0,2

Also number of nondegenerate triangles that can be made from rods of length 1,2,3,4,...,n (Alfred Bruckstein, freddy(AT)cs.technion.ac.il).

Also number of circumscribable (or escrible) quadrilaterals that can be made from rods of length 1,2,3,4,....,n (xpolakis(AT)otenet.gr, Antreas P. Hatzipolakis)

Also number of 2 X n binary matrices up to row and column permutation [see the link: Binary matrices up to row and column permutations ]. - Vladeta Jovovic (vladeta(AT)eunet.rs).

Also partial sum of alternate triangular numbers (1, 3, 1+6, 3+10, 1+6+15, 3+10+21, etc.); and also number of triangles pointing in opposite direction to largest triangle in triangular matchstick arrangement of side n+2 [cf. A002717, also the Larsen article] - Henry Bottomley (se16(AT)btinternet.com), Aug 08 2000

Also Molien series for certain 4-D representation of cyclic group of order 2.

Number of non-congruent non-parallelogram trapezoids with positive integer sides (trapezints) and perimeter 2n+5. Also with perimeter 2n+8. - Michael Somos May 12 2005

a(n) = A108561(n+4,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 10 2005

Also number of nonisomorphic planes with n points and 2 lines. E.g. a(0)=1 because with no points, we just have two empty lines. a(1)=3 because the one point may belong to 0, 1 or 2 lines. a(2)=7 because there are 7 ways to determine which of 2 points belong to which of 2 lines, up to isomorphism, i.e. up to a bijection f on the sets of points and a bijection g on the sets of lines, such that A belongs to a iff f(A) belongs to g(a). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005

a(n-2) is the number of ways to pick two non-overlapping subwords of equal nonzero length from a word of length n. - Michael Somos Oct 22 2006

Partial sums of A002620. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007

Comment from Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007: (Start) Also number of squares of any size in a staircase of n steps built with unit squares:

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For a staircase of 3 steps 6 squares of size 1 and 1 square of size 2, hence c(3)=7. (End)

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).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.

P. Diaconis, R. L. Graham and B. Sturmfels, Primitive partition identities, in Combinatorics: Paul Erdos is Eighty, Vol. 2, Bolyai Soc. Math. Stud., 2, 1996, pp. 173-192.

E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.

H. Gupta, Partitions of $j$-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.

M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.

I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.

J. Silverman, V. E. Vickers and J. M. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853.

Brian OSullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=2]

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for sequences related to linear recurrences with constant coefficients

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 203

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 413

Vladeta Jovovic, Binary matrices up to row and column permutations

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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.

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

Index entries for Molien series

a(n+1) = a(n) + {(k-1)*k if n=2*k} or {k*k if n=2*k+1}.

a(n) = a(n-2)+A000217(n+1) = A002717(n+2)-A000292(n+1)

Also: a(n)=C(n, 3)-a(n-1) with a(0)=0 and A002623(0)=a(3); a(n)=A002623(n-3). - Labos E. (labos(AT)ana.hu), Apr 26 2003

Sum{k=0..n, (-1)^(n-k)C(k+3, 3) } - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003

The signed version 1, -3, 7, .... has a(n)=(4n^3+30n^2+68n+45)(-1)^n/48+1/16. This is the partial sums of the signed version of A000292. - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003

a(n)=sum{k=0..n, floor((k+2)^2/4)}; a(n)=sum{k=0..n, sum{j=0..k, sum{i=0..j, (1+(-1)^i)/2 }}}; - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003

Ordered union of A002412(n+1) and A016061(n+1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 13 2003

Comment from Radu Grigore (radugrigore(AT)gmail.com), Jun 19 2004: a(n) = floor( (n+2)*(n+4)*(2n+3) / 24). E.g. a(2) = floor(4*6*7/24) = 7 because there are 7 upside down triangles (6 of size one and 1 of size two) in the matchstick figure:

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a[n] == a[n - 2] + (n*(n - 1))/2, a[1] == 0, a[2] == 1; (3*(-1)^n - 3*(-1)^(2*n) + 8*n - 12*(-1)^(2*n)*n + 12* n^2 - 6*(-1)^(2*n)*n^2 + 4*n^3)/48 - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004

a(n) = ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4; a(n) = sum(floor(i^2/4), i=2..n+1) - Jerry W. Lewis (JLewis(AT)wyeth.com), Mar 23 2005

a(n) = 2*a(n-1) - a(n-2) + 1 + floor(n/2). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005

a(-5-n)=-a(n). - Michael Somos Sep 04 2006

Euler transform of length 2 sequence [ 3, 1]. - Michael Somos Sep 04 2006

a(n) = ceiling( (n+3)(n+1)(2n+7) ) - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007

Let P(i,k) be the number of integer partitions of n into k parts, then with k=2 we have a(n) = sum_{m=1}^{n} sum_{i=k}^{m} P(i,k). For k=1 we get A000217 = triangular numbers. - Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 18 2007

a(n) = (n+(1+(-1)^n)/2)*(n + (5+(-1)^n)/2)*(2*n+3+2*(-1)^n)/24. - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007

a(n) = sum of row n+1 of triangle A134446. Also, binomial transform of [1, 2, 2, 0, 1, -2, 4, -8, 16, -32,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2007

a(5- 2)=a(3)=13 since the word 12345 of length 5 has the following subword pairs: 1,2; 1,3; 1,4; 1,5; 2,3; 2,4; 2,5; 3,4; 3,5; 4,5; 12,34; 12,45; 23,45.

A002623 := n->(1/16)*(1+(-1)^n)+(n+1)/8+binomial(n+2, 2)/4+binomial(n+3, 3)/2;

seq( ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4, n=1..47); (Lewis)

A002623:=1/(z+1)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]

(PARI) a(n)=(8+34/3*n+5*n^2+2/3*n^3)\8

A002620(n+3)=a(n+1)-a(n).

Cf. A002717 (a companion sequence), A002727, A006148.

Partial sums of A002620. Sums of 2 consecutive terms give A000292.

Cf. A000217, A057524, A134446.

Adjacent sequences: A002620 A002621 A002622 this_sequence A002624 A002625 A002626

Sequence in context: A136219 A078582 A051336 this_sequence A081662 A091652 A134197

nonn,easy,nice

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

PARI formula and more terms from Michael Somos