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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
A000203
a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).
[ "1", "3", "4", "7", "6", "12", "8", "15", "13", "18", "12", "28", "14", "24", "24", "31", "18", "39", "20" ]
42
1
A000204
Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.
[ "1", "3", "4", "7", "11", "18", "29", "47", "76", "123", "199", "322", "521", "843", "1364", "2207", "3571", "5778", "9349" ]
15127
1
A000205
Number of positive integers <= 2^n of form x^2 + 3 y^2.
[ "1", "1", "3", "4", "8", "14", "25", "45", "82", "151", "282", "531", "1003", "1907", "3645", "6993", "13456", "25978", "50248" ]
97446
0
A000206
Even sequences with period 2n.
[ "1", "1", "3", "4", "12", "22", "71", "181", "618", "1957", "6966", "24367", "89010", "324766", "1204815", "4482400", "16802826", "63195016", "238711285" ]
904338163
1
A000207
Number of inequivalent ways of dissecting a regular (n+2)-gon into n triangles by n-1 non-intersecting diagonals under rotations and reflections; also the number of (unlabeled) maximal outerplanar graphs on n+2 vertices.
[ "1", "1", "1", "3", "4", "12", "27", "82", "228", "733", "2282", "7528", "24834", "83898", "285357", "983244", "3412420", "11944614", "42080170" ]
149197152
1
A000208
Number of even sequences with period 2n.
[ "1", "1", "3", "4", "12", "28", "94", "298", "1044", "3658", "13164", "47710", "174948", "645436", "2397342", "8948416", "33556500", "126324496", "477225962" ]
1808414182
1
A000209
Nearest integer to tan n.
[ "0", "2", "-2", "0", "1", "-3", "0", "1", "-7", "0", "1", "-226", "-1", "0", "7", "-1", "0", "3", "-1" ]
0
0
A000210
A Beatty sequence: floor(n*(e-1)).
[ "1", "3", "5", "6", "8", "10", "12", "13", "15", "17", "18", "20", "22", "24", "25", "27", "29", "30", "32" ]
34
0
A000211
a(n) = a(n-1) + a(n-2) - 2, a(0) = 4, a(1) = 3.
[ "4", "3", "5", "6", "9", "13", "20", "31", "49", "78", "125", "201", "324", "523", "845", "1366", "2209", "3573", "5780" ]
9351
1
A000212
a(n) = floor(n^2/3).
[ "0", "0", "1", "3", "5", "8", "12", "16", "21", "27", "33", "40", "48", "56", "65", "75", "85", "96", "108" ]
120
1
A000213
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=a(2)=1.
[ "1", "1", "1", "3", "5", "9", "17", "31", "57", "105", "193", "355", "653", "1201", "2209", "4063", "7473", "13745", "25281" ]
46499
1
A000214
Number of equivalence classes of Boolean functions of n variables under action of AG(n,2).
[ "3", "5", "10", "32", "382", "15768919", "16224999167506438730294" ]
84575066435667906978109556031081616704183639810103015118
1
A000215
Fermat numbers: a(n) = 2^(2^n) + 1.
[ "3", "5", "17", "257", "65537", "4294967297", "18446744073709551617", "340282366920938463463374607431768211457" ]
115792089237316195423570985008687907853269984665640564039457584007913129639937
1
A000216
Take sum of squares of digits of previous term, starting with 2.
[ "2", "4", "16", "37", "58", "89", "145", "42", "20", "4", "16", "37", "58", "89", "145", "42", "20", "4", "16" ]
37
1
A000217
Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.
[ "0", "1", "3", "6", "10", "15", "21", "28", "36", "45", "55", "66", "78", "91", "105", "120", "136", "153", "171" ]
190
1
A000218
Take sum of squares of digits of previous term; start with 3.
[ "3", "9", "81", "65", "61", "37", "58", "89", "145", "42", "20", "4", "16", "37", "58", "89", "145", "42", "20" ]
4
1
A000219
Number of plane partitions (or planar partitions) of n.
