christopher/oeis · Datasets at Hugging Face

a-number stringlengths 7 7	sequence sequencelengths 1 377	description stringlengths 3 852
A001001	[ "1", "7", "13", "35", "31", "91", "57", "155", "130", "217", "133", "455", "183", "399", "403", "651", "307", "910", "381", "1085", "741", "931", "553", "2015", "806", "1281", "1210", "1995", "871", "2821", "993", "2667", "1729", "2149", "1767", "4550", "1407", "2667", "2379", "4805", "1723", "5187", "1893", "4655", "4030", "3871", "2257", "8463", "2850", "5642", "3991", "6405", "2863" ]	Number of sublattices of index n in generic 3-dimensional lattice.
A001002	[ "1", "1", "3", "10", "38", "154", "654", "2871", "12925", "59345", "276835", "1308320", "6250832", "30142360", "146510216", "717061938", "3530808798", "17478955570", "86941210950", "434299921440", "2177832612120", "10959042823020", "55322023332420", "280080119609550", "1421744205767418", "7234759677699954" ]	Number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.
A001003	[ "1", "1", "3", "11", "45", "197", "903", "4279", "20793", "103049", "518859", "2646723", "13648869", "71039373", "372693519", "1968801519", "10463578353", "55909013009", "300159426963", "1618362158587", "8759309660445", "47574827600981", "259215937709463", "1416461675464871" ]	Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.
A001004	[ "1", "1", "2", "3", "9", "20", "75", "262", "1117", "4783", "21971", "102249", "489077", "2370142", "11654465", "57916324", "290693391", "1471341341", "7504177738", "38532692207", "199076194985", "1034236705992", "5400337050086", "28329240333758", "149244907249629" ]	Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.
A001005	[ "1", "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" ]	Number of ways of partitioning n points on a circle into subsets only of sizes 2 and 3.
A001006	[ "1", "1", "2", "4", "9", "21", "51", "127", "323", "835", "2188", "5798", "15511", "41835", "113634", "310572", "853467", "2356779", "6536382", "18199284", "50852019", "142547559", "400763223", "1129760415", "3192727797", "9043402501", "25669818476", "73007772802", "208023278209", "593742784829" ]	Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.
A001007	[ "1", "2", "15", "42", "421", "1331", "15119", "618765", "2155578", "98032875", "1290154807", "4682196239", "63117678751", "3186252107917", "164886529617695", "616630679090258", "32763760653353135", "467443761039641135", "1768227793278781667", "96699391743949360451", "1402535447576150395335" ]	a(n) = ( Sum C(p,i); i=1,...,floor(2p/3) ) / p^2, where p = prime(n).
A001008	[ "1", "3", "11", "25", "137", "49", "363", "761", "7129", "7381", "83711", "86021", "1145993", "1171733", "1195757", "2436559", "42142223", "14274301", "275295799", "55835135", "18858053", "19093197", "444316699", "1347822955", "34052522467", "34395742267", "312536252003", "315404588903", "9227046511387" ]	Numerators of harmonic numbers H(n) = Sum_{i=1..n} 1/i.
A001009	[ "1", "1", "1", "1", "1", "1", "1", "3", "4", "4", "1", "11", "46", "56", "56", "1", "53", "1064", "6552", "9408", "9408", "1", "309", "35792", "1293216", "11270400", "16942080", "16942080", "1", "2119", "1673792", "420909504", "27206658048", "335390189568", "535281401856", "535281401856", "1", "16687", "103443808" ]	Triangle giving number L(n,k) of normalized k X n Latin rectangles.
A001010	[ "1", "2", "2", "4", "6", "8", "18", "20", "56", "48", "178", "132", "574", "348", "1870", "1008", "6144", "2812", "20314", "8420", "67534", "24396", "225472", "74756", "755672", "222556", "2540406", "693692", "8564622", "2107748", "28941258", "6656376", "98011464", "20548932" ]	Number of symmetric foldings of a strip of n stamps.
A001011	[ "1", "1", "2", "5", "14", "38", "120", "353", "1148", "3527", "11622", "36627", "121622", "389560", "1301140", "4215748", "14146335", "46235800", "155741571", "512559195", "1732007938", "5732533570", "19423092113", "64590165281", "219349187968", "732358098471", "2492051377341", "8349072895553", "28459491475593" ]	Number of ways to fold a strip of n blank stamps.
A001012	[ "1", "1", "1", "1", "2", "2", "22", "563", "1676257" ]	Erroneous version of A040082.
A001013	[ "1", "2", "4", "6", "8", "12", "16", "24", "32", "36", "48", "64", "72", "96", "120", "128", "144", "192", "216", "240", "256", "288", "384", "432", "480", "512", "576", "720", "768", "864", "960", "1024", "1152", "1296", "1440", "1536", "1728", "1920", "2048", "2304", "2592", "2880", "3072", "3456", "3840", "4096", "4320", "4608", "5040", "5184", "5760" ]	Jordan-Polya numbers: products of factorial numbers A000142.
A001014	[ "0", "1", "64", "729", "4096", "15625", "46656", "117649", "262144", "531441", "1000000", "1771561", "2985984", "4826809", "7529536", "11390625", "16777216", "24137569", "34012224", "47045881", "64000000", "85766121", "113379904", "148035889", "191102976", "244140625", "308915776", "387420489", "481890304" ]	Sixth powers: a(n) = n^6.
A001015	[ "0", "1", "128", "2187", "16384", "78125", "279936", "823543", "2097152", "4782969", "10000000", "19487171", "35831808", "62748517", "105413504", "170859375", "268435456", "410338673", "612220032", "893871739", "1280000000", "1801088541", "2494357888", "3404825447", "4586471424", "6103515625", "8031810176" ]	Seventh powers: a(n) = n^7.
A001016	[ "0", "1", "256", "6561", "65536", "390625", "1679616", "5764801", "16777216", "43046721", "100000000", "214358881", "429981696", "815730721", "1475789056", "2562890625", "4294967296", "6975757441", "11019960576", "16983563041", "25600000000", "37822859361", "54875873536", "78310985281", "110075314176" ]	Eighth powers: a(n) = n^8.
A001017	[ "0", "1", "512", "19683", "262144", "1953125", "10077696", "40353607", "134217728", "387420489", "1000000000", "2357947691", "5159780352", "10604499373", "20661046784", "38443359375", "68719476736", "118587876497", "198359290368", "322687697779", "512000000000", "794280046581", "1207269217792" ]	Ninth powers: a(n) = n^9.
A001018	[ "1", "8", "64", "512", "4096", "32768", "262144", "2097152", "16777216", "134217728", "1073741824", "8589934592", "68719476736", "549755813888", "4398046511104", "35184372088832", "281474976710656", "2251799813685248", "18014398509481984", "144115188075855872", "1152921504606846976", "9223372036854775808", "73786976294838206464", "590295810358705651712", "4722366482869645213696" ]	Powers of 8: a(n) = 8^n.
A001019	[ "1", "9", "81", "729", "6561", "59049", "531441", "4782969", "43046721", "387420489", "3486784401", "31381059609", "282429536481", "2541865828329", "22876792454961", "205891132094649", "1853020188851841", "16677181699666569", "150094635296999121", "1350851717672992089", "12157665459056928801" ]	Powers of 9: a(n) = 9^n.
A001020	[ "1", "11", "121", "1331", "14641", "161051", "1771561", "19487171", "214358881", "2357947691", "25937424601", "285311670611", "3138428376721", "34522712143931", "379749833583241", "4177248169415651", "45949729863572161", "505447028499293771", "5559917313492231481", "61159090448414546291" ]	Powers of 11: a(n) = 11^n.
A001021	[ "1", "12", "144", "1728", "20736", "248832", "2985984", "35831808", "429981696", "5159780352", "61917364224", "743008370688", "8916100448256", "106993205379072", "1283918464548864", "15407021574586368", "184884258895036416", "2218611106740436992" ]	Powers of 12.
A001022	[ "1", "13", "169", "2197", "28561", "371293", "4826809", "62748517", "815730721", "10604499373", "137858491849", "1792160394037", "23298085122481", "302875106592253", "3937376385699289", "51185893014090757", "665416609183179841", "8650415919381337933", "112455406951957393129", "1461920290375446110677", "19004963774880799438801" ]	Powers of 13.
A001023	[ "1", "14", "196", "2744", "38416", "537824", "7529536", "105413504", "1475789056", "20661046784", "289254654976", "4049565169664", "56693912375296", "793714773254144", "11112006825558016", "155568095557812224", "2177953337809371136", "30491346729331195904", "426878854210636742656", "5976303958948914397184", "83668255425284801560576" ]	Powers of 14.
A001024	[ "1", "15", "225", "3375", "50625", "759375", "11390625", "170859375", "2562890625", "38443359375", "576650390625", "8649755859375", "129746337890625", "1946195068359375", "29192926025390625", "437893890380859375", "6568408355712890625", "98526125335693359375", "1477891880035400390625", "22168378200531005859375", "332525673007965087890625" ]	Powers of 15.
A001025	[ "1", "16", "256", "4096", "65536", "1048576", "16777216", "268435456", "4294967296", "68719476736", "1099511627776", "17592186044416", "281474976710656", "4503599627370496", "72057594037927936", "1152921504606846976", "18446744073709551616", "295147905179352825856", "4722366482869645213696", "75557863725914323419136", "1208925819614629174706176" ]	Powers of 16: a(n) = 16^n.
A001026	[ "1", "17", "289", "4913", "83521", "1419857", "24137569", "410338673", "6975757441", "118587876497", "2015993900449", "34271896307633", "582622237229761", "9904578032905937", "168377826559400929", "2862423051509815793", "48661191875666868481", "827240261886336764177", "14063084452067724991009", "239072435685151324847153", "4064231406647572522401601" ]	Powers of 17.
