107", "458", "2058", "9498", "44947", "216598", "1059952", "5251806", "26297238", "132856766", "676398395", "3466799104", "17873508798", "92630098886", "482292684506" ]
2521610175006
0
A000152
Number of ways of writing n as a sum of 16 squares.
[ "1", "32", "480", "4480", "29152", "140736", "525952", "1580800", "3994080", "8945824", "18626112", "36714624", "67978880", "118156480", "197120256", "321692928", "509145568", "772845120", "1143441760" ]
1681379200
1
A000153
a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.
[ "0", "1", "2", "7", "32", "181", "1214", "9403", "82508", "808393", "8743994", "103459471", "1328953592", "18414450877", "273749755382", "4345634192131", "73362643649444", "1312349454922513", "24796092486996338" ]
493435697986613143
1
A000154
Erroneous version of A003713.
[ "1", "1", "2", "7", "35", "228", "1834", "17382", "195866", "2487832", "35499576", "562356672", "9794156448", "186025364016", "3826961710272", "84775065603888" ]
2011929826983504
0
A000155
Nearest integer to modified Bessel function K_n(1).
[ "0", "1", "2", "7", "44", "361", "3654", "44207", "622552", "10005041", "180713290", "3624270839", "79914671748", "1921576392793", "50040900884366", "1403066801155039", "42142044935535536", "1349948504738292193" ]
45940391206037470098
0
A000156
Number of ways of writing n as a sum of 24 squares.
[ "1", "48", "1104", "16192", "170064", "1362336", "8662720", "44981376", "195082320", "721175536", "2319457632", "6631997376", "17231109824", "41469483552", "93703589760", "200343312768", "407488018512", "793229226336", "1487286966928" ]
2697825744960
1
A000157
Number of Boolean functions of n variables.
[ "1", "2", "7", "111", "308063", "100126976263592", "131867858014413288241233435594064" ]
2804519150441879896741320496043335469582478995067528608051651559815168
0
A000158
Number of partitions into non-integral powers.
[ "1", "2", "8", "19", "41", "78", "134", "218", "339", "506", "730", "1023", "1397", "1884", "2477", "3218", "4118", "5192", "6486" ]
8010
0
A000159
Coefficients of ménage hit polynomials.
[ "2", "8", "20", "152", "994", "7888", "70152", "695760", "7603266", "90758872", "1174753372", "16386899368", "245046377410", "3910358788256", "66323124297872", "1191406991067168", "22596344660865282", "451208920617687720", "9461897733571886372" ]
207894669895136763704
0
A000160
Number of partitions into non-integral powers.
[ "1", "2", "8", "21", "48", "99", "186", "330", "556", "895", "1397", "2107", "3097", "4459", "6264", "8644", "11760", "15742", "20790" ]
27128
0
A000161
Number of partitions of n into 2 squares.
[ "1", "1", "1", "0", "1", "1", "0", "0", "1", "1", "1", "0", "0", "1", "0", "0", "1", "1", "1" ]
0
1
A000162
Number of 3-dimensional polyominoes (or polycubes) with n cells.
[ "1", "1", "2", "8", "29", "166", "1023", "6922", "48311", "346543", "2522522", "18598427", "138462649", "1039496297", "7859514470", "59795121480", "457409613979", "3516009200564", "27144143923583" ]
210375361379518
0
A000163
Series-parallel numbers.
[ "2", "8", "34", "136", "538", "2080", "7970", "30224", "113874", "426520", "1590338", "5906640", "21866794", "80725280", "297292610", "1092539736", "4007553922", "14675748416", "53663197698" ]
195962433328
0
A000164
Number of partitions of n into 3 squares (allowing part zero).
[ "1", "1", "1", "1", "1", "1", "1", "0", "1", "2", "1", "1", "1", "1", "1", "0", "1", "2", "2" ]
1
0
A000165
Double factorial of even numbers: (2n)!! = 2^n*n!.
[ "1", "2", "8", "48", "384", "3840", "46080", "645120", "10321920", "185794560", "3715891200", "81749606400", "1961990553600", "51011754393600", "1428329123020800", "42849873690624000", "1371195958099968000", "46620662575398912000", "1678343852714360832000" ]
63777066403145711616000
1
A000166
Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.
[ "1", "0", "1", "2", "9", "44", "265", "1854", "14833", "133496", "1334961", "14684570", "176214841", "2290792932", "32071101049", "481066515734", "7697064251745", "130850092279664", "2355301661033953" ]
44750731559645106
1
A000167
Nearest integer to modified Bessel function K_n(2).
[ "0", "0", "0", "1", "2", "9", "49", "306", "2188", "17810", "162482", "1642635", "18231462", "220420179", "2883693795", "40592133316", "611765693528", "9828843229764", "167702100599524" ]
3028466654021205
0
A000168
a(n) = 2*3^n*(2*n)!/(n!*(n+2)!).