[ "1", "1", "3", "6", "13", "24", "48", "86", "160", "282", "500", "859", "1479", "2485", "4167", "6879", "11297", "18334", "29601" ]
47330
1
A000220
Number of asymmetric trees with n nodes (also called identity trees).
[ "1", "0", "0", "0", "0", "0", "1", "1", "3", "6", "15", "29", "67", "139", "310", "667", "1480", "3244", "7241" ]
16104
0
A000221
Take sum of squares of digits of previous term; start with 5.
[ "5", "25", "29", "85", "89", "145", "42", "20", "4", "16", "37", "58", "89", "145", "42", "20", "4", "16", "37" ]
58
1
A000222
Coefficients of ménage hit polynomials.
[ "0", "0", "1", "3", "6", "38", "213", "1479", "11692", "104364", "1036809", "11344859", "135548466", "1755739218", "24504637741", "366596136399", "5852040379224", "99283915922264", "1783921946910417" ]
33840669046326579
0
A000223
Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives (nearest integer to, I believe) P(A000092(n)).
[ "3", "7", "10", "19", "32", "34", "37", "51", "81", "119", "122", "134", "157", "160", "161", "174", "221", "252", "254" ]
294
0
A000224
Number of squares mod n.
[ "1", "2", "2", "2", "3", "4", "4", "3", "4", "6", "6", "4", "7", "8", "6", "4", "9", "8", "10" ]
6
1
A000225
a(n) = 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)
[ "0", "1", "3", "7", "15", "31", "63", "127", "255", "511", "1023", "2047", "4095", "8191", "16383", "32767", "65535", "131071", "262143" ]
524287
1
A000226
Number of n-node unlabeled connected graphs with one cycle of length 3.
[ "1", "1", "3", "7", "18", "44", "117", "299", "793", "2095", "5607", "15047", "40708", "110499", "301541", "825784", "2270211", "6260800", "17319689" ]
48042494
0
A000227
Nearest integer to e^n.
[ "1", "3", "7", "20", "55", "148", "403", "1097", "2981", "8103", "22026", "59874", "162755", "442413", "1202604", "3269017", "8886111", "24154953", "65659969" ]
178482301
1
A000228
Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.
[ "1", "1", "3", "7", "22", "82", "333", "1448", "6572", "30490", "143552", "683101", "3274826", "15796897", "76581875", "372868101", "1822236628", "8934910362", "43939164263" ]
216651036012
0
A000229
a(n) is the least number m such that the n-th prime is the least quadratic nonresidue modulo m.
[ "3", "7", "23", "71", "311", "479", "1559", "5711", "10559", "18191", "31391", "422231", "701399", "366791", "3818929", "9257329", "22000801", "36415991", "48473881" ]
175244281
0
A000230
a(0)=2; for n>=1, a(n) = smallest prime p such that there is a gap of exactly 2n between p and next prime, or -1 if no such prime exists.
[ "2", "3", "7", "23", "89", "139", "199", "113", "1831", "523", "887", "1129", "1669", "2477", "2971", "4297", "5591", "1327", "9551" ]
30593
0
A000231
Number of inequivalent Boolean functions of n variables under action of complementing group.
[ "2", "3", "7", "46", "4336", "134281216", "288230380379570176", "2658455991569831764110243006194384896" ]
452312848583266388373324160190187140390789016525312000869601987902398529536
1
A000232
Construct a triangle as in A036262. Sequence is one less than the position of the first number larger than 2 in the n-th row (n-th difference).
[ "3", "8", "14", "14", "25", "24", "23", "22", "25", "59", "98", "97", "98", "97", "174", "176", "176", "176", "176" ]
291
0
A000233
Generalized class numbers c_(n,1).
[ "1", "3", "8", "16", "30", "46", "64", "96", "126", "158", "216", "256", "302", "396", "448", "512", "636", "702", "792" ]
960
1
A000234
Partitions into non-integral powers (see Comments for precise definition).
[ "1", "3", "8", "18", "37", "72", "136", "251", "445", "770", "1312", "2202", "3632", "5908", "9501", "15111", "23781", "37083", "57293" ]
87813
0
A000235
Number of n-node rooted trees of height 3.