A001027	[ "1", "18", "324", "5832", "104976", "1889568", "34012224", "612220032", "11019960576", "198359290368", "3570467226624", "64268410079232", "1156831381426176", "20822964865671168", "374813367582081024", "6746640616477458432", "121439531096594251776", "2185911559738696531968", "39346408075296537575424", "708235345355337676357632", "12748236216396078174437376" ]	Powers of 18.
A001028	[ "1", "1", "2", "7", "37", "269", "2535", "29738", "421790", "7076459", "138061343", "3089950076", "78454715107", "2238947459974", "71253947372202", "2511742808382105", "97495087989736907", "4145502184671892500", "192200099033324115855", "9676409879981926733908", "527029533717566423156698" ]	E.g.f. satisfies A'(x) = 1 + A(A(x)), A(0)=0.
A001029	[ "1", "19", "361", "6859", "130321", "2476099", "47045881", "893871739", "16983563041", "322687697779", "6131066257801", "116490258898219", "2213314919066161", "42052983462257059", "799006685782884121", "15181127029874798299", "288441413567621167681", "5480386857784802185939", "104127350297911241532841", "1978419655660313589123979", "37589973457545958193355601" ]	Powers of 19.
A001030	[ "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2" ]	Fixed under 1 -> 21, 2 -> 211.
A001031	[ "1", "2", "2", "2", "2", "2", "3", "2", "3", "3", "3", "4", "3", "2", "4", "3", "4", "4", "3", "3", "5", "4", "4", "6", "4", "3", "6", "3", "4", "7", "4", "5", "6", "3", "5", "7", "6", "5", "7", "5", "5", "9", "5", "4", "10", "4", "5", "7", "4", "6", "9", "6", "6", "9", "7", "7", "11", "6", "6", "12", "4", "5", "10", "4", "7", "10", "6", "5", "9", "8", "8", "11", "6", "5", "13", "5", "8", "11", "6", "8", "10", "6", "6", "14", "9", "6", "12", "7", "7", "15", "7", "8", "13", "5", "8", "12", "8", "9" ]	Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).
A001032	[ "1", "2", "11", "23", "24", "26", "33", "47", "49", "50", "59", "73", "74", "88", "96", "97", "107", "121", "122", "146", "169", "177", "184", "191", "193", "194", "218", "239", "241", "242", "249", "289", "297", "299", "311", "312", "313", "337", "338", "347", "352", "361", "362", "376", "383", "393", "407", "409", "431", "443", "457", "458", "479", "481", "491", "506" ]	Numbers k such that sum of squares of k consecutive integers >= 1 is a square.
A001033	[ "1", "16", "25", "33", "49", "52", "64", "73", "97", "100", "121", "148", "169", "177", "193", "196", "241", "244", "249", "256", "276", "289", "292", "297", "313", "337", "361", "388", "393", "400", "409", "457", "481", "484", "528", "529", "537", "577", "592", "625", "628", "649", "673", "676", "708", "724", "753", "772", "784", "793", "832", "841", "852", "897", "913", "961", "964", "976", "996" ]	Numbers n such that the sum of the squares of n consecutive positive odd numbers x^2 + (x+2)^2 + ... + (x+2n-2)^2 = k^2 for some integer k. The least values of x and k for each n are in A056131 and A056132, respectively.
A001034	[ "60", "168", "360", "504", "660", "1092", "2448", "2520", "3420", "4080", "5616", "6048", "6072", "7800", "7920", "9828", "12180", "14880", "20160", "25308", "25920", "29120", "32736", "34440", "39732", "51888", "58800", "62400", "74412", "95040", "102660", "113460", "126000", "150348", "175560", "178920" ]	Orders of noncyclic simple groups (without repetition).
A001035	[ "1", "1", "3", "19", "219", "4231", "130023", "6129859", "431723379", "44511042511", "6611065248783", "1396281677105899", "414864951055853499", "171850728381587059351", "98484324257128207032183", "77567171020440688353049939", "83480529785490157813844256579", "122152541250295322862941281269151", "241939392597201176602897820148085023" ]	Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).
A001036	[ "1", "2", "4", "7", "13", "22", "40", "70", "126", "225", "411", "746", "1376", "2537", "4719", "8799", "16509", "31041", "58635", "111012", "210870", "401427", "766149", "1465019", "2807195", "5387990", "10358998", "19945393", "38458183", "74248450", "143522116", "277737796", "538038782", "1043325197" ]	Partial sums of A001037, omitting A001037(1).
A001037	[ "1", "2", "1", "2", "3", "6", "9", "18", "30", "56", "99", "186", "335", "630", "1161", "2182", "4080", "7710", "14532", "27594", "52377", "99858", "190557", "364722", "698870", "1342176", "2580795", "4971008", "9586395", "18512790", "35790267", "69273666", "134215680", "260300986", "505286415", "981706806", "1908866960", "3714566310", "7233615333", "14096302710", "27487764474" ]	Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n.
A001038	[ "2", "2", "10", "52246", "2631645209645100680144", "312242081385925594286511113384607360432260178128338777217975928751832" ]	Invertible Boolean functions with GL(n,2) acting on the domain and range.
A001039	[ "3", "13", "781", "137257", "28531167061", "25239592216021", "51702516367896047761", "109912203092239643840221", "949112181811268728834319677753", "91703076898614683377208150526107718802981" ]	a(n) = (p^p-1)/(p-1) where p = prime(n).
A001040	[ "0", "1", "1", "3", "10", "43", "225", "1393", "9976", "81201", "740785", "7489051", "83120346", "1004933203", "13147251985", "185066460993", "2789144166880", "44811373131073", "764582487395121", "13807296146243251", "263103209266016890", "5275871481466581051", "111056404320064218961", "2448516766522879398193" ]	a(n+1) = n*a(n) + a(n-1) with a(0)=0, a(1)=1.
A001041	[ "12", "24", "72", "360", "2520", "27720", "360360", "6126120", "116396280", "2677114440", "77636318760", "2406725881560", "89048857617720", "3651003162326520", "156993135980040360", "7378677391061896920", "391069901726280536760", "23073124201850551668840" ]	a(0)=12; thereafter a(n) = 12 times the product of the first n primes.
A001042	[ "1", "2", "3", "5", "16", "231", "53105", "2820087664", "7952894429824835871", "63248529811938901240357985099443351745", "4000376523371723941902615329287219027543200136435757892789536976747706216384" ]	a(n) = a(n-1)^2 - a(n-2)^2.
A001043	[ "5", "8", "12", "18", "24", "30", "36", "42", "52", "60", "68", "78", "84", "90", "100", "112", "120", "128", "138", "144", "152", "162", "172", "186", "198", "204", "210", "216", "222", "240", "258", "268", "276", "288", "300", "308", "320", "330", "340", "352", "360", "372", "384", "390", "396", "410", "434", "450", "456", "462", "472", "480", "492", "508", "520" ]	Numbers that are the sum of 2 successive primes.
A001044	[ "1", "1", "4", "36", "576", "14400", "518400", "25401600", "1625702400", "131681894400", "13168189440000", "1593350922240000", "229442532802560000", "38775788043632640000", "7600054456551997440000", "1710012252724199424000000", "437763136697395052544000000", "126513546505547170185216000000" ]	a(n) = (n!)^2.
A001045	[ "0", "1", "1", "3", "5", "11", "21", "43", "85", "171", "341", "683", "1365", "2731", "5461", "10923", "21845", "43691", "87381", "174763", "349525", "699051", "1398101", "2796203", "5592405", "11184811", "22369621", "44739243", "89478485", "178956971", "357913941", "715827883", "1431655765", "2863311531", "5726623061" ]	Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer to 2^n/3.
A001046	[ "1", "1", "2", "7", "44", "447", "6749", "142176", "3987677", "143698548", "6470422337", "356016927083", "23503587609815", "1833635850492653", "166884365982441238", "17524692064006822643", "2103129932046801158398", "286043195450428964364771", "43766712033847678348968361" ]	a(n) = n(n-1)a(n-1)/2 + a(n-2), a(0) = a(1) = 1.
A001047	[ "0", "1", "5", "19", "65", "211", "665", "2059", "6305", "19171", "58025", "175099", "527345", "1586131", "4766585", "14316139", "42981185", "129009091", "387158345", "1161737179", "3485735825", "10458256051", "31376865305", "94134790219", "282412759265", "847255055011", "2541798719465", "7625463267259", "22876524019505" ]	a(n) = 3^n - 2^n.
A001048	[ "2", "3", "8", "30", "144", "840", "5760", "45360", "403200", "3991680", "43545600", "518918400", "6706022400", "93405312000", "1394852659200", "22230464256000", "376610217984000", "6758061133824000", "128047474114560000", "2554547108585472000", "53523844179886080000", "1175091669949317120000" ]	a(n) = n! + (n-1)!.
A001049	[ "8", "14", "23", "28", "33", "42", "51", "59", "68", "77", "86", "96", "103", "110", "116", "125" ]	Numbered stops in Manhattan on the Lexington Avenue subway.
A001050	[ "5", "4", "5", "5", "5", "5", "5", "9", "9", "8", "8", "10", "11", "11", "11", "11", "11", "15", "15", "14", "13", "17", "18", "18", "18", "18", "18", "22", "22", "21", "13", "17", "18", "18", "18", "18", "18", "22", "22", "21", "13", "17", "18", "18", "18", "18", "18", "22", "22", "21", "13", "17", "18", "18", "18", "18", "18", "22", "22", "21", "13", "17", "18", "18", "18", "18", "18" ]	Number of letters in n (in Finnish).
A001051	[ "1", "3", "1", "5", "1", "5", "1", "7", "1", "5", "1", "8", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "10", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "8", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "8", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "8" ]	Number of subgroups of order n in orthogonal group O(3).
