0,3

Number of edges in complete graph of order n, K_n.

Number of legal ways to insert a pair of parentheses in a string of n letters. E.g. there are 6 ways for three letters: (a)bc, (ab)c, (abc), a(b)c, a(bc), ab(c). [Proof: there are C(n+2,2) ways to choose where the parentheses might go, but n+1 of them are illegal because the parentheses are adjacent.] Cf. A002415.

For n >= 1 a(n)=n(n+1)/2 is also the genus of a nonsingular curve of degree n+2 like the Fermat curve x^(n+2) + y^(n+2) = 1 - Ahmed Fares (ahmedfares(AT)my_deja.com), Feb 21 2001

From Harnack's theorem (1876), the number of branches of a non-singuliar curve of order n is bounded by a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 29 2002

Number of tiles in the set of double-n dominoes. - Scott A. Brown (scottbrown(AT)neo.rr.com), Sep 24 2002

Number of ways a chain of n non-identical links can be be broken up. This is based on a similar problem in the field of proteomics: the number of ways a peptide of n amino acid residues can be be broken up in a mass spectrometer. In general each amino acid has a different mass, so AB and BC would have different masses. - James Raymond (raymond(AT)unlv.edu), Apr 08 2003

Maximum number of intersections of n+1 lines which may only have 2 lines per intersection point. Maximal number of closed regions when n+1 lines are maximally 2-intersected in given by T(n-1). Using n+1 lines with k>1 parallel lines, the maximum number of 2-intersections is given by T(n)-T(k-1). - Jon Perry (perry(AT)globalnet.co.uk), Jun 11 2003

Number of distinct straight lines that can pass through n points in 3-dimensional space. - Cino Hilliard (hillcino368(AT)gmail.com), Aug 12 2003

Triangular numbers - odd numbers = triangular numbers: 0,1,3,6,10,15,21... - 0,1,3,5,7,9,11... = 0,0,0,1,3,6,10... - Xavier Acloque Oct 31 2003

Centered polygonal numbers are the result of [number of sides * A000217 + 1]. E.g. centered pentagonal numbers (1,6,16,31...)= 5 * (0,1,3,6...) + 1. Centered heptagonal numbers (1,8,22,43...)= 7 * (0,1,3,6...) + 1. - Xavier Acloque Oct 31 2003

Maximum number of lines formed by the intersection of n+1 planes. - Ronald R. King (king_ron(AT)asdk12.org), Mar 29 2004

Number of permutations of [n] which avoid the pattern 132 and have exactly 1 descent. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004

a(n) == 1 mod (n+2) if n is odd and == n/2+2 mod (n+2) if n is even. - Jon Perry (perry(AT)globalnet.co.uk), Dec 16 2004

Number of ways two different numbers can be selected from the set {0,1,2,...,n} without repetition, or, number of ways two different numbers can be selected from the set {1,2,...,n} with repetition.

1, 6, 120 are the only numbers which are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005

a(n) = A108299(n+3,4) = -A108299(n+4,5). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005

In 24-bit RGB color cube, the number of color-lattice-points in r+g+b = n planes at n < 256 equals the triangular numbers. For n = 256, ..., 765 the number of legitimate color partitions is less than A000217(n) because {r,g,b} components cannot exceed 255. For n=256,..,511, the number of non-color partitions are computable with A045943(n-255), while for n = 512-765, the number of color points in r+g+b planes equals A000217(765-n). - Labos E. (labos(AT)ana.sote.hu), Jun 20 2005

A110560/A110561 = numerator/denominator of the coefficients of the exponential generating function. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Jul 27 2005

Binomial transform is {0, 1, 5, 18, 56, 160, 432, ... }, A001793 with one leading zero . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2005

a(n) = A111808(n,2) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 17 2005

Each pair of neighboring terms adds to a perfect square. - Zak Seidov (zakseidov(AT)yahoo.com), Mar 21 2006

a(n)*a(n+1) = A006011(n) = n^2*(n^2-1)/4 = 3*A002415(n) = 1/2*a(n^2+2*n). a(n-1)*a(n) = 1/2*a(n^2-1). - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 13 2006