[ "1", "2", "9", "54", "378", "2916", "24057", "208494", "1876446", "17399772", "165297834", "1602117468", "15792300756", "157923007560", "1598970451545", "16365932856990", "169114639522230", "1762352559231660", "18504701871932430" ]
195621134074714260
1
A000169
Number of labeled rooted trees with n nodes: n^(n-1).
[ "1", "2", "9", "64", "625", "7776", "117649", "2097152", "43046721", "1000000000", "25937424601", "743008370688", "23298085122481", "793714773254144", "29192926025390625", "1152921504606846976", "48661191875666868481", "2185911559738696531968", "104127350297911241532841" ]
5242880000000000000000000
1
A000170
Number of ways of placing n nonattacking queens on an n X n board.
[ "1", "1", "0", "0", "2", "10", "4", "40", "92", "352", "724", "2680", "14200", "73712", "365596", "2279184", "14772512", "95815104", "666090624" ]
4968057848
0
A000171
Number of self-complementary graphs with n nodes.
[ "1", "0", "0", "1", "2", "0", "0", "10", "36", "0", "0", "720", "5600", "0", "0", "703760", "11220000", "0", "0" ]
9168331776
0
A000172
The Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.
[ "1", "2", "10", "56", "346", "2252", "15184", "104960", "739162", "5280932", "38165260", "278415920", "2046924400", "15148345760", "112738423360", "843126957056", "6332299624282", "47737325577620", "361077477684436" ]
2739270870994736
1
A000173
Unitary-sociable numbers (smallest member of each cycle).
[ "30", "1482", "2418", "24180", "35238", "263820", "395730", "473298", "698130", "763620", "2212026", "2233554", "172459210", "209524210", "341354790", "384121920", "525150234", "530946330", "582129630" ]
1799281330
0
A000174
Number of partitions of n into 5 squares.
[ "1", "1", "1", "1", "2", "2", "1", "1", "2", "2", "2", "2", "2", "3", "2", "1", "3", "3", "3" ]
3
0
A000175
Related to zeros of Bessel function.
[ "1", "1", "2", "11", "38", "946", "4580", "202738", "3786092", "261868876", "1992367192", "2381255244240", "21411255538848", "2902625722978656", "451716954504285504", "319933105641374465472" ]
3761845343198709705600
0
A000176
Generalized tangent numbers d_(n,2).
[ "2", "11", "46", "128", "272", "522", "904", "1408", "2160", "3154", "4306", "5888", "7888", "10012", "12888", "16384", "19680", "24354", "29866" ]
34816
0
A000177
Number of partitions of n into 6 squares.
[ "1", "1", "1", "1", "2", "2", "2", "1", "2", "3", "2", "2", "3", "3", "3", "2", "3", "4", "4" ]
3
0
A000178
Superfactorials: product of first n factorials.
[ "1", "1", "2", "12", "288", "34560", "24883200", "125411328000", "5056584744960000", "1834933472251084800000", "6658606584104736522240000000", "265790267296391946810949632000000000", "127313963299399416749559771247411200000000000" ]
792786697595796795607377086400871488552960000000000000
1
A000179
Ménage numbers: a(0) = 1, a(1) = -1, and for n >= 2, a(n) = number of permutations s of [0, ..., n-1] such that s(i) != i and s(i) != i+1 (mod n) for all i.
[ "1", "-1", "0", "1", "2", "13", "80", "579", "4738", "43387", "439792", "4890741", "59216642", "775596313", "10927434464", "164806435783", "2649391469058", "45226435601207", "817056406224416" ]
15574618910994665
1
A000180
Expansion of E.g.f. exp(-x)/(1-3x).
[ "1", "2", "13", "116", "1393", "20894", "376093", "7897952", "189550849", "5117872922", "153536187661", "5066694192812", "182400990941233", "7113638646708086", "298772823161739613", "13444777042278282584", "645349298029357564033" ]
32912814199497235765682
1
A000181
Coefficients of ménage hit polynomials.
[ "2", "15", "60", "469", "3660", "32958", "328920", "3614490", "43341822", "563144725", "7880897892", "118177520295", "1890389939000", "32130521850972", "578260307815920", "10985555094348948", "219687969344126490", "4613039009310624795", "101479234383619208204" ]
2333872309936442446905
0
A000182
Tangent (or "Zag") numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).
[ "1", "2", "16", "272", "7936", "353792", "22368256", "1903757312", "209865342976", "29088885112832", "4951498053124096", "1015423886506852352", "246921480190207983616", "70251601603943959887872", "23119184187809597841473536", "8713962757125169296170811392" ]
3729407703720529571097509625856
1
A000183
Number of discordant permutations of length n.
