0,3

The number of balls in a triangular pyramid in which each edge contains n+1 balls. The sum of the first n triangular numbers (A000217).

Also (1/6)*(n^3+3*n^2+2*n) is the number of ways to color vertices of a triangle using <= n colors, allowing rotations and reflections. Group is the dihedral group D_6 with cycle index (x1^3+2*x3+3*x1*x2)/6.

Also the convolution of the natural numbers with themselves - Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001

Connected with the Eulerian numbers (1,4,1) via 1*a(x-2)+4*a(x-1)+1*a(x) = x^3. - Gottfried Helms (helms(AT)uni-kassel.de), Apr 15 2002

a(n) = sum |i-j| for all 1 <= i <= j <= n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 05 2002

a(n) = sum of the all possible products p*q where (p,q) are ordered pairs and p+q = n+1. a(5) = 5 + 8 + 9 + 8 + 5 = 35. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 29 2003

Number of labeled graphs on n+3 nodes that are triangles. - Jon Perry (perry(AT)globalnet.co.uk), Jun 14 2003

Number of permutations of n+3 which have exactly 1 descent and avoid the pattern 1324. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Nov 05 2004

Schlaefli symbol for this polyhedron: {3,3}

Transform of n^2 under the Riordan array (1/(1-x^2),x). - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005

a(n) = -A108299(n+5,6) = A108299(n+6,7). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005

a(n) = -A110555(n+4,3). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005

a(n) is a perfect square only for n = {1, 2, 48}. a(48) = 19600 = 140^2. - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 24 2006

a(n+1) is the number of terms in the expansion of (a_1+a_2+a_3+a_4)^n - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Feb 12 2007. (Corrected by Graeme McRae (g_m(AT)mcraefamily.com), Aug 28 2007)

This is also the average "permutation entropy", sum((pi(n)-n)^2)/n!, over the set of all possible n! permutations pi. - Jeff Boscole (jazzerciser(AT)hotmail.com), Mar 20 2007

a(n)=diff(S(n,x),x)|_{x=2}. First derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. W. Lang, Apr 04 2007.

If X is an n-set and Y a fixed (n-1)-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Aug 15 2007

Number of n-permutations (n=4) of 2 objects u, v, with repetition allowed, containing exactly three (3) u's. Example: a(2)=4 because we have: uuuv, uuvu, uvuu and vuuu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2008

Complement of A145397; A023533(a(n))=1; A014306(a(n))=0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 14 2008]

Equals row sums of triangle A152205 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]

a(n) is the number of gifts received from the lyricist's true love up to and including day n in the song "The Twelve Days of Christmas". a(12)=364, almost the number of days in the year. [From Bernard Hill (bernard(AT)braeburn.co.uk), Dec 05 2008]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009: (Start)

Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF2 denominators of A156925. See A157703 for background information.

(End)

Starting with "1" = row sums of triangle A158823 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]

Wiener index of the path graph P_n [From Eric W. Weisstein (eric(AT)weisstein.com), Apr 30 2009]

Contribution from Peter Luschny (peter(AT)luschny.de), Jul 14 2009: (Start)

This is a 'Matryoshka doll' sequence with alpha=0, the multiplicative counterpart is A000178

seq(add(add(i,i=alpha..k),k=alpha..n),n=alpha..50); (End)

a(n) is the number of non-decreasing, three-element permutations of n distinct numbers. [From Samuel Savitz, Sep 12 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 83.

H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.

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. 4.

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (1).

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

A. Szenes, The combinatorics of the Verlinde formulas (N.J. Hitchin et al., ed.), in Vector bundles in algebraic geometry, Cambridge, 1995.

D. Wells, The Penguin Dictionary of Curious and interesting Numbers, pp. 126-7 Penguin Books 1987.

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

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

Milan Janjic, Two Enumerative Functions

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

O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.

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

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

R. Jovanovic, First 2500 Tetrahedral numbers

Hyun Kwang Kim, On Regular Polytope Numbers

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

N. J. A. Sloane, Illustration of initial terms

N. J. A. Sloane, Pyramid of 20 balls corresponding to a(3)=20.

G. Villemin's Almanach of Numbers, Nombres Tetraedriques

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).

Index entries for "core" sequences

Eric Weisstein's World of Mathematics, Wiener Index [From Eric W. Weisstein (eric(AT)weisstein.com), Apr 30 2009]

Partial sums of the triangular numbers (A000217).