[ "0", "0", "0", "1", "3", "8", "18", "38", "76", "147", "277", "509", "924", "1648", "2912", "5088", "8823", "15170", "25935" ]
44042
0
A000236
Maximum m such that there are no two adjacent elements belonging to the same n-th power residue class modulo some prime p in the sequence 1,2,...,m (equivalently, there is no n-th power residue modulo p in the sequence 1/2,2/3,...,(m-1)/m).
[ "3", "8", "20", "44", "80", "343", "288", "608", "1023", "2848", "4095", "40959", "16383", "32768", "11375", "655360", "262143", "3670016", "1048575" ]
2097151
0
A000237
Number of mixed Husimi trees with n nodes; or rooted polygonal cacti with bridges.
[ "0", "1", "1", "3", "8", "26", "84", "297", "1066", "3976", "15093", "58426", "229189", "910127", "3649165", "14756491", "60103220", "246357081", "1015406251" ]
4205873378
1
A000238
Number of oriented trees with n nodes.
[ "1", "1", "3", "8", "27", "91", "350", "1376", "5743", "24635", "108968", "492180", "2266502", "10598452", "50235931", "240872654", "1166732814", "5702001435", "28088787314" ]
139354922608
0
A000239
One-half of number of permutations of [n] with exactly one run of adjacent symbols differing by 1.
[ "1", "1", "3", "8", "28", "143", "933", "7150", "62310", "607445", "6545935", "77232740", "989893248", "13692587323", "203271723033", "3223180454138", "54362625941818", "971708196867905", "18347779304380995" ]
364911199401630640
0
A000240
Rencontres numbers: number of permutations of [n] with exactly one fixed point.
[ "1", "0", "3", "8", "45", "264", "1855", "14832", "133497", "1334960", "14684571", "176214840", "2290792933", "32071101048", "481066515735", "7697064251744", "130850092279665", "2355301661033952", "44750731559645107" ]
895014631192902120
1
A000241
Crossing number of complete graph with n nodes.
[ "0", "0", "0", "0", "0", "1", "3", "9", "18", "36", "60", "100", "150", "225" ]
315
0
A000242
3rd power of rooted tree enumerator; number of linear forests of 3 rooted trees.
[ "1", "3", "9", "25", "69", "186", "503", "1353", "3651", "9865", "26748", "72729", "198447", "543159", "1491402", "4107152", "11342826", "31408719", "87189987" ]
242603970
1
A000243
Number of trees with n nodes, 2 of which are labeled.
[ "1", "3", "9", "26", "75", "214", "612", "1747", "4995", "14294", "40967", "117560", "337830", "972027", "2800210", "8075889", "23315775", "67380458", "194901273" ]
564239262
1
A000244
Powers of 3: a(n) = 3^n.
[ "1", "3", "9", "27", "81", "243", "729", "2187", "6561", "19683", "59049", "177147", "531441", "1594323", "4782969", "14348907", "43046721", "129140163", "387420489" ]
1162261467
1
A000245
a(n) = 3*(2*n)!/((n+2)!*(n-1)!).
[ "0", "1", "3", "9", "28", "90", "297", "1001", "3432", "11934", "41990", "149226", "534888", "1931540", "7020405", "25662825", "94287120", "347993910", "1289624490" ]
4796857230
1
A000246
Number of permutations in the symmetric group S_n that have odd order.
[ "1", "1", "1", "3", "9", "45", "225", "1575", "11025", "99225", "893025", "9823275", "108056025", "1404728325", "18261468225", "273922023375", "4108830350625", "69850115960625", "1187451971330625" ]
22561587455281875
1
A000247
a(n) = 2^n - n - 2.
[ "0", "3", "10", "25", "56", "119", "246", "501", "1012", "2035", "4082", "8177", "16368", "32751", "65518", "131053", "262124", "524267", "1048554" ]
2097129
1
A000248
Expansion of e.g.f. exp(x*exp(x)).
[ "1", "1", "3", "10", "41", "196", "1057", "6322", "41393", "293608", "2237921", "18210094", "157329097", "1436630092", "13810863809", "139305550066", "1469959371233", "16184586405328", "185504221191745" ]
2208841954063318
1
A000249
Nearest integer to modified Bessel function K_n(5).