A001052	[ "1", "2", "3", "11", "69", "701", "10584", "222965", "6253604", "225352709", "10147125509", "558317255704", "36859086001973", "2875567025409598", "261713458398275391", "27482788698844325653", "3298196357319717353751", "448582187384180404435789", "68636372866136921596029468" ]	a(n) = n(n-1)a(n-1)/2 + a(n-2), a(0) = 1, a(1) = 2.
A001053	[ "1", "0", "1", "2", "7", "30", "157", "972", "6961", "56660", "516901", "5225670", "57999271", "701216922", "9173819257", "129134686520", "1946194117057", "31268240559432", "533506283627401", "9634381345852650", "183586751854827751", "3681369418442407670", "77492344539145388821", "1708512949279640961732" ]	a(n+1) = n*a(n) + a(n-1) with a(0)=1, a(1)=0.
A001054	[ "0", "1", "-1", "-2", "1", "-3", "-4", "11", "-45", "-496", "22319", "-11070225", "-247076351776", "2735190806339469599", "-675800965841611881515781657825", "-1848444588685310753420392017318175868503407962176" ]	a(n) = a(n-1)*a(n-2) - 1.
A001055	[ "1", "1", "1", "2", "1", "2", "1", "3", "2", "2", "1", "4", "1", "2", "2", "5", "1", "4", "1", "4", "2", "2", "1", "7", "2", "2", "3", "4", "1", "5", "1", "7", "2", "2", "2", "9", "1", "2", "2", "7", "1", "5", "1", "4", "4", "2", "1", "12", "2", "4", "2", "4", "1", "7", "2", "7", "2", "2", "1", "11", "1", "2", "4", "11", "2", "5", "1", "4", "2", "5", "1", "16", "1", "2", "4", "4", "2", "5", "1", "12", "5", "2", "1", "11", "2", "2", "2", "7", "1", "11", "2", "4", "2", "2", "2", "19", "1", "4", "4", "9", "1", "5", "1" ]	The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).
A001056	[ "1", "3", "4", "13", "53", "690", "36571", "25233991", "922832284862", "23286741570717144243", "21489756930695820973683319349467", "500426416062641238759467086706254193219790764168482", "10754042042885415070816603338436200915110904821126871858491675028294447933424899095" ]	a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.
A001057	[ "0", "1", "-1", "2", "-2", "3", "-3", "4", "-4", "5", "-5", "6", "-6", "7", "-7", "8", "-8", "9", "-9", "10", "-10", "11", "-11", "12", "-12", "13", "-13", "14", "-14", "15", "-15", "16", "-16", "17", "-17", "18", "-18", "19", "-19", "20", "-20", "21", "-21", "22", "-22", "23", "-23", "24", "-24", "25", "-25", "26", "-26", "27", "-27", "28", "-28", "29", "-29", "30", "-30", "31", "-31" ]	Canonical enumeration of integers: interleaved positive and negative integers with zero prepended.
A001058	[ "0", "2", "3", "6", "7", "1", "9", "4", "5", "8", "22", "23", "26", "27", "21", "29", "24", "25", "28", "20", "12", "32", "33", "36", "37", "31", "39", "34", "35", "38", "30", "13", "10", "62", "63", "66", "67", "61", "69", "64", "65", "68", "60", "16", "72", "73", "76", "77", "71", "79", "74", "75", "78", "70", "17", "92", "93", "96", "97", "91", "99", "94", "95", "98", "90", "19", "14", "42", "43", "46", "47", "41" ]	1-digit numbers in reverse alphabetical order, then 2-digit numbers, etc.
A001059	[ "1", "1", "5", "59", "1263", "42713", "2094399", "140434335", "12340275539", "1375857855221", "189751578038547", "31714568837559539", "6316261763436325285", "1477890415844440910325", "401400487846091289175217", "125247016772173387008904623", "44493481073675052201518261955" ]	Number of doubly labeled heap-ordered trees.
A001060	[ "2", "5", "7", "12", "19", "31", "50", "81", "131", "212", "343", "555", "898", "1453", "2351", "3804", "6155", "9959", "16114", "26073", "42187", "68260", "110447", "178707", "289154", "467861", "757015", "1224876", "1981891", "3206767", "5188658", "8395425", "13584083", "21979508", "35563591", "57543099", "93106690", "150649789" ]	a(n) = a(n-1) + a(n-2) with a(0)=2, a(1)=5. Sometimes called the Evangelist Sequence.
A001061	[ "8", "3", "1", "5", "9", "0", "6", "7", "4", "2", "88", "38", "58", "98", "68", "78", "48", "28", "18", "80", "30", "83", "33", "53", "93", "63", "73", "43", "23", "13", "81", "31", "51", "91", "61", "71", "41", "21", "11", "85", "35", "55", "95", "65", "75", "45", "25", "15", "50", "89", "39", "59", "99", "69", "79", "49", "29", "19", "90" ]	1-, 2-, 3-, ... digit numbers in alphabetical order in German.
A001062	[ "5", "2", "8", "9", "4", "7", "6", "3", "1", "0", "50", "55", "52", "51", "58", "59", "54", "57", "56", "53", "10", "18", "19", "17", "12", "11", "40", "45", "42", "41", "48", "49", "44", "47", "46", "43", "14", "80", "85", "82", "90", "98", "99", "97", "92", "88", "89", "91", "94", "84", "95", "96", "87", "86", "93", "83", "81", "15", "16", "60", "70", "78", "79", "77", "72", "71", "74", "75", "76", "73", "13", "30" ]	1-, 2-, 3- ... digit numbers in alphabetical order in French (incorrect version, see A187876 for the correct version).
A001063	[ "1", "1", "1", "3", "15", "111", "1131", "15081", "253473", "5220225", "128886921", "3749014251", "126648293391", "4909623331023", "216189866951235", "10718939718977121", "593865369943409601", "36520856568972350721", "2478236630512178688273", "184588566642520989171795", "15020141103053997234030351" ]	E.g.f. satisfies A'(x) = A(x/(1-x)).
A001064	[ "1", "1", "0", "1", "1", "1", "2", "3", "7", "23", "164", "3779", "619779", "2342145005", "1451612289057674", "3399886472013047316638149", "4935316984175079105557291745555191750431", "16779517449593082173916263081219908459297087421776218065830849893" ]	a(n) = a(n-1)*a(n-2) + a(n-3).
A001065	[ "0", "1", "1", "3", "1", "6", "1", "7", "4", "8", "1", "16", "1", "10", "9", "15", "1", "21", "1", "22", "11", "14", "1", "36", "6", "16", "13", "28", "1", "42", "1", "31", "15", "20", "13", "55", "1", "22", "17", "50", "1", "54", "1", "40", "33", "26", "1", "76", "8", "43", "21", "46", "1", "66", "17", "64", "23", "32", "1", "108", "1", "34", "41", "63", "19", "78", "1", "58", "27", "74", "1", "123", "1", "40", "49", "64", "19", "90", "1", "106" ]	Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.
A001066	[ "3", "6", "8", "10", "14", "15", "16", "20", "21", "24", "28", "30", "35", "36", "42", "45", "48", "52", "55", "56", "63", "66", "70", "72", "78", "80", "90", "91", "96", "99", "104", "105", "110", "120", "126", "132", "133", "136", "143", "153", "156", "160", "168", "171", "182", "190", "195", "198", "210", "224", "231", "240", "248", "253", "255", "266", "272", "276", "286", "288", "300", "306" ]	Dimensions (sorted, with duplicates removed) of real simple Lie algebras.
A001067	[ "1", "-1", "1", "-1", "1", "-691", "1", "-3617", "43867", "-174611", "77683", "-236364091", "657931", "-3392780147", "1723168255201", "-7709321041217", "151628697551", "-26315271553053477373", "154210205991661", "-261082718496449122051", "1520097643918070802691", "-2530297234481911294093" ]	Numerator of Bernoulli(2n)/(2n).
A001068	[ "0", "1", "2", "3", "5", "6", "7", "8", "10", "11", "12", "13", "15", "16", "17", "18", "20", "21", "22", "23", "25", "26", "27", "28", "30", "31", "32", "33", "35", "36", "37", "38", "40", "41", "42", "43", "45", "46", "47", "48", "50", "51", "52", "53", "55", "56", "57", "58", "60", "61", "62", "63", "65", "66", "67", "68", "70", "71", "72", "73", "75", "76", "77", "78", "80", "81", "82", "83", "85", "86", "87", "88" ]	a(n) = floor(5*n/4), numbers that are congruent to {0, 1, 2, 3} mod 5.
A001069	[ "0", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3" ]	Log2*(n) (version 2): take log_2 of n this many times to get a number < 2.
A001070	[ "0", "0", "1", "2", "36", "4704", "8501760", "267533746176", "188809932117639168", "3790336726450693283512320" ]	Number of normalized Latin squares with second row even.
A001071	[ "2", "1", "4", "10", "36", "108", "392", "1363", "5000", "18223", "67792", "252938", "952540", "3602478", "13699554", "52296713", "200406388", "770411478", "2970401696", "11482395526", "44491881090", "172766311857", "672186650116" ]	Number of one-sided chessboard polyominoes with n cells.
A001072	[ "1", "1", "3", "4", "11", "23", "63", "159", "459", "1331", "4083", "12750" ]	Number of minimally 2-edge-connected non-isomorphic graphs with n nodes.
A001073	[ "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "1", "0", "1", "2", "1", "4", "1", "6", "1", "8", "2", "0", "2", "2", "2", "4", "2", "6", "2", "8", "3", "0", "3", "2", "3", "4", "3", "6", "3", "8", "4", "0", "4", "2", "4", "4", "4", "6", "4", "8", "5", "0", "5", "2", "5", "4", "5", "6", "5", "8", "6", "0", "6", "2", "6", "4", "6", "6", "6", "8", "7", "0", "7", "2", "7", "4", "7", "6", "7", "8", "8", "0", "8", "2", "8", "4", "8", "6", "8", "8", "9", "0", "9", "2", "9", "4", "9", "6", "9", "8", "1", "0", "0", "1", "0", "3" ]	Label a 1-cm ruler with digits 1 cm wide.
A001074	[ "1", "4", "7", "8", "9", "13", "19", "25", "27", "28", "31", "32", "36", "37", "43", "49", "52", "56", "61", "63", "64", "67", "72", "73", "76", "79", "91", "97", "100", "103", "104", "108", "109", "117", "121", "124", "125", "127", "133", "139", "148", "151", "152", "157", "163", "169", "171", "172", "175", "181", "189", "193", "196", "199", "200", "211", "216", "217" ]	Numbers m such that Sum_{k=0..m-1} exp(2Pii*k^3/m) != 0.
A001075	[ "1", "2", "7", "26", "97", "362", "1351", "5042", "18817", "70226", "262087", "978122", "3650401", "13623482", "50843527", "189750626", "708158977", "2642885282", "9863382151", "36810643322", "137379191137", "512706121226", "1913445293767", "7141075053842", "26650854921601", "99462344632562", "371198523608647" ]	a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - a(n-2).