G.f.: x/(1-x)^4. a(-4-n)=-a(n).

a(n)=(n+3)/n*a(n-1) - Ralf Stephan, Apr 26 2003

Sums of three consecutive terms give A006003. - Ralf Stephan, Apr 26 2003

a(n)=C[1, 2, ]+C[2, 2]+...+C[n-1, 2]+C[n, 2]; n=5: a(5)=0+1+3+6+10=20. - Labos E. (labos(AT)ana.sote.hu), May 09 2003

a(n)=sum{k=0..n, k(n-k)} (offset 1). - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003

Determinant of the n X n symmetric Pascal matrix M_(i, j)=C(i+j+2, i) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2003

The sum of a series constructed by the products of the index and the length of the series (n) minus the index (i): a(n) = sum[i(n-i)]. Also the sum of n terms of A000217. - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005

a(n)=sum{k=0..floor((n-1)/2), (n-2k)^2} [offset 0]; a(n+1)=sum{k=0..n, k^2*(1-(-1)^(n+k-1))/2} [offset 0]; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005

C(3+n,3)-C(2+n,2) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2006

Values of the Verlinde formula for SL_2, with g=2: a(n)=sum(j=1, n-1, n/(2*sin^2(j*Pi/n))) - Simone Severini (ss54(AT)york.ac.uk), Sep 25 2006

a(n) = Sum[ Sum[ k, {k,1,m} ], {m,1,n} ]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 28 2006

a(n)=Sum{k=1..n} binomial(n*k+1,n*k-1), with a(0)=0. - Paolo P. Lava (ppl(AT)spl.at), Apr 13 2007

a(n)=numbperm(n,3)/6, n>=2 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

a(n-1) = 1/(1!*2!)*sum {1 <= x_1, x_2 <= n} |det V(x_1,x_2)| = 1/2*sum {1 <= i,j <= n} |i-j|, where V(x_1,x_2} is the Vandermonde matrix of order 2. Column 2 of A133112. - Peter Bala (pbala(AT)toucansurf.com), Sep 13 2007

Starting with "1", = binomial transform of [1, 3, 3, 1,...]; e.g. a(4) = 20 = (1, 3, 3, 1) dot (1, 3, 3, 1) = (1 + 9 + 9 + 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 04 2007

a(n) = A006503(n) - A002378(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 24 2008]

a(0)=0, a(1)=1, a(2)=4, a(3)=10, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>=4. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 18 2008]

sum_{n=1..infinity} 1/a(n) = 3/2, case x=1 in Gradstein-Ryshik 1.513.7. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 27 2009]

E.g.f.:((x^3)/6+x^2+x)*exp(x) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Feb 21 2009]

a(2) = 3*4*5/6 = 10, the number of balls in a pyramid of 3 layers of balls, 6 in a triangle at the bottom, 3 in the middle layer and 1 on top.

Consider the square array

1 2 3 4 5 6...

2 4 6 8 10 12...

3 6 9 12 16 20...

4 8 12 16 20 24...

5 10 15 20 25 30...

...

then a(n) = sum of n-th antidiagonal. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 06 2003

A000292 := n->binomial(n+3, 3);

Or, f:=n->(1/6)*(n^3+3*n^2+2*n);

a:=n->sum ((j+n)*(n+2)/9, j=0..n): seq(a(n), n=0..44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 17 2006

Table[((n^3 - n)/6), {n, 1, 45}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007

ZL := [S, {S=Prod(B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=3..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007

a:=n->sum(numbperm (n, 2)/6, j=0..n): seq(a(n), n=1..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007

seq(numbperm (n, 3)/6, n=2..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

seq(sum(binomial(n, k+1), k=2..2), n=2..33); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2007

a:=n->sum(j^2-j, j=0..n): seq(a(n)/2, n=1..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2008

seq(binomial(n+3, 3)*1^n, n=-1..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2008

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009: (Start)

nmax:=44; for n from 0 to nmax do fz(n):=product((1-m*z)^(n+1-m), m=1..n); c(n):= abs(coeff(fz(n), z, 1)); end do: a:=n-> c(n): seq(a(n), n=0..nmax);

(End)

restart: G(x):=x^3*exp(x): f[0]:=G(x): for n from 1 to 56 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/3!, n=2..46); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]

Rest[FoldList[Plus, 0, Rest[FoldList[Plus, 0, Range[50]]]]]

Table[Sum[(n - i)*i, {i, 0, n}], {n, 1, 55}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]

Rest[FoldList[Plus, 0, Rest[FoldList[Plus, 0, Range[0, 44]]]]] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]

lst = {}; Do[AppendTo[lst, GegenbauerC[n, 2, 1]], {n, -1, 43}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 14 2009]

a=0; Table[(a=n^2-n+a)/2, {n, 90}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 18 2009]

(PARI) a(n)=(n)*(n+1)*(n+2)/6

Sums of 2 consecutive terms give A000330.

a(3n-3)=A006566(n). A000447(n)=a(2n-2). A002492(n)=a(2n+1).

First differences give triangular numbers.

Column 0 of triangle A094415.

Cf. A000217, A001044, A003991, A061552.

Cf. A040977, A133111, A133112.

A152205 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 2009: (Start)

Cf. A156925, A157703.

(End)

A158823 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]

Sequence in context: A138778 A038409 A090579 this_sequence A101552 A038419 A057319

Adjacent sequences: A000289 A000290 A000291 this_sequence A000293 A000294 A000295

nonn,core,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Michael Somos

Corrected PARI program. - Harry J. Smith (hjsmithh(AT)sbcglobal.net), Dec 22 2008

Multiplied g.f. with x to match the offset R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 23 2009