Number of transpositions in the symmetric group of n+1 letters i.e. the number of permutations that leave all but two elements fixed. - Geoffrey Critzer (geoffreycritzer(AT)yahoo.com), Jun 23 2006

Beginning from a(3), a(n) is the number of way to get a semiprime from n primes. Example: From 2 and 3 the number of semiprimes is 3: 2*2, 3*3, 2*3; from 2 and 3 and 5 the number of semiprimes is 6: 2*2, 3*3, 5*5, 2*3, 2*5, 3*5. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Sep 17 2006

With rho(n):=exp(i*2*Pi/n) (an n-th root of 1) one has, for n>=1, rho(n)^a(n)=(-1)^(n+1). Just use the triviality a(2*k+1)=0(mod (2*k+1)) and a(2*k)=k(mod 2*k).

Comments from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 18 2006: (Start)

a:=n->sum(j + 1,j=-1..n): seq(a(n),n=-1..50);

a:=n->sum(j + 2,j=0..n): seq(a(n),n=-1..51); => A000096 = this sequence + 1*A001477

a:=n->sum(j + 2,j=1..n): seq(a(n),n=0..48); => A055998 = this sequence + 2*A001477

a:=n->sum(j + 2,j=2..n):seq(a(n),n=1..50); => A055999 = this sequence + 3*A001477

a:=n->sum(j + 2,j=3..n):seq(a(n),n=2..52); => A056000 = this sequence + 4*A001477

a:=n->sum(j + 2,j=4..n):seq(a(n),n=3..53); => A056115 = this sequence + 5*A001477

a:=n->sum(j + 2,j=5..n):seq(a(n),n=4..54); => A056119 = this sequence + 6*A001477

a:=n->sum(j + 2,j=6..n):seq(a(n),n=5..50); => A056121 = this sequence + 7*A001477

a:=n->sum(j + 2,j=7..n):seq(a(n),n=6..56); => A056126 = this sequence + 8*A001477

a:=n->sum(j + 2,j=8..n):seq(a(n),n=7..56); => A051942 = this sequence + 9*A001477

a:=n->sum(j + 2,j=9..n):seq(a(n),n=8..59); => A101859 = this sequence + 10*A001477 (End)

a(n) = A126890(n,0). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Dec 30 2006

a(n) is the number of terms in the expansion of (a_1+a_2+a_3)^n - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007

(sqrt(8 a(n) + 1) - 1)/2 = n. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 26 2007

The number of distinct handshakes in a room with n people (n>=2). - Mohammad K. Azarian (azarian(AT)evansville.edu), Apr 12 2007

Equal to the rank (minimal cardinality of a generating set) of the semigroup PT_n\S_n, where PT_n and S_n denote the partial transformation semigroup and symmetric group on [n]. - James East (james.east(AT)latrobe.edu.au), May 03 2007

Gives the total number of triangles found when cevians are drawn from a single vertex on a triangle to the side opposite that vertex, where n=the number of cevians drawn+1. For instance, with 1 cevian drawn, n=1+1=2 and a(n)=2(2+1)/2=3 so there is a total of 3 triangles in the figure. If 2 cevians are drawn from one point to the opposite side, then n=1+2=3 and a(n)=3(3+1)/2=6 so there is a total of 6 triangles in the figure. - Noah Priluck (npriluck(AT)gmail.com), Apr 30 2007

a(n), n>=1, is the number of ways in which n-1 can be written as a sum of three positive integers if representations differing in the order of the terms are considered to be different. In other words a(n),n>=1, is the number of positive integral solutions of the equation x + y + z = n-1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 22 2001

a(n+1), n>=0, is the number of levels with energy n+3/2 (in units of h*f0, with Planck's constant h and the oscillator frequency f0) of the three dimensional isotropic harmonic quantum oscillator. See the comment by A. Murthy above: n=n1+n2+n3 with positive integers and ordered. Proof from the o.g.f. See the A. Messiah reference. W. Lang, Jun 29 2007.