A001076	[ "0", "1", "4", "17", "72", "305", "1292", "5473", "23184", "98209", "416020", "1762289", "7465176", "31622993", "133957148", "567451585", "2403763488", "10182505537", "43133785636", "182717648081", "774004377960", "3278735159921", "13888945017644", "58834515230497", "249227005939632", "1055742538989025" ]	Denominators of continued fraction convergents to sqrt(5).
A001077	[ "1", "2", "9", "38", "161", "682", "2889", "12238", "51841", "219602", "930249", "3940598", "16692641", "70711162", "299537289", "1268860318", "5374978561", "22768774562", "96450076809", "408569081798", "1730726404001", "7331474697802", "31056625195209" ]	Numerators of continued fraction convergents to sqrt(5).
A001078	[ "0", "2", "20", "198", "1960", "19402", "192060", "1901198", "18819920", "186298002", "1844160100", "18255302998", "180708869880", "1788833395802", "17707625088140", "175287417485598", "1735166549767840", "17176378080192802", "170028614252160180", "1683109764441408998" ]	a(n) = 10*a(n-1) - a(n-2) with a(0) = 0, a(1) = 2.
A001079	[ "1", "5", "49", "485", "4801", "47525", "470449", "4656965", "46099201", "456335045", "4517251249", "44716177445", "442644523201", "4381729054565", "43374646022449", "429364731169925", "4250272665676801", "42073361925598085", "416483346590304049" ]	a(n) = 10*a(n-1) - a(n-2); a(0) = 1, a(1) = 5.
A001080	[ "0", "3", "48", "765", "12192", "194307", "3096720", "49353213", "786554688", "12535521795", "199781794032", "3183973182717", "50743789129440", "808716652888323", "12888722657083728", "205410845860451325", "3273684811110137472", "52173546131901748227", "831503053299317834160" ]	a(n) = 16*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.
A001081	[ "1", "8", "127", "2024", "32257", "514088", "8193151", "130576328", "2081028097", "33165873224", "528572943487", "8424001222568", "134255446617601", "2139663144659048", "34100354867927167", "543466014742175624", "8661355881006882817" ]	a(n) = 16*a(n-1) - a(n-2).
A001082	[ "0", "1", "5", "8", "16", "21", "33", "40", "56", "65", "85", "96", "120", "133", "161", "176", "208", "225", "261", "280", "320", "341", "385", "408", "456", "481", "533", "560", "616", "645", "705", "736", "800", "833", "901", "936", "1008", "1045", "1121", "1160", "1240", "1281", "1365", "1408", "1496", "1541", "1633", "1680", "1776", "1825", "1925", "1976" ]	Generalized octagonal numbers: k(3k-2), k=0, +- 1, +- 2, +-3, ...
A001083	[ "1", "2", "2", "3", "5", "7", "10", "15", "23", "34", "50", "75", "113", "170", "255", "382", "574", "863", "1293", "1937", "2903", "4353", "6526", "9789", "14688", "22029", "33051", "49577", "74379", "111580", "167388", "251090", "376631", "564932", "847376", "1271059", "1906628", "2859984" ]	Length of one version of Kolakoski sequence {A000002(i)} at n-th growth stage.
A001084	[ "0", "3", "60", "1197", "23880", "476403", "9504180", "189607197", "3782639760", "75463188003", "1505481120300", "30034159217997", "599177703239640", "11953519905574803", "238471220408256420", "4757470888259553597", "94910946544782815520", "1893461460007396756803", "37774318253603152320540" ]	a(n) = 20*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.
A001085	[ "1", "10", "199", "3970", "79201", "1580050", "31521799", "628855930", "12545596801", "250283080090", "4993116004999", "99612037019890", "1987247624392801", "39645340450836130", "790919561392329799", "15778745887395759850", "314783998186522867201", "6279901217843061584170" ]	a(n) = 20*a(n-1) - a(n-2).
A001086	[ "0", "3", "4", "1", "1", "5", "2", "168", "46793", "1", "7", "1", "51", "1", "7", "1", "6", "2", "1", "1", "1", "10", "1", "2", "10", "1", "2", "11", "16", "3", "1", "1", "1", "1", "4", "1", "1", "3", "1", "1", "5", "5", "25", "1", "34", "10", "2", "18", "10", "585", "1", "2", "3", "1", "1", "440", "1", "1", "7", "2", "1", "4", "6", "16", "5", "2", "3", "2", "5", "1", "1", "77", "1" ]	Continued fraction associated with y(y+1) = x(x^2 -1).
A001087	[ "1", "2", "3", "4", "5", "7", "11", "12", "13", "18", "26", "31", "49", "62", "80", "81", "82", "101", "126", "167", "215", "295", "417", "436", "602", "887", "1371", "1454", "2332", "2479", "2645", "2646" ]	Related to S(n), the number of self-dual monotone Boolean functions of n variables (A001206): 2^n-th term is S(n).
A001088	[ "1", "1", "2", "4", "16", "32", "192", "768", "4608", "18432", "184320", "737280", "8847360", "53084160", "424673280", "3397386240", "54358179840", "326149079040", "5870683422720", "46965467381760", "563585608581120", "5635856085811200", "123988833887846400", "991910671102771200", "19838213422055424000" ]	Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).
A001089	[ "0", "0", "0", "0", "3", "24", "133", "635", "2807", "11864", "48756", "196707", "783750", "3095708", "12152855", "47500635", "185082495", "719559600", "2793121080", "10830450780", "41965864794", "162539516448", "629399492330", "2437072038302", "9437097796918" ]	Number of permutations of [n] containing exactly 2 increasing subsequences of length 3.
A001090	[ "0", "1", "8", "63", "496", "3905", "30744", "242047", "1905632", "15003009", "118118440", "929944511", "7321437648", "57641556673", "453811015736", "3572846569215", "28128961537984", "221458845734657", "1743541804339272", "13726875588979519", "108071462907496880", "850844827670995521", "6698687158460467288" ]	a(n) = 8*a(n-1) - a(n-2); a(0) = 0, a(1) = 1.
A001091	[ "1", "4", "31", "244", "1921", "15124", "119071", "937444", "7380481", "58106404", "457470751", "3601659604", "28355806081", "223244789044", "1757602506271", "13837575261124", "108942999582721", "857706421400644", "6752708371622431", "53163960551578804" ]	a(n) = 8*a(n-1) - a(n-2); a(0) = 1, a(1) = 4.
A001092	[ "1", "2", "3", "4", "5", "7", "9", "11", "13", "17", "19", "23", "25", "27", "29", "31", "41", "43", "47", "49", "59", "61", "71", "73", "79", "81", "83", "101", "103", "107", "109", "125", "127", "137", "139", "149", "151", "167", "169", "179", "181", "191", "193", "197", "199", "227", "229", "239", "241", "243", "269", "271", "281", "283", "311", "313", "347", "349", "359" ]	Union of all numbers {p, q} where p and q are both primes or powers of primes and q = p+2.
A001093	[ "0", "1", "2", "9", "28", "65", "126", "217", "344", "513", "730", "1001", "1332", "1729", "2198", "2745", "3376", "4097", "4914", "5833", "6860", "8001", "9262", "10649", "12168", "13825", "15626", "17577", "19684", "21953", "24390", "27001", "29792", "32769", "35938", "39305", "42876", "46657", "50654", "54873", "59320" ]	a(n) = n^3 + 1.
A001094	[ "0", "1", "2", "3", "28", "125", "366", "847", "1688", "3033", "5050", "7931", "11892", "17173", "24038", "32775", "43696", "57137", "73458", "93043", "116300", "143661", "175582", "212543", "255048", "303625", "358826", "421227", "491428", "570053", "657750", "755191", "863072", "982113", "1113058" ]	a(n) = n + n(n-1)(n-2)*(n-3).
A001095	[ "0", "1", "2", "3", "4", "125", "726", "2527", "6728", "15129", "30250", "55451", "95052", "154453", "240254", "360375", "524176", "742577", "1028178", "1395379", "1860500", "2441901", "3160102", "4037903", "5100504", "6375625", "7893626", "9687627", "11793628", "14250629", "17100750", "20389351" ]	a(n) = n + n(n-1)(n-2)(n-3)(n-4).
A001096	[ "0", "1", "2", "3", "4", "5", "726", "5047", "20168", "60489", "151210", "332651", "665292", "1235533", "2162174", "3603615", "5765776", "8910737", "13366098", "19535059", "27907220", "39070101", "53721382", "72681863", "96909144", "127512025", "165765626", "213127227", "271252828", "342014429" ]	a(n) = n + n(n-1)(n-2)(n-3)(n-4)*(n-5).
A001097	[ "3", "5", "7", "11", "13", "17", "19", "29", "31", "41", "43", "59", "61", "71", "73", "101", "103", "107", "109", "137", "139", "149", "151", "179", "181", "191", "193", "197", "199", "227", "229", "239", "241", "269", "271", "281", "283", "311", "313", "347", "349", "419", "421", "431", "433", "461", "463", "521", "523", "569", "571", "599", "601", "617", "619", "641", "643" ]	Twin primes.
A001098	[ "1", "10", "10100", "100111011101000", "101101100001100111010010100000100111001010010000" ]	Multiply previous term by 2 and write in binary.
A001099	[ "1", "3", "24", "232", "2893", "43763", "779780", "15997436", "371423053", "9628576947", "275683093664", "8640417354592", "294234689237661", "10817772136320355", "427076118244539020", "18019667955465012596", "809220593930871751581", "38537187481365665823843", "1939882468178947923300136" ]	a(n) = n^n - a(n-1), with a(1) = 1.
A001100	[ "1", "0", "2", "0", "4", "2", "2", "10", "10", "2", "14", "40", "48", "16", "2", "90", "230", "256", "120", "22", "2", "646", "1580", "1670", "888", "226", "28", "2", "5242", "12434", "12846", "7198", "2198", "366", "34", "2", "47622", "110320", "112820", "64968", "22120", "4448", "540", "40", "2", "479306", "1090270", "1108612", "650644", "236968", "54304", "7900", "748", "46", "2" ]	Triangle read by rows: T(n,k) = number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1.