[ "0", "0", "0", "0", "0", "0", "0", "0", "1", "3", "10", "42", "193", "966", "5215", "30170", "186234", "1222065", "8496274" ]
62395234
0
A000250
Number of symmetric reflexive relations on n nodes: (1/2)*A000666.
[ "1", "3", "10", "45", "272", "2548", "39632", "1104306", "56871880", "5463113568", "978181717680", "326167542296048", "202701136710498400", "235284321080559981952", "511531711735594715527360", "2089424601541011618029114896", "16084004145036771186002041099712", "234026948449058790311618594954430848" ]
6454432593140577452393525511509194184320
0
A000251
Number of trees of diameter 6.
[ "1", "3", "11", "29", "74", "167", "367", "755", "1515", "2931", "5551", "10263", "18677", "33409", "59024", "102984", "177915", "304458", "516939" ]
871180
0
A000252
Number of invertible 2 X 2 matrices mod n.
[ "1", "6", "48", "96", "480", "288", "2016", "1536", "3888", "2880", "13200", "4608", "26208", "12096", "23040", "24576", "78336", "23328", "123120" ]
46080
1
A000253
a(n) = 2*a(n-1) - a(n-2) + a(n-3) + 2^(n-1).
[ "0", "1", "4", "11", "27", "63", "142", "312", "673", "1432", "3015", "6295", "13055", "26926", "55284", "113081", "230572", "468883", "951347" ]
1926527
0
A000254
Unsigned Stirling numbers of first kind, s(n+1,2): a(n+1) = (n+1)*a(n) + n!.
[ "0", "1", "3", "11", "50", "274", "1764", "13068", "109584", "1026576", "10628640", "120543840", "1486442880", "19802759040", "283465647360", "4339163001600", "70734282393600", "1223405590579200", "22376988058521600" ]
431565146817638400
1
A000255
a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
[ "1", "1", "3", "11", "53", "309", "2119", "16687", "148329", "1468457", "16019531", "190899411", "2467007773", "34361893981", "513137616783", "8178130767479", "138547156531409", "2486151753313617", "47106033220679059" ]
939765362752547227
1
A000256
Number of simple triangulations of the plane with n nodes.
[ "1", "1", "0", "1", "3", "12", "52", "241", "1173", "5929", "30880", "164796", "897380", "4970296", "27930828", "158935761", "914325657", "5310702819", "31110146416" ]
183634501753
0
A000257
Number of rooted bicubic maps: a(n) = (8*n-4)*a(n-1)/(n+2) for n >= 2, a(0) = a(1) = 1.
[ "1", "1", "3", "12", "56", "288", "1584", "9152", "54912", "339456", "2149888", "13891584", "91287552", "608583680", "4107939840", "28030648320", "193100021760", "1341536993280", "9390758952960" ]
66182491668480
1
A000258
Expansion of e.g.f. exp(exp(exp(x)-1)-1).
[ "1", "1", "3", "12", "60", "358", "2471", "19302", "167894", "1606137", "16733779", "188378402", "2276423485", "29367807524", "402577243425", "5840190914957", "89345001017415", "1436904211547895", "24227076487779802" ]
427187837301557598
1
A000259
Number of certain rooted planar maps.
[ "1", "3", "13", "63", "326", "1761", "9808", "55895", "324301", "1908878", "11369744", "68395917", "414927215", "2535523154", "15592255913", "96419104103", "599176447614", "3739845108057", "23435007764606" ]
147374772979438
0
A000260
Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.
[ "1", "1", "3", "13", "68", "399", "2530", "16965", "118668", "857956", "6369883", "48336171", "373537388", "2931682810", "23317105140", "187606350645", "1524813969276", "12504654858828", "103367824774012" ]
860593023907540
1
A000261
a(n) = n*a(n-1) + (n-3)*a(n-2), with a(1) = 0, a(2) = 1.