Numbers m>=0 such that round(sqrt(2m+1))-round(sqrt(2m))=1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007

Numbers m>=0 such that ceiling(2*sqrt(2m+1))-1=1+floor(2*sqrt(2m)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007

Numbers m>=0 such that fract(sqrt(2m+1))>1/2 and fract(sqrt(2m))<1/2, where fract(x) is the fractional part of x (i.e. x-floor(x), x>=0). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007

Each term, except for the initial 0, is a sum of digits of terms in A007908. - Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 01 2007

Sequence allows us to find X values of the equation: 8*X^3 + X^2 = Y^2. To find Y values: b(n)=n(n+1)(2n+1)/2. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007

If Y and Z are 3-blocks of an n-set X then, for n>=6, a(n-1) is the number of (n-2)-subsets of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Nov 09 2007

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 109ff.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1.

Tomislav Doslic, Maximum Product Over Partitions Into Distinct Parts, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.8.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Labos E.: On the number of RGB-colors we can distinguish. Partition Spectra. Lecture at 7th Hungarian Conference on Biometry and Biomathematics. Budapest. Jul 06,2005.

A. Messiah, Quantum Mechanics, Vol.1, North Holland, Amsterdam, 1965, p. 457.

J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

T. Trotter, Some Identities for the Triangular Numbers, Journal of Recreational Mathematics, Spring 1973, 6(2).

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 91-3 Penguin Books 1987.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

H. Bottomley, Illustration of initial terms of A000217, A002378

Scott A. Brown, Brown's Math Page, etc.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

S. S. Gupta, Fascinating Triangular Numbers

C. Hamberg, Triangular Numbers Are Everywhere

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 253

Milan Janjic, Two Enumerative Functions

R. Jovanovic, Triangular numbers

R. Jovanovic, First 2500 Triangular numbers

H. K. Kim, "On Regular polytope numbers".

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

J. Koller, Triangular Numbers

A. J. F. Leatherland, Triangle Numbers on Ulam Spiral

Ivars Peterson, Triangular Numbers and Magic Squares.

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

F. Richman, Triangle numbers

James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326

Thesaurus.maths.org, Triangular Numbers

T. Trotter, Some Identities for the Triangular Numbers, J. Rec. Math. vol. 6, no. 2 Spring 1973.

G. Villemin's Almanach of Numbers, Nombres Triangulaires

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (3).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (4).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (5).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (6).

Eric Weisstein's World of Mathematics, Line Line Picking

Eric Weisstein's World of Mathematics, Trinomial Coefficient

Eric Weisstein's World of Mathematics, Wiener Index

Index entries for "core" sequences

Index entries for related partition-counting sequences

a(n)=a(n-1)+n. - Zak Seidov (zakseidov(AT)yahoo.com), Mar 06 2005

a(n) + a(n-1)*a(n+1) = a(n)^2. - Terry Trotter (ttrotter(AT)telesal.net), Apr 08, 2002

a(n)=(-1)^n*sum(k=1, n, (-1)^k*k^2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 29 2002

a(n)=(n+2)/n*a(n-1)

Sum(n=1..infinity, 1/a(n)) = 2. - Jon Perry (perry(AT)globalnet.co.uk), Jul 13 2003

For n>0, a(n)=A001109(n)-(sum_{k=0...n-1}((2k+1)*A001652(n-1-k))) e.g. 10=204-(1*119+3*20+5*3+7*0) - Charlie Marion (charliem(AT)bestweb.net), Jul 18 2003

G.f.: x/(1-x)^3. E.g.f.: exp(x)(x+x^2/2). a(n)=a(-1-n).

With interpolated zeros, this is n(n+2)/8*(1+(-1)^n)/2=sum{k=0..n, sum{j=0..k, floor(k^2/4)}}. - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2003

a(n+1) is the determinant of the n X n symmetric Pascal matrix M_(i, j)=C(i+j+1, i) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2003

a(n)=[(n^3-(n-1)^3)-(n^1-(n-1)^1)]/(2^3-2^1)= (n^3-(n-1)^3-1)/6 - Xavier Acloque Oct 24 2003

a(n) = a(n-1) + (1 + sqrt[1 + 8*a(n-1)])/2. E.g. a(4) = a(3) + (1 + sqrt[1 + 8*a(3)])/2 = 6 + (1 + sqrt[49])/2 = 6+8/2 = 10. This recursive relation is inverted when taking the negative branch of the square root, i.e. a(n) is transformed into a(n-1) rather than a(n+1). - Carl R. White (cyrek(AT)cyreksoft.yorks.com), Nov 04 2003