A001001

[ "1", "7", "13", "35", "31", "91", "57", "155", "130", "217", "133", "455", "183", "399", "403", "651", "307", "910", "381", "1085", "741", "931", "553", "2015", "806", "1281", "1210", "1995", "871", "2821", "993", "2667", "1729", "2149", "1767", "4550", "1407", "2667", "2379", "4805", "1723", "5187", "1893", "4655", "4030", "3871", "2257", "8463", "2850", "5642", "3991", "6405", "2863" ]

Number of sublattices of index n in generic 3-dimensional lattice.

A001002

[ "1", "1", "3", "10", "38", "154", "654", "2871", "12925", "59345", "276835", "1308320", "6250832", "30142360", "146510216", "717061938", "3530808798", "17478955570", "86941210950", "434299921440", "2177832612120", "10959042823020", "55322023332420", "280080119609550", "1421744205767418", "7234759677699954" ]

Number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.

A001003

[ "1", "1", "3", "11", "45", "197", "903", "4279", "20793", "103049", "518859", "2646723", "13648869", "71039373", "372693519", "1968801519", "10463578353", "55909013009", "300159426963", "1618362158587", "8759309660445", "47574827600981", "259215937709463", "1416461675464871" ]

Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.

A001004

[ "1", "1", "2", "3", "9", "20", "75", "262", "1117", "4783", "21971", "102249", "489077", "2370142", "11654465", "57916324", "290693391", "1471341341", "7504177738", "38532692207", "199076194985", "1034236705992", "5400337050086", "28329240333758", "149244907249629" ]

Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.

A001005

[ "1", "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" ]

Number of ways of partitioning n points on a circle into subsets only of sizes 2 and 3.

A001006

[ "1", "1", "2", "4", "9", "21", "51", "127", "323", "835", "2188", "5798", "15511", "41835", "113634", "310572", "853467", "2356779", "6536382", "18199284", "50852019", "142547559", "400763223", "1129760415", "3192727797", "9043402501", "25669818476", "73007772802", "208023278209", "593742784829" ]

Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.

A001007

[ "1", "2", "15", "42", "421", "1331", "15119", "618765", "2155578", "98032875", "1290154807", "4682196239", "63117678751", "3186252107917", "164886529617695", "616630679090258", "32763760653353135", "467443761039641135", "1768227793278781667", "96699391743949360451", "1402535447576150395335" ]

a(n) = ( Sum C(p,i); i=1,...,floor(2p/3) ) / p^2, where p = prime(n).

A001008

[ "1", "3", "11", "25", "137", "49", "363", "761", "7129", "7381", "83711", "86021", "1145993", "1171733", "1195757", "2436559", "42142223", "14274301", "275295799", "55835135", "18858053", "19093197", "444316699", "1347822955", "34052522467", "34395742267", "312536252003", "315404588903", "9227046511387" ]

Numerators of harmonic numbers H(n) = Sum_{i=1..n} 1/i.

A001009

[ "1", "1", "1", "1", "1", "1", "1", "3", "4", "4", "1", "11", "46", "56", "56", "1", "53", "1064", "6552", "9408", "9408", "1", "309", "35792", "1293216", "11270400", "16942080", "16942080", "1", "2119", "1673792", "420909504", "27206658048", "335390189568", "535281401856", "535281401856", "1", "16687", "103443808" ]

Triangle giving number L(n,k) of normalized k X n Latin rectangles.

A001010

[ "1", "2", "2", "4", "6", "8", "18", "20", "56", "48", "178", "132", "574", "348", "1870", "1008", "6144", "2812", "20314", "8420", "67534", "24396", "225472", "74756", "755672", "222556", "2540406", "693692", "8564622", "2107748", "28941258", "6656376", "98011464", "20548932" ]

Number of symmetric foldings of a strip of n stamps.

A001011

[ "1", "1", "2", "5", "14", "38", "120", "353", "1148", "3527", "11622", "36627", "121622", "389560", "1301140", "4215748", "14146335", "46235800", "155741571", "512559195", "1732007938", "5732533570", "19423092113", "64590165281", "219349187968", "732358098471", "2492051377341", "8349072895553", "28459491475593" ]

Number of ways to fold a strip of n blank stamps.

A001012

[ "1", "1", "1", "1", "2", "2", "22", "563", "1676257" ]

Erroneous version of A040082.

A001013

[ "1", "2", "4", "6", "8", "12", "16", "24", "32", "36", "48", "64", "72", "96", "120", "128", "144", "192", "216", "240", "256", "288", "384", "432", "480", "512", "576", "720", "768", "864", "960", "1024", "1152", "1296", "1440", "1536", "1728", "1920", "2048", "2304", "2592", "2880", "3072", "3456", "3840", "4096", "4320", "4608", "5040", "5184", "5760" ]

Jordan-Polya numbers: products of factorial numbers A000142.

A001014

[ "0", "1", "64", "729", "4096", "15625", "46656", "117649", "262144", "531441", "1000000", "1771561", "2985984", "4826809", "7529536", "11390625", "16777216", "24137569", "34012224", "47045881", "64000000", "85766121", "113379904", "148035889", "191102976", "244140625", "308915776", "387420489", "481890304" ]

Sixth powers: a(n) = n^6.

A001015

[ "0", "1", "128", "2187", "16384", "78125", "279936", "823543", "2097152", "4782969", "10000000", "19487171", "35831808", "62748517", "105413504", "170859375", "268435456", "410338673", "612220032", "893871739", "1280000000", "1801088541", "2494357888", "3404825447", "4586471424", "6103515625", "8031810176" ]

Seventh powers: a(n) = n^7.

A001016

[ "0", "1", "256", "6561", "65536", "390625", "1679616", "5764801", "16777216", "43046721", "100000000", "214358881", "429981696", "815730721", "1475789056", "2562890625", "4294967296", "6975757441", "11019960576", "16983563041", "25600000000", "37822859361", "54875873536", "78310985281", "110075314176" ]

Eighth powers: a(n) = n^8.

A001017

[ "0", "1", "512", "19683", "262144", "1953125", "10077696", "40353607", "134217728", "387420489", "1000000000", "2357947691", "5159780352", "10604499373", "20661046784", "38443359375", "68719476736", "118587876497", "198359290368", "322687697779", "512000000000", "794280046581", "1207269217792" ]

Ninth powers: a(n) = n^9.

A001018

[ "1", "8", "64", "512", "4096", "32768", "262144", "2097152", "16777216", "134217728", "1073741824", "8589934592", "68719476736", "549755813888", "4398046511104", "35184372088832", "281474976710656", "2251799813685248", "18014398509481984", "144115188075855872", "1152921504606846976", "9223372036854775808", "73786976294838206464", "590295810358705651712", "4722366482869645213696" ]

Powers of 8: a(n) = 8^n.

A001019

[ "1", "9", "81", "729", "6561", "59049", "531441", "4782969", "43046721", "387420489", "3486784401", "31381059609", "282429536481", "2541865828329", "22876792454961", "205891132094649", "1853020188851841", "16677181699666569", "150094635296999121", "1350851717672992089", "12157665459056928801" ]

Powers of 9: a(n) = 9^n.

A001020

[ "1", "11", "121", "1331", "14641", "161051", "1771561", "19487171", "214358881", "2357947691", "25937424601", "285311670611", "3138428376721", "34522712143931", "379749833583241", "4177248169415651", "45949729863572161", "505447028499293771", "5559917313492231481", "61159090448414546291" ]

Powers of 11: a(n) = 11^n.

A001021

[ "1", "12", "144", "1728", "20736", "248832", "2985984", "35831808", "429981696", "5159780352", "61917364224", "743008370688", "8916100448256", "106993205379072", "1283918464548864", "15407021574586368", "184884258895036416", "2218611106740436992" ]

Powers of 12.

A001022

[ "1", "13", "169", "2197", "28561", "371293", "4826809", "62748517", "815730721", "10604499373", "137858491849", "1792160394037", "23298085122481", "302875106592253", "3937376385699289", "51185893014090757", "665416609183179841", "8650415919381337933", "112455406951957393129", "1461920290375446110677", "19004963774880799438801" ]

Powers of 13.

A001023

[ "1", "14", "196", "2744", "38416", "537824", "7529536", "105413504", "1475789056", "20661046784", "289254654976", "4049565169664", "56693912375296", "793714773254144", "11112006825558016", "155568095557812224", "2177953337809371136", "30491346729331195904", "426878854210636742656", "5976303958948914397184", "83668255425284801560576" ]

Powers of 14.

A001024

[ "1", "15", "225", "3375", "50625", "759375", "11390625", "170859375", "2562890625", "38443359375", "576650390625", "8649755859375", "129746337890625", "1946195068359375", "29192926025390625", "437893890380859375", "6568408355712890625", "98526125335693359375", "1477891880035400390625", "22168378200531005859375", "332525673007965087890625" ]

Powers of 15.

A001025

[ "1", "16", "256", "4096", "65536", "1048576", "16777216", "268435456", "4294967296", "68719476736", "1099511627776", "17592186044416", "281474976710656", "4503599627370496", "72057594037927936", "1152921504606846976", "18446744073709551616", "295147905179352825856", "4722366482869645213696", "75557863725914323419136", "1208925819614629174706176" ]

Powers of 16: a(n) = 16^n.

A001026

[ "1", "17", "289", "4913", "83521", "1419857", "24137569", "410338673", "6975757441", "118587876497", "2015993900449", "34271896307633", "582622237229761", "9904578032905937", "168377826559400929", "2862423051509815793", "48661191875666868481", "827240261886336764177", "14063084452067724991009", "239072435685151324847153", "4064231406647572522401601" ]

Powers of 17.

A001027

[ "1", "18", "324", "5832", "104976", "1889568", "34012224", "612220032", "11019960576", "198359290368", "3570467226624", "64268410079232", "1156831381426176", "20822964865671168", "374813367582081024", "6746640616477458432", "121439531096594251776", "2185911559738696531968", "39346408075296537575424", "708235345355337676357632", "12748236216396078174437376" ]

Powers of 18.