[ "0", "1", "3", "13", "71", "465", "3539", "30637", "296967", "3184129", "37401155", "477471021", "6581134823", "97388068753", "1539794649171", "25902759280525", "461904032857319", "8702813980639617", "172743930157869827" ]
3602826440828270029
0
A000262
Number of "sets of lists": number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset.
[ "1", "1", "3", "13", "73", "501", "4051", "37633", "394353", "4596553", "58941091", "824073141", "12470162233", "202976401213", "3535017524403", "65573803186921", "1290434218669921", "26846616451246353", "588633468315403843" ]
13564373693588558173
1
A000263
Number of partitions into non-integral powers.
[ "3", "14", "39", "91", "173", "307", "502", "779", "1150", "1651", "2280", "3090", "4090", "5313", "6787", "8564", "10643", "13103", "15948" ]
19235
0
A000264
Number of 3-edge-connected rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle.
[ "1", "1", "3", "14", "80", "518", "3647", "27274", "213480", "1731652", "14455408", "123552488", "1077096124", "9548805240", "85884971043", "782242251522", "7203683481720", "66989439309452", "628399635777936" ]
5940930064989720
0
A000265
Remove all factors of 2 from n; or largest odd divisor of n; or odd part of n.
[ "1", "1", "3", "1", "5", "3", "7", "1", "9", "5", "11", "3", "13", "7", "15", "1", "17", "9", "19" ]
5
1
A000266
Expansion of e.g.f. exp(-x^2/2) / (1-x).
[ "1", "1", "1", "3", "15", "75", "435", "3045", "24465", "220185", "2200905", "24209955", "290529855", "3776888115", "52876298475", "793144477125", "12690313661025", "215735332237425", "3883235945814225" ]
73781482970470275
0
A000267
Integer part of square root of 4n+1.
[ "1", "2", "3", "3", "4", "4", "5", "5", "5", "6", "6", "6", "7", "7", "7", "7", "8", "8", "8" ]
8
1
A000268
E.g.f.: -log(1+log(1+log(1-x))).
[ "1", "3", "15", "105", "947", "10472", "137337", "2085605", "36017472", "697407850", "14969626900", "352877606716", "9064191508018", "252024567201300", "7542036496650006", "241721880399970938", "8261159383595659128", "299916384730043070880", "11526945327529620432872" ]
467583770376898192016104
0
A000269
Number of trees with n nodes, 3 of which are labeled.
[ "3", "16", "67", "251", "888", "3023", "10038", "32722", "105228", "334836", "1056611", "3311784", "10322791", "32026810", "98974177", "304835956", "936147219", "2867586542", "8764280567" ]
26733395986
1
A000270
For n >= 2, a(n) = b(n+1)+b(n)+b(n-1), where the b(i) are the ménage numbers A000179; a(0)=a(1)=1.
[ "1", "1", "0", "3", "16", "95", "672", "5397", "48704", "487917", "5373920", "64547175", "839703696", "11762247419", "176509466560", "2825125339305", "48040633506048", "864932233294681", "16436901752820288" ]
328791893988472843
0
A000271
Sums of ménage numbers.
[ "1", "0", "0", "1", "3", "16", "96", "675", "5413", "48800", "488592", "5379333", "64595975", "840192288", "11767626752", "176574062535", "2825965531593", "48052401132800", "865108807357216" ]
16439727718351881
1
A000272
Number of trees on n labeled nodes: n^(n-2) with a(0)=1.
[ "1", "1", "1", "3", "16", "125", "1296", "16807", "262144", "4782969", "100000000", "2357947691", "61917364224", "1792160394037", "56693912375296", "1946195068359375", "72057594037927936", "2862423051509815793", "121439531096594251776" ]
5480386857784802185939
1
A000273
Number of unlabeled simple digraphs with n nodes.
[ "1", "1", "3", "16", "218", "9608", "1540944", "882033440", "1793359192848", "13027956824399552", "341260431952972580352", "32522909385055886111197440", "11366745430825400574433894004224", "14669085692712929869037096075316220928" ]
70315656615234999521385506555979904091217920
0
A000274
Number of permutations of length n with 2 consecutive ascending pairs.