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

a(n)=a(n-2)+2n-1. - Paul Barry (pbarry(AT)wit.ie), Jul 17 2004

a(n) = Sqrt[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 24 2004

a(n) = Sqrt[Sqrt[Sum[Sum[(i*j)^3, {i, 1, n}], {j, 1, n}]]]. a(n) = Sum[Sum[Sum[(i*j*k)^3, {i, 1, n}], {j, 1, n}], {k, 1, n}]^(1/6) - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 26 2004

a(0)=0, a(1)=1, a(n)=2*a(n-1)-a(n-2)+1 - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 09 2005

a(n) = Sum_{k = 1...n} phi(k)*floor(n/k) = Sum{k = 1...n} A000010(k)*A010766(n, k) (R. Dedekind). - Vladeta Jovovic = (vladeta(AT)Eunet.yu), Feb 05 2004

a(n)=floor((2n+1)^2/8) - Paul Barry (pbarry(AT)wit.ie), May 29 2006

For positive n,we have a(8*a(n))/a(n) = 4*(2n+1)^2 = (4n+2)^2,i.e.,a(A033996(n))/a(n) = 4*A016754(n) = (A016825(n))^2 = A016826(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 29 2006

[a(n)]^2+[a(n+1)]^2=a((n+1)^2) [R B Nelsen, Math Mag 70 (2) (1997) p 130]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 22 2006

a(n) = A023896(n) + A067392(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Mar 02 2007

a(n)=sum(sum(j-k, j=1..n),k=0..n), n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

Sum_{k, 0<=k<=n}a(k)*A039599(n,k)=A002457(n-1), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 10 2007

a(n) = (n+1)^2 - a(n+1) - Eugeny Yakimovitch (Eugeny.Yakimovitch(AT)gmail.com), Jan 21 2008

When n=3, a(3) = 4*3/2 = 6.

Example(a(4)=10): ABCD where A, B, C and D are different links in a chain or different amino acids in a peptide possible fragments: A, B, C, D, AB, ABC, ABCD, BC, BCD, CD = 10

A000217 := proc(n) n*(n+1)/2; end; [ seq(n*(n+1)/2, n=0..100)];

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+1 od: seq(a[n], n=0..50); (Kristof)

[seq (stirling2(n+1, n) , n=1..53)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo), Dec 06 2006

ZL := [S, {S=Prod(B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=2..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2007

a:=n->sum(n+2*j, j=0..n)/4: seq(a(n), n=0..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2007

a:=n->sum(sum(j-k, j=1..n), k=0..n): seq(a(n), n=0..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

seq(sum(mul(gcd(j, k), j=0..n), k=0..n), n=0..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2007

A000217:=-1/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

seq(sum(binomial(n, k+1), k=1..1), n=1..54); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2007

with(combinat):a:=n->sum(fibonacci(2, i), i=0..n): seq(a(n), n=0..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008

Table[Sqrt[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}]], {n, 0, 10}]

Table[(m^2 - m)/2, {m, 54}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2007

Table[Sqrt[StirlingS2[i+1, i]*(-StirlingS1[i+1, i])] , {i, 0, 53}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2007

(PARI) a(n)=n*(n+1)/2

Cf. A007318, A002024, A000096, A000124, A002378, A000292, A000330.

a(2k-1)=A000384(k), a(2k)=A014105(k), k>0.

A diagonal of A008291. a(n) = A110555(n+2,2).

a(n) = A110449(n,0).

Cf. A006011, A002415.

Cf. A000096, A055998, A055999, A056000, A056115, A056119, A056121, A056126, A051942, A101859, A001477.

Cf. A010054.

Cf. A005563, A046092, A001082, A002378, A036666, A062717, A028347, A087475.

Adjacent sequences: A000214 A000215 A000216 this_sequence A000218 A000219 A000220

Sequence in context: A105338 A105339 A089594 this_sequence A105340 A109811 A025747

nonn,core,easy,nice

njas