A001028

[ "1", "1", "2", "7", "37", "269", "2535", "29738", "421790", "7076459", "138061343", "3089950076", "78454715107", "2238947459974", "71253947372202", "2511742808382105", "97495087989736907", "4145502184671892500", "192200099033324115855", "9676409879981926733908", "527029533717566423156698" ]

E.g.f. satisfies A'(x) = 1 + A(A(x)), A(0)=0.

A001029

[ "1", "19", "361", "6859", "130321", "2476099", "47045881", "893871739", "16983563041", "322687697779", "6131066257801", "116490258898219", "2213314919066161", "42052983462257059", "799006685782884121", "15181127029874798299", "288441413567621167681", "5480386857784802185939", "104127350297911241532841", "1978419655660313589123979", "37589973457545958193355601" ]

Powers of 19.

A001030

[ "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "2", "1", "1", "2", "1", "2", "1", "1", "2", "1", "2" ]

Fixed under 1 -> 21, 2 -> 211.

A001031

[ "1", "2", "2", "2", "2", "2", "3", "2", "3", "3", "3", "4", "3", "2", "4", "3", "4", "4", "3", "3", "5", "4", "4", "6", "4", "3", "6", "3", "4", "7", "4", "5", "6", "3", "5", "7", "6", "5", "7", "5", "5", "9", "5", "4", "10", "4", "5", "7", "4", "6", "9", "6", "6", "9", "7", "7", "11", "6", "6", "12", "4", "5", "10", "4", "7", "10", "6", "5", "9", "8", "8", "11", "6", "5", "13", "5", "8", "11", "6", "8", "10", "6", "6", "14", "9", "6", "12", "7", "7", "15", "7", "8", "13", "5", "8", "12", "8", "9" ]

Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).

A001032

[ "1", "2", "11", "23", "24", "26", "33", "47", "49", "50", "59", "73", "74", "88", "96", "97", "107", "121", "122", "146", "169", "177", "184", "191", "193", "194", "218", "239", "241", "242", "249", "289", "297", "299", "311", "312", "313", "337", "338", "347", "352", "361", "362", "376", "383", "393", "407", "409", "431", "443", "457", "458", "479", "481", "491", "506" ]

Numbers k such that sum of squares of k consecutive integers >= 1 is a square.

A001033

[ "1", "16", "25", "33", "49", "52", "64", "73", "97", "100", "121", "148", "169", "177", "193", "196", "241", "244", "249", "256", "276", "289", "292", "297", "313", "337", "361", "388", "393", "400", "409", "457", "481", "484", "528", "529", "537", "577", "592", "625", "628", "649", "673", "676", "708", "724", "753", "772", "784", "793", "832", "841", "852", "897", "913", "961", "964", "976", "996" ]

Numbers n such that the sum of the squares of n consecutive positive odd numbers x^2 + (x+2)^2 + ... + (x+2n-2)^2 = k^2 for some integer k. The least values of x and k for each n are in A056131 and A056132, respectively.

A001034

[ "60", "168", "360", "504", "660", "1092", "2448", "2520", "3420", "4080", "5616", "6048", "6072", "7800", "7920", "9828", "12180", "14880", "20160", "25308", "25920", "29120", "32736", "34440", "39732", "51888", "58800", "62400", "74412", "95040", "102660", "113460", "126000", "150348", "175560", "178920" ]

Orders of noncyclic simple groups (without repetition).

A001035

[ "1", "1", "3", "19", "219", "4231", "130023", "6129859", "431723379", "44511042511", "6611065248783", "1396281677105899", "414864951055853499", "171850728381587059351", "98484324257128207032183", "77567171020440688353049939", "83480529785490157813844256579", "122152541250295322862941281269151", "241939392597201176602897820148085023" ]

Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).

A001036

[ "1", "2", "4", "7", "13", "22", "40", "70", "126", "225", "411", "746", "1376", "2537", "4719", "8799", "16509", "31041", "58635", "111012", "210870", "401427", "766149", "1465019", "2807195", "5387990", "10358998", "19945393", "38458183", "74248450", "143522116", "277737796", "538038782", "1043325197" ]

Partial sums of A001037, omitting A001037(1).

A001037

[ "1", "2", "1", "2", "3", "6", "9", "18", "30", "56", "99", "186", "335", "630", "1161", "2182", "4080", "7710", "14532", "27594", "52377", "99858", "190557", "364722", "698870", "1342176", "2580795", "4971008", "9586395", "18512790", "35790267", "69273666", "134215680", "260300986", "505286415", "981706806", "1908866960", "3714566310", "7233615333", "14096302710", "27487764474" ]

Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n.

A001038

[ "2", "2", "10", "52246", "2631645209645100680144", "312242081385925594286511113384607360432260178128338777217975928751832" ]

Invertible Boolean functions with GL(n,2) acting on the domain and range.

A001039

[ "3", "13", "781", "137257", "28531167061", "25239592216021", "51702516367896047761", "109912203092239643840221", "949112181811268728834319677753", "91703076898614683377208150526107718802981" ]

a(n) = (p^p-1)/(p-1) where p = prime(n).

A001040

[ "0", "1", "1", "3", "10", "43", "225", "1393", "9976", "81201", "740785", "7489051", "83120346", "1004933203", "13147251985", "185066460993", "2789144166880", "44811373131073", "764582487395121", "13807296146243251", "263103209266016890", "5275871481466581051", "111056404320064218961", "2448516766522879398193" ]

a(n+1) = n*a(n) + a(n-1) with a(0)=0, a(1)=1.

A001041

[ "12", "24", "72", "360", "2520", "27720", "360360", "6126120", "116396280", "2677114440", "77636318760", "2406725881560", "89048857617720", "3651003162326520", "156993135980040360", "7378677391061896920", "391069901726280536760", "23073124201850551668840" ]

a(0)=12; thereafter a(n) = 12 times the product of the first n primes.

A001042

[ "1", "2", "3", "5", "16", "231", "53105", "2820087664", "7952894429824835871", "63248529811938901240357985099443351745", "4000376523371723941902615329287219027543200136435757892789536976747706216384" ]

a(n) = a(n-1)^2 - a(n-2)^2.

A001043

[ "5", "8", "12", "18", "24", "30", "36", "42", "52", "60", "68", "78", "84", "90", "100", "112", "120", "128", "138", "144", "152", "162", "172", "186", "198", "204", "210", "216", "222", "240", "258", "268", "276", "288", "300", "308", "320", "330", "340", "352", "360", "372", "384", "390", "396", "410", "434", "450", "456", "462", "472", "480", "492", "508", "520" ]

Numbers that are the sum of 2 successive primes.

A001044

[ "1", "1", "4", "36", "576", "14400", "518400", "25401600", "1625702400", "131681894400", "13168189440000", "1593350922240000", "229442532802560000", "38775788043632640000", "7600054456551997440000", "1710012252724199424000000", "437763136697395052544000000", "126513546505547170185216000000" ]

a(n) = (n!)^2.

A001045

[ "0", "1", "1", "3", "5", "11", "21", "43", "85", "171", "341", "683", "1365", "2731", "5461", "10923", "21845", "43691", "87381", "174763", "349525", "699051", "1398101", "2796203", "5592405", "11184811", "22369621", "44739243", "89478485", "178956971", "357913941", "715827883", "1431655765", "2863311531", "5726623061" ]

Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer to 2^n/3.

A001046

[ "1", "1", "2", "7", "44", "447", "6749", "142176", "3987677", "143698548", "6470422337", "356016927083", "23503587609815", "1833635850492653", "166884365982441238", "17524692064006822643", "2103129932046801158398", "286043195450428964364771", "43766712033847678348968361" ]

a(n) = n*(n-1)*a(n-1)/2 + a(n-2), a(0) = a(1) = 1.

A001047

[ "0", "1", "5", "19", "65", "211", "665", "2059", "6305", "19171", "58025", "175099", "527345", "1586131", "4766585", "14316139", "42981185", "129009091", "387158345", "1161737179", "3485735825", "10458256051", "31376865305", "94134790219", "282412759265", "847255055011", "2541798719465", "7625463267259", "22876524019505" ]

a(n) = 3^n - 2^n.

A001048

[ "2", "3", "8", "30", "144", "840", "5760", "45360", "403200", "3991680", "43545600", "518918400", "6706022400", "93405312000", "1394852659200", "22230464256000", "376610217984000", "6758061133824000", "128047474114560000", "2554547108585472000", "53523844179886080000", "1175091669949317120000" ]

a(n) = n! + (n-1)!.

A001049

[ "8", "14", "23", "28", "33", "42", "51", "59", "68", "77", "86", "96", "103", "110", "116", "125" ]

Numbered stops in Manhattan on the Lexington Avenue subway.

A001050

[ "5", "4", "5", "5", "5", "5", "5", "9", "9", "8", "8", "10", "11", "11", "11", "11", "11", "15", "15", "14", "13", "17", "18", "18", "18", "18", "18", "22", "22", "21", "13", "17", "18", "18", "18", "18", "18", "22", "22", "21", "13", "17", "18", "18", "18", "18", "18", "22", "22", "21", "13", "17", "18", "18", "18", "18", "18", "22", "22", "21", "13", "17", "18", "18", "18", "18", "18" ]

Number of letters in n (in Finnish).

A001051

[ "1", "3", "1", "5", "1", "5", "1", "7", "1", "5", "1", "8", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "10", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "8", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "8", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "7", "1", "5", "1", "8" ]

Number of subgroups of order n in orthogonal group O(3).

A001052

[ "1", "2", "3", "11", "69", "701", "10584", "222965", "6253604", "225352709", "10147125509", "558317255704", "36859086001973", "2875567025409598", "261713458398275391", "27482788698844325653", "3298196357319717353751", "448582187384180404435789", "68636372866136921596029468" ]

a(n) = n*(n-1)*a(n-1)/2 + a(n-2), a(0) = 1, a(1) = 2.

A001053

[ "1", "0", "1", "2", "7", "30", "157", "972", "6961", "56660", "516901", "5225670", "57999271", "701216922", "9173819257", "129134686520", "1946194117057", "31268240559432", "533506283627401", "9634381345852650", "183586751854827751", "3681369418442407670", "77492344539145388821", "1708512949279640961732" ]

a(n+1) = n*a(n) + a(n-1) with a(0)=1, a(1)=0.