[ "0", "0", "1", "3", "18", "110", "795", "6489", "59332", "600732", "6674805", "80765135", "1057289046", "14890154058", "224497707343", "3607998868005", "61576514013960", "1112225784377144", "21197714949305577" ]
425131949816628507
1
A000275
Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden.
[ "1", "1", "3", "19", "211", "3651", "90921", "3081513", "136407699", "7642177651", "528579161353", "44237263696473", "4405990782649369", "515018848029036937", "69818743428262376523", "10865441556038181291819", "1923889742567310611949459", "384565973956329859109177427" ]
86180438505835750284241676121
0
A000276
Associated Stirling numbers.
[ "3", "20", "130", "924", "7308", "64224", "623376", "6636960", "76998240", "967524480", "13096736640", "190060335360", "2944310342400", "48503818137600", "846795372595200", "15618926924697600", "303517672703078400", "6198400928176128000", "132720966600284160000" ]
2973385109386137600000
0
A000277
3*n - 2*floor(sqrt(4*n+5)) + 5.
[ "1", "2", "5", "6", "9", "10", "13", "16", "17", "20", "23", "24", "27", "30", "33", "34", "37", "40", "43" ]
44
1
A000278
a(n) = a(n-1) + a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.
[ "0", "1", "1", "2", "3", "7", "16", "65", "321", "4546", "107587", "20773703", "11595736272", "431558332068481", "134461531248108526465", "186242594112190847520182173826", "18079903385772308300945867582153787570051" ]
34686303861638264961101080464895364211215702792496667048327
0
A000279
Card matching: coefficients B[n,1] of t in the reduced hit polynomial A[n,n,n](t).
[ "3", "24", "216", "1824", "15150", "124416", "1014888", "8241792", "66724398", "538990800", "4346692680", "35009591040", "281699380560", "2264868936960", "18198009147600", "146142982814208", "1173123636533454", "9413509300965936", "75513633110271264" ]
605598295606296000
0
A000280
a(n) = a(n-1) + a(n-2)^3.
[ "0", "1", "1", "2", "3", "11", "38", "1369", "56241", "2565782650", "177895665388171", "16891164530321501264425013171" ]
5629840598310484749297545401724540333537382
0
A000281
Expansion of cos(x)/cos(2x).
[ "1", "3", "57", "2763", "250737", "36581523", "7828053417", "2309644635483", "898621108880097", "445777636063460643", "274613643571568682777", "205676334188681975553003", "184053312545818735778213457" ]
193944394596325636374396208563
1
A000282
Finite automata.
[ "3", "70", "3783", "338475", "40565585", "6061961733", "1083852977811", "225615988054171", "53595807366038234", "14308700593468127485", "4241390625289880226714", "1382214286200071777573643", "491197439886557439295166226", "189044982636675290371386547592", "78334771617452038208125184627931" ]
34771576300926271400714044414858372
0
A000283
a(n) = a(n-1)^2 + a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.
[ "0", "1", "1", "2", "5", "29", "866", "750797", "563696885165", "317754178345286893212434" ]
100967717855888389973004846476977145423449281581
1
A000284
a(n) = a(n-1)^3 + a(n-2) with a(0)=0, a(1)=1.
[ "0", "1", "1", "2", "9", "731", "390617900", "59601394712394173339000731" ]
211723599072542785377729319366442939995427829921816290889198752331804918235791
1
A000285
a(0) = 1, a(1) = 4, and a(n) = a(n-1) + a(n-2) for n >= 2.
[ "1", "4", "5", "9", "14", "23", "37", "60", "97", "157", "254", "411", "665", "1076", "1741", "2817", "4558", "7375", "11933" ]
19308
1
A000286
Number of positive integers <= 2^n of form 2 x^2 + 5 y^2.
[ "0", "1", "1", "4", "5", "11", "20", "36", "65", "119", "218", "412", "770", "1466", "2784", "5322", "10226", "19691", "38048" ]
73665
0
A000287
Number of rooted polyhedral graphs with n edges.