[ "0", "0", "0", "1", "2", "20", "144", "1265", "12072", "126565", "1445100", "17875140", "238282730", "3407118041", "52034548064", "845569542593", "14570246018686", "265397214435860", "5095853023109484" ]
102877234050493609
1
A000184
Number of genus 0 rooted maps with 3 faces with n vertices.
[ "2", "22", "164", "1030", "5868", "31388", "160648", "795846", "3845020", "18211380", "84876152", "390331292", "1775032504", "7995075960", "35715205136", "158401506118", "698102372988", "3059470021316", "13341467466520" ]
57918065919924
0
A000185
Coefficients of ménage hit polynomials.
[ "2", "24", "140", "1232", "11268", "115056", "1284360", "15596208", "204710454", "2888897032", "43625578836", "702025263328", "11993721979336", "216822550325472", "4135337882588880", "82986434235959712", "1747976804189353962", "38559791049947726328" ]
889047923669760546140
0
A000186
Number of 3 X n Latin rectangles in which the first row is in order.
[ "1", "0", "0", "2", "24", "552", "21280", "1073760", "70299264", "5792853248", "587159944704", "71822743499520", "10435273503677440", "1776780700509416448", "350461958856515690496", "79284041282622163140608", "20392765404792755583221760", "5917934230798104348783083520" ]
1924427226324694427836833857536
1
A000187
Generalized Euler numbers, c(5,n).
[ "2", "30", "3522", "1066590", "604935042", "551609685150", "737740947722562", "1360427147514751710", "3308161927353377294082", "10256718523496425979562270", "39490468691102039103925777602", "184856411587530526077816051412830" ]
1033888847501229495999134528615701122
1
A000188
(1) Number of solutions to x^2 == 0 (mod n). (2) Also square root of largest square dividing n. (3) Also max_{ d divides n } gcd(d, n/d).
[ "1", "1", "1", "2", "1", "1", "1", "2", "3", "1", "1", "2", "1", "1", "1", "4", "1", "3", "1" ]
2
1
A000189
Number of solutions to x^3 == 0 (mod n).
[ "1", "1", "1", "2", "1", "1", "1", "4", "3", "1", "1", "2", "1", "1", "1", "4", "1", "3", "1" ]
2
1
A000190
Number of solutions to x^4 == 0 (mod n).
[ "1", "1", "1", "2", "1", "1", "1", "4", "3", "1", "1", "2", "1", "1", "1", "8", "1", "3", "1" ]
2
1
A000191
Generalized tangent numbers d(3, n).
[ "2", "46", "3362", "515086", "135274562", "54276473326", "30884386347362", "23657073914466766", "23471059057478981762", "29279357851856595135406", "44855282210826271011257762", "82787899853638102222862479246", "181184428895772987376073015175362" ]
463938847087789978515380344866258286
1
A000192
Generalized Euler numbers c(6,n).
[ "2", "46", "7970", "3487246", "2849229890", "3741386059246", "7205584123783010", "19133892392367261646", "67000387673723462963330", "299131045427247559446422446", "1658470810032820740402966226850", "11179247066648898992009055586869646" ]
90035623994788132387893239340761189570
1
A000193
Nearest integer to log n.
[ "0", "1", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3" ]
3
1
A000194
n appears 2n times, for n >= 1; also nearest integer to square root of n.
[ "0", "1", "1", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "4", "4", "4", "4", "4", "4" ]
4
1
A000195
a(n) = floor(log(n)).
[ "0", "0", "1", "1", "1", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2" ]
2
1
A000196
Integer part of square root of n. Or, number of positive squares <= n. Or, n appears 2n+1 times.
[ "0", "1", "1", "1", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "4", "4", "4" ]
4
1
A000198
Largest order of automorphism group of a tournament with n nodes.
[ "1", "1", "3", "3", "5", "9", "21", "21", "81", "81", "81", "243", "243", "441", "1215", "1701", "1701", "6561", "6561" ]
6561
0
A000199
Coefficient of q^(2n-1) in the series expansion of Ramanujan's mock theta function f(q).
[ "1", "3", "3", "7", "6", "12", "13", "20", "21", "34", "36", "51", "58", "78", "89", "118", "131", "171", "197" ]
245
0
A000200
Number of bicentered hydrocarbons with n atoms.
[ "0", "0", "1", "0", "1", "1", "3", "3", "9", "15", "38", "73", "174", "380", "915", "2124", "5134", "12281", "30010" ]
73401
0
A000201
Lower Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi), where phi = (1+sqrt(5))/2 = A001622.
[ "1", "3", "4", "6", "8", "9", "11", "12", "14", "16", "17", "19", "21", "22", "24", "25", "27", "29", "30" ]
32
1