A001054

[ "0", "1", "-1", "-2", "1", "-3", "-4", "11", "-45", "-496", "22319", "-11070225", "-247076351776", "2735190806339469599", "-675800965841611881515781657825", "-1848444588685310753420392017318175868503407962176" ]

a(n) = a(n-1)*a(n-2) - 1.

A001055

[ "1", "1", "1", "2", "1", "2", "1", "3", "2", "2", "1", "4", "1", "2", "2", "5", "1", "4", "1", "4", "2", "2", "1", "7", "2", "2", "3", "4", "1", "5", "1", "7", "2", "2", "2", "9", "1", "2", "2", "7", "1", "5", "1", "4", "4", "2", "1", "12", "2", "4", "2", "4", "1", "7", "2", "7", "2", "2", "1", "11", "1", "2", "4", "11", "2", "5", "1", "4", "2", "5", "1", "16", "1", "2", "4", "4", "2", "5", "1", "12", "5", "2", "1", "11", "2", "2", "2", "7", "1", "11", "2", "4", "2", "2", "2", "19", "1", "4", "4", "9", "1", "5", "1" ]

The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).

A001056

[ "1", "3", "4", "13", "53", "690", "36571", "25233991", "922832284862", "23286741570717144243", "21489756930695820973683319349467", "500426416062641238759467086706254193219790764168482", "10754042042885415070816603338436200915110904821126871858491675028294447933424899095" ]

a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.

A001057

[ "0", "1", "-1", "2", "-2", "3", "-3", "4", "-4", "5", "-5", "6", "-6", "7", "-7", "8", "-8", "9", "-9", "10", "-10", "11", "-11", "12", "-12", "13", "-13", "14", "-14", "15", "-15", "16", "-16", "17", "-17", "18", "-18", "19", "-19", "20", "-20", "21", "-21", "22", "-22", "23", "-23", "24", "-24", "25", "-25", "26", "-26", "27", "-27", "28", "-28", "29", "-29", "30", "-30", "31", "-31" ]

Canonical enumeration of integers: interleaved positive and negative integers with zero prepended.

A001058

[ "0", "2", "3", "6", "7", "1", "9", "4", "5", "8", "22", "23", "26", "27", "21", "29", "24", "25", "28", "20", "12", "32", "33", "36", "37", "31", "39", "34", "35", "38", "30", "13", "10", "62", "63", "66", "67", "61", "69", "64", "65", "68", "60", "16", "72", "73", "76", "77", "71", "79", "74", "75", "78", "70", "17", "92", "93", "96", "97", "91", "99", "94", "95", "98", "90", "19", "14", "42", "43", "46", "47", "41" ]

1-digit numbers in reverse alphabetical order, then 2-digit numbers, etc.

A001059

[ "1", "1", "5", "59", "1263", "42713", "2094399", "140434335", "12340275539", "1375857855221", "189751578038547", "31714568837559539", "6316261763436325285", "1477890415844440910325", "401400487846091289175217", "125247016772173387008904623", "44493481073675052201518261955" ]

Number of doubly labeled heap-ordered trees.

A001060

[ "2", "5", "7", "12", "19", "31", "50", "81", "131", "212", "343", "555", "898", "1453", "2351", "3804", "6155", "9959", "16114", "26073", "42187", "68260", "110447", "178707", "289154", "467861", "757015", "1224876", "1981891", "3206767", "5188658", "8395425", "13584083", "21979508", "35563591", "57543099", "93106690", "150649789" ]

a(n) = a(n-1) + a(n-2) with a(0)=2, a(1)=5. Sometimes called the Evangelist Sequence.

A001061

[ "8", "3", "1", "5", "9", "0", "6", "7", "4", "2", "88", "38", "58", "98", "68", "78", "48", "28", "18", "80", "30", "83", "33", "53", "93", "63", "73", "43", "23", "13", "81", "31", "51", "91", "61", "71", "41", "21", "11", "85", "35", "55", "95", "65", "75", "45", "25", "15", "50", "89", "39", "59", "99", "69", "79", "49", "29", "19", "90" ]

1-, 2-, 3-, ... digit numbers in alphabetical order in German.

A001062

[ "5", "2", "8", "9", "4", "7", "6", "3", "1", "0", "50", "55", "52", "51", "58", "59", "54", "57", "56", "53", "10", "18", "19", "17", "12", "11", "40", "45", "42", "41", "48", "49", "44", "47", "46", "43", "14", "80", "85", "82", "90", "98", "99", "97", "92", "88", "89", "91", "94", "84", "95", "96", "87", "86", "93", "83", "81", "15", "16", "60", "70", "78", "79", "77", "72", "71", "74", "75", "76", "73", "13", "30" ]

1-, 2-, 3- ... digit numbers in alphabetical order in French (incorrect version, see A187876 for the correct version).

A001063

[ "1", "1", "1", "3", "15", "111", "1131", "15081", "253473", "5220225", "128886921", "3749014251", "126648293391", "4909623331023", "216189866951235", "10718939718977121", "593865369943409601", "36520856568972350721", "2478236630512178688273", "184588566642520989171795", "15020141103053997234030351" ]

E.g.f. satisfies A'(x) = A(x/(1-x)).

A001064

[ "1", "1", "0", "1", "1", "1", "2", "3", "7", "23", "164", "3779", "619779", "2342145005", "1451612289057674", "3399886472013047316638149", "4935316984175079105557291745555191750431", "16779517449593082173916263081219908459297087421776218065830849893" ]

a(n) = a(n-1)*a(n-2) + a(n-3).

A001065

[ "0", "1", "1", "3", "1", "6", "1", "7", "4", "8", "1", "16", "1", "10", "9", "15", "1", "21", "1", "22", "11", "14", "1", "36", "6", "16", "13", "28", "1", "42", "1", "31", "15", "20", "13", "55", "1", "22", "17", "50", "1", "54", "1", "40", "33", "26", "1", "76", "8", "43", "21", "46", "1", "66", "17", "64", "23", "32", "1", "108", "1", "34", "41", "63", "19", "78", "1", "58", "27", "74", "1", "123", "1", "40", "49", "64", "19", "90", "1", "106" ]

Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

A001066

[ "3", "6", "8", "10", "14", "15", "16", "20", "21", "24", "28", "30", "35", "36", "42", "45", "48", "52", "55", "56", "63", "66", "70", "72", "78", "80", "90", "91", "96", "99", "104", "105", "110", "120", "126", "132", "133", "136", "143", "153", "156", "160", "168", "171", "182", "190", "195", "198", "210", "224", "231", "240", "248", "253", "255", "266", "272", "276", "286", "288", "300", "306" ]

Dimensions (sorted, with duplicates removed) of real simple Lie algebras.

A001067

[ "1", "-1", "1", "-1", "1", "-691", "1", "-3617", "43867", "-174611", "77683", "-236364091", "657931", "-3392780147", "1723168255201", "-7709321041217", "151628697551", "-26315271553053477373", "154210205991661", "-261082718496449122051", "1520097643918070802691", "-2530297234481911294093" ]

Numerator of Bernoulli(2*n)/(2*n).

A001068

[ "0", "1", "2", "3", "5", "6", "7", "8", "10", "11", "12", "13", "15", "16", "17", "18", "20", "21", "22", "23", "25", "26", "27", "28", "30", "31", "32", "33", "35", "36", "37", "38", "40", "41", "42", "43", "45", "46", "47", "48", "50", "51", "52", "53", "55", "56", "57", "58", "60", "61", "62", "63", "65", "66", "67", "68", "70", "71", "72", "73", "75", "76", "77", "78", "80", "81", "82", "83", "85", "86", "87", "88" ]

a(n) = floor(5*n/4), numbers that are congruent to {0, 1, 2, 3} mod 5.

A001069

[ "0", "1", "1", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "2", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3", "3" ]

Log2*(n) (version 2): take log_2 of n this many times to get a number < 2.

A001070

[ "0", "0", "1", "2", "36", "4704", "8501760", "267533746176", "188809932117639168", "3790336726450693283512320" ]

Number of normalized Latin squares with second row even.

A001071

[ "2", "1", "4", "10", "36", "108", "392", "1363", "5000", "18223", "67792", "252938", "952540", "3602478", "13699554", "52296713", "200406388", "770411478", "2970401696", "11482395526", "44491881090", "172766311857", "672186650116" ]

Number of one-sided chessboard polyominoes with n cells.

A001072

[ "1", "1", "3", "4", "11", "23", "63", "159", "459", "1331", "4083", "12750" ]

Number of minimally 2-edge-connected non-isomorphic graphs with n nodes.

A001073

[ "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "1", "0", "1", "2", "1", "4", "1", "6", "1", "8", "2", "0", "2", "2", "2", "4", "2", "6", "2", "8", "3", "0", "3", "2", "3", "4", "3", "6", "3", "8", "4", "0", "4", "2", "4", "4", "4", "6", "4", "8", "5", "0", "5", "2", "5", "4", "5", "6", "5", "8", "6", "0", "6", "2", "6", "4", "6", "6", "6", "8", "7", "0", "7", "2", "7", "4", "7", "6", "7", "8", "8", "0", "8", "2", "8", "4", "8", "6", "8", "8", "9", "0", "9", "2", "9", "4", "9", "6", "9", "8", "1", "0", "0", "1", "0", "3" ]

Label a 1-cm ruler with digits 1 cm wide.

A001074

[ "1", "4", "7", "8", "9", "13", "19", "25", "27", "28", "31", "32", "36", "37", "43", "49", "52", "56", "61", "63", "64", "67", "72", "73", "76", "79", "91", "97", "100", "103", "104", "108", "109", "117", "121", "124", "125", "127", "133", "139", "148", "151", "152", "157", "163", "169", "171", "172", "175", "181", "189", "193", "196", "199", "200", "211", "216", "217" ]

Numbers m such that Sum_{k=0..m-1} exp(2*Pi*i*k^3/m) != 0.

A001075

[ "1", "2", "7", "26", "97", "362", "1351", "5042", "18817", "70226", "262087", "978122", "3650401", "13623482", "50843527", "189750626", "708158977", "2642885282", "9863382151", "36810643322", "137379191137", "512706121226", "1913445293767", "7141075053842", "26650854921601", "99462344632562", "371198523608647" ]

a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - a(n-2).