[ "1", "0", "4", "6", "24", "66", "214", "676", "2209", "7296", "24460", "82926", "284068", "981882", "3421318", "12007554", "42416488", "150718770", "538421590" ]
1932856590
0
A000288
Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) with a(0) = a(1) = a(2) = a(3) = 1.
[ "1", "1", "1", "1", "4", "7", "13", "25", "49", "94", "181", "349", "673", "1297", "2500", "4819", "9289", "17905", "34513" ]
66526
1
A000289
A nonlinear recurrence: a(n) = a(n-1)^2 - 3*a(n-1) + 3 (for n>1).
[ "1", "4", "7", "31", "871", "756031", "571580604871", "326704387862983487112031", "106735757048926752040856495274871386126283608871" ]
11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068031
1
A000290
The squares: a(n) = n^2.
[ "0", "1", "4", "9", "16", "25", "36", "49", "64", "81", "100", "121", "144", "169", "196", "225", "256", "289", "324" ]
361
1
A000291
Number of bipartite partitions of n white objects and 2 black ones.
[ "2", "4", "9", "16", "29", "47", "77", "118", "181", "267", "392", "560", "797", "1111", "1541", "2106", "2863", "3846", "5142" ]
6808
0
A000292
Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.
[ "0", "1", "4", "10", "20", "35", "56", "84", "120", "165", "220", "286", "364", "455", "560", "680", "816", "969", "1140" ]
1330
1
A000293
a(n) = number of solid (i.e., three-dimensional) partitions of n.
[ "1", "1", "4", "10", "26", "59", "140", "307", "684", "1464", "3122", "6500", "13426", "27248", "54804", "108802", "214071", "416849", "805124" ]
1541637
0
A000294
Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).
[ "1", "1", "4", "10", "26", "59", "141", "310", "692", "1483", "3162", "6583", "13602", "27613", "55579", "110445", "217554", "424148", "820294" ]
1572647
1
A000295
Eulerian numbers (Euler's triangle: column k=2 of A008292, column k=1 of A173018).
[ "0", "0", "1", "4", "11", "26", "57", "120", "247", "502", "1013", "2036", "4083", "8178", "16369", "32752", "65519", "131054", "262125" ]
524268
1
A000296
Set partitions without singletons: number of partitions of an n-set into blocks of size > 1. Also number of cyclically spaced (or feasible) partitions.
[ "1", "0", "1", "1", "4", "11", "41", "162", "715", "3425", "17722", "98253", "580317", "3633280", "24011157", "166888165", "1216070380", "9264071767", "73600798037" ]
608476008122
1
A000297
a(n) = (n+1)*(n+3)*(n+8)/6.
[ "0", "4", "12", "25", "44", "70", "104", "147", "200", "264", "340", "429", "532", "650", "784", "935", "1104", "1292", "1500" ]
1729
1
A000298
Number of partitions into non-integral powers.
[ "1", "4", "12", "30", "70", "159", "339", "706", "1436", "2853", "5551", "10622", "19975", "37043", "67811", "122561", "219090", "387578", "678977" ]
1178769
0
A000299
Number of n-node rooted trees of height 4.
[ "0", "0", "0", "0", "1", "4", "13", "36", "93", "225", "528", "1198", "2666", "5815", "12517", "26587", "55933", "116564", "241151" ]
495417
0
A000300
4th power of rooted tree enumerator: linear forests of 4 rooted trees.
[ "1", "4", "14", "44", "133", "388", "1116", "3168", "8938", "25100", "70334", "196824", "550656", "1540832", "4314190", "12089368", "33911543", "95228760", "267727154" ]
753579420
0
A000301
a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).
[ "1", "2", "2", "4", "8", "32", "256", "8192", "2097152", "17179869184", "36028797018963968", "618970019642690137449562112", "22300745198530623141535718272648361505980416" ]
13803492693581127574869511724554050904902217944340773110325048447598592
1
A000302
Powers of 4: a(n) = 4^n.
[ "1", "4", "16", "64", "256", "1024", "4096", "16384", "65536", "262144", "1048576", "4194304", "16777216", "67108864", "268435456", "1073741824", "4294967296", "17179869184", "68719476736" ]
274877906944
1