A001076

[ "0", "1", "4", "17", "72", "305", "1292", "5473", "23184", "98209", "416020", "1762289", "7465176", "31622993", "133957148", "567451585", "2403763488", "10182505537", "43133785636", "182717648081", "774004377960", "3278735159921", "13888945017644", "58834515230497", "249227005939632", "1055742538989025" ]

Denominators of continued fraction convergents to sqrt(5).

A001077

[ "1", "2", "9", "38", "161", "682", "2889", "12238", "51841", "219602", "930249", "3940598", "16692641", "70711162", "299537289", "1268860318", "5374978561", "22768774562", "96450076809", "408569081798", "1730726404001", "7331474697802", "31056625195209" ]

Numerators of continued fraction convergents to sqrt(5).

A001078

[ "0", "2", "20", "198", "1960", "19402", "192060", "1901198", "18819920", "186298002", "1844160100", "18255302998", "180708869880", "1788833395802", "17707625088140", "175287417485598", "1735166549767840", "17176378080192802", "170028614252160180", "1683109764441408998" ]

a(n) = 10*a(n-1) - a(n-2) with a(0) = 0, a(1) = 2.

A001079

[ "1", "5", "49", "485", "4801", "47525", "470449", "4656965", "46099201", "456335045", "4517251249", "44716177445", "442644523201", "4381729054565", "43374646022449", "429364731169925", "4250272665676801", "42073361925598085", "416483346590304049" ]

a(n) = 10*a(n-1) - a(n-2); a(0) = 1, a(1) = 5.

A001080

[ "0", "3", "48", "765", "12192", "194307", "3096720", "49353213", "786554688", "12535521795", "199781794032", "3183973182717", "50743789129440", "808716652888323", "12888722657083728", "205410845860451325", "3273684811110137472", "52173546131901748227", "831503053299317834160" ]

a(n) = 16*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.

A001081

[ "1", "8", "127", "2024", "32257", "514088", "8193151", "130576328", "2081028097", "33165873224", "528572943487", "8424001222568", "134255446617601", "2139663144659048", "34100354867927167", "543466014742175624", "8661355881006882817" ]

a(n) = 16*a(n-1) - a(n-2).

A001082

[ "0", "1", "5", "8", "16", "21", "33", "40", "56", "65", "85", "96", "120", "133", "161", "176", "208", "225", "261", "280", "320", "341", "385", "408", "456", "481", "533", "560", "616", "645", "705", "736", "800", "833", "901", "936", "1008", "1045", "1121", "1160", "1240", "1281", "1365", "1408", "1496", "1541", "1633", "1680", "1776", "1825", "1925", "1976" ]

Generalized octagonal numbers: k*(3*k-2), k=0, +- 1, +- 2, +-3, ...

A001083

[ "1", "2", "2", "3", "5", "7", "10", "15", "23", "34", "50", "75", "113", "170", "255", "382", "574", "863", "1293", "1937", "2903", "4353", "6526", "9789", "14688", "22029", "33051", "49577", "74379", "111580", "167388", "251090", "376631", "564932", "847376", "1271059", "1906628", "2859984" ]

Length of one version of Kolakoski sequence {A000002(i)} at n-th growth stage.

A001084

[ "0", "3", "60", "1197", "23880", "476403", "9504180", "189607197", "3782639760", "75463188003", "1505481120300", "30034159217997", "599177703239640", "11953519905574803", "238471220408256420", "4757470888259553597", "94910946544782815520", "1893461460007396756803", "37774318253603152320540" ]

a(n) = 20*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.

A001085

[ "1", "10", "199", "3970", "79201", "1580050", "31521799", "628855930", "12545596801", "250283080090", "4993116004999", "99612037019890", "1987247624392801", "39645340450836130", "790919561392329799", "15778745887395759850", "314783998186522867201", "6279901217843061584170" ]

a(n) = 20*a(n-1) - a(n-2).

A001086

[ "0", "3", "4", "1", "1", "5", "2", "168", "46793", "1", "7", "1", "51", "1", "7", "1", "6", "2", "1", "1", "1", "10", "1", "2", "10", "1", "2", "11", "16", "3", "1", "1", "1", "1", "4", "1", "1", "3", "1", "1", "5", "5", "25", "1", "34", "10", "2", "18", "10", "585", "1", "2", "3", "1", "1", "440", "1", "1", "7", "2", "1", "4", "6", "16", "5", "2", "3", "2", "5", "1", "1", "77", "1" ]

Continued fraction associated with y(y+1) = x(x^2 -1).

A001087

[ "1", "2", "3", "4", "5", "7", "11", "12", "13", "18", "26", "31", "49", "62", "80", "81", "82", "101", "126", "167", "215", "295", "417", "436", "602", "887", "1371", "1454", "2332", "2479", "2645", "2646" ]

Related to S(n), the number of self-dual monotone Boolean functions of n variables (A001206): 2^n-th term is S(n).

A001088

[ "1", "1", "2", "4", "16", "32", "192", "768", "4608", "18432", "184320", "737280", "8847360", "53084160", "424673280", "3397386240", "54358179840", "326149079040", "5870683422720", "46965467381760", "563585608581120", "5635856085811200", "123988833887846400", "991910671102771200", "19838213422055424000" ]

Product of totient function: a(n) = Product_{k=1..n} phi(k) (cf. A000010).

A001089

[ "0", "0", "0", "0", "3", "24", "133", "635", "2807", "11864", "48756", "196707", "783750", "3095708", "12152855", "47500635", "185082495", "719559600", "2793121080", "10830450780", "41965864794", "162539516448", "629399492330", "2437072038302", "9437097796918" ]

Number of permutations of [n] containing exactly 2 increasing subsequences of length 3.

A001090

[ "0", "1", "8", "63", "496", "3905", "30744", "242047", "1905632", "15003009", "118118440", "929944511", "7321437648", "57641556673", "453811015736", "3572846569215", "28128961537984", "221458845734657", "1743541804339272", "13726875588979519", "108071462907496880", "850844827670995521", "6698687158460467288" ]

a(n) = 8*a(n-1) - a(n-2); a(0) = 0, a(1) = 1.

A001091

[ "1", "4", "31", "244", "1921", "15124", "119071", "937444", "7380481", "58106404", "457470751", "3601659604", "28355806081", "223244789044", "1757602506271", "13837575261124", "108942999582721", "857706421400644", "6752708371622431", "53163960551578804" ]

a(n) = 8*a(n-1) - a(n-2); a(0) = 1, a(1) = 4.

A001092

[ "1", "2", "3", "4", "5", "7", "9", "11", "13", "17", "19", "23", "25", "27", "29", "31", "41", "43", "47", "49", "59", "61", "71", "73", "79", "81", "83", "101", "103", "107", "109", "125", "127", "137", "139", "149", "151", "167", "169", "179", "181", "191", "193", "197", "199", "227", "229", "239", "241", "243", "269", "271", "281", "283", "311", "313", "347", "349", "359" ]

Union of all numbers {p, q} where p and q are both primes or powers of primes and q = p+2.

A001093

[ "0", "1", "2", "9", "28", "65", "126", "217", "344", "513", "730", "1001", "1332", "1729", "2198", "2745", "3376", "4097", "4914", "5833", "6860", "8001", "9262", "10649", "12168", "13825", "15626", "17577", "19684", "21953", "24390", "27001", "29792", "32769", "35938", "39305", "42876", "46657", "50654", "54873", "59320" ]

a(n) = n^3 + 1.

A001094

[ "0", "1", "2", "3", "28", "125", "366", "847", "1688", "3033", "5050", "7931", "11892", "17173", "24038", "32775", "43696", "57137", "73458", "93043", "116300", "143661", "175582", "212543", "255048", "303625", "358826", "421227", "491428", "570053", "657750", "755191", "863072", "982113", "1113058" ]

a(n) = n + n*(n-1)*(n-2)*(n-3).

A001095

[ "0", "1", "2", "3", "4", "125", "726", "2527", "6728", "15129", "30250", "55451", "95052", "154453", "240254", "360375", "524176", "742577", "1028178", "1395379", "1860500", "2441901", "3160102", "4037903", "5100504", "6375625", "7893626", "9687627", "11793628", "14250629", "17100750", "20389351" ]

a(n) = n + n*(n-1)*(n-2)*(n-3)*(n-4).

A001096

[ "0", "1", "2", "3", "4", "5", "726", "5047", "20168", "60489", "151210", "332651", "665292", "1235533", "2162174", "3603615", "5765776", "8910737", "13366098", "19535059", "27907220", "39070101", "53721382", "72681863", "96909144", "127512025", "165765626", "213127227", "271252828", "342014429" ]

a(n) = n + n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5).

A001097

[ "3", "5", "7", "11", "13", "17", "19", "29", "31", "41", "43", "59", "61", "71", "73", "101", "103", "107", "109", "137", "139", "149", "151", "179", "181", "191", "193", "197", "199", "227", "229", "239", "241", "269", "271", "281", "283", "311", "313", "347", "349", "419", "421", "431", "433", "461", "463", "521", "523", "569", "571", "599", "601", "617", "619", "641", "643" ]

Twin primes.

A001098

[ "1", "10", "10100", "100111011101000", "101101100001100111010010100000100111001010010000" ]

Multiply previous term by 2 and write in binary.

A001099

[ "1", "3", "24", "232", "2893", "43763", "779780", "15997436", "371423053", "9628576947", "275683093664", "8640417354592", "294234689237661", "10817772136320355", "427076118244539020", "18019667955465012596", "809220593930871751581", "38537187481365665823843", "1939882468178947923300136" ]

a(n) = n^n - a(n-1), with a(1) = 1.

A001100

[ "1", "0", "2", "0", "4", "2", "2", "10", "10", "2", "14", "40", "48", "16", "2", "90", "230", "256", "120", "22", "2", "646", "1580", "1670", "888", "226", "28", "2", "5242", "12434", "12846", "7198", "2198", "366", "34", "2", "47622", "110320", "112820", "64968", "22120", "4448", "540", "40", "2", "479306", "1090270", "1108612", "650644", "236968", "54304", "7900", "748", "46", "2" ]

Triangle read by rows: T(n,k) = number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1.