A002623 Expansion of 1/((1-x)^4*(1+x)).

1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715, 825, 946, 1078, 1222, 1378, 1547, 1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843, 8372, 8924, 9500

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Also a(n) is the number of nondegenerate triangles that can be made from rods of lengths 1 to n+1. - Alfred Bruckstein; corrected by Hans Rudolf Widmer, Nov 02 2023

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

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

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, Aug 08 2000

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

Also Molien series for certain 4-D representation of cyclic group of order 2. - N. J. A. Sloane, Jun 12 2004

From Radu Grigore (radugrigore(AT)gmail.com), Jun 19 2004: (Start)

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:

/\

/\/\

/\/\/\

/\/\/\/\

(End)

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, 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. E.g., 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. - Michael Somos, Oct 22 2006

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

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:

||__

||__|

||__||

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

Columns sums of:

1 3 6 10 15 21 28 ...

1 3 6 10 15 ...

1 3 6 ...

1 ...

---------------------

1 3 7 13 22 34 50 ...

(End)

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, Oct 25 2007

Let b(n) be the number of 4-tuples (w,x,y,z) having all terms in {1,...,n} and 2w=x+y+z+n; then b(n+3) = a(n) for n >= 0. - Clark Kimberling, May 08 2012

a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w >= x+y and x <= y. - Clark Kimberling, Jun 04 2012

Also, number of unlabeled bipartite graphs with two left vertices and n right vertices. - Yavuz Oruc, Jan 14 2018

Also number of triples (x,y,z) with 0 < x <= y <= z <= n + 1, x + y > z. - Ralf Steiner, Feb 06 2020

Bisections A002412 and A016061: a(2*k) = k*(k+1)*(4*k-1)/3! and a(2*k+1) = (k+1)*(k+2)*(4*k+9)/3!, for k >= 0. See the Woolhouse link, II. Solution by Stephen Watson, p. 65, with index shifts. - Mo Li, Apr 02 2020

Also, Wiener index of the square of the path graph P_(n+2). - Allan Bickle, Aug 01 2020

Maximum Wiener index of all maximal 2-degenerate graphs with n+2 vertices. (A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices.) The extremal graphs are squares of paths, so the bound also applies to 2-trees and maximal outerplanar graphs. - Allan Bickle, Sep 15 2022

Examples

			G.f. = 1 + 3*x + 7*x^2 + 13*x^3 + 22*x^4 + 34*x^5 + 50*x^6 + 70*x^7 + 95*x^8 + ...
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
• P. Diaconis, R. L. Graham and B. Sturmfels, Primitive partition identities, in Combinatorics: Paul Erdős is Eighty, Vol. 2, Bolyai Soc. Math. Stud., 2, 1996, pp. 173-192.
• 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).
• 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.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• A. Atmaca and A. Yavuz Oruç, On the size of two families of unlabeled bipartite graphs, AKCE International Journal of Graphs and Combinatorics, 2017.
• M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, Lemma 6 5th line.
• Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Shalosh B. Ekhad and Doron Zeilberger, Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary partitions mod m (and doing MUCH more!), arXiv preprint arXiv:1511.06791 [math.CO], 2015.
• E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
• E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
• Enrique González-Jiménez and Xavier Xarles, On a conjecture of Rudin on squares in Arithmetic Progressions, arXiv preprint arXiv:1301.5122 [math.NT], 2013.
• H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
• M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649.
• M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
• 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
• A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]. See page 79.
• W. Lanssens, B. Demoen, and P.-L. Nguyen, The Diagonal Latin Tableau and the Redundancy of its Disequalities, Report CW 666, July 2014, Department of Computer Science, KU Leuven
• M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.
• P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
• Math StackExchange, cycle index for S_2 X S_4, Apr. 2021
• B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218. See page 217.
• G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=2]
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Giovanni Resta, Illustration for a(8)=70.
• Marko Riedel, The O.g.f. of inequivalent colorings of a 2xN board with C colors in N is (1/2) 1/(1-w)^(C^2) + (1/2) 1/(1+w)^(C(C-1)/2)*1/(1-w)^(C(C+1)/2) by PET
• Sebastian Sassi, Aula Al-Adulrazzaq, Matti Heikinheimo, and Kimmo Tuominen, Fast numerical evaluation of dark matter direct detection event rates, arXiv:2504.19714 [hep-ph], 2025. See p. 7.
• Atsuto Seko, Tutorial: Systematic development of polynomial machine learning potentials for elemental and alloy systems, J. Appl. Phys. (2023) Vol. 133, 011101.
• 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.
• Eric Weisstein's World of Mathematics, Triangle Counting.
• W. S. B. Woolhouse, Problem 2420. On the probability of the number of triangles, Mathematical questions with their solutions, v. 9 (June 1868), pp. 63-65.
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A002620 (first differences), A000292, A001752 (partial sums), A062109 (binomial transf.).

Bisections A002412, A016061.

Cf. also A002717 (a companion sequence), A002727, A006148, A057524, A134446, A014125, A122046, A122047.

Cf. A000217, A006918, A058187, A108561.

The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623 (this sequence), A014125, A122046, A122047, A175724, respectively.

Programs

Maple

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
a := n -> ((-1)^n*3 + 45 + 68*n + 30*n^2 + 4*n^3) / 48:
seq(a(n), n=0..46); # Peter Luschny, Jan 22 2018

Mathematica

CoefficientList[Series[1/((1-x)^3(1-x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,3,7,13,22},50] (* Harvey P. Dale, Jul 19 2011 *)
Table[((2 n^3 + 15 n^2 + 34 n + 45 / 2 + (3/2) (-1)^n) / 24), {n, 0, 100}] (* Vincenzo Librandi, Jan 15 2018 *)
a[ n_] := Floor[(n + 2)*(n + 4)*(2*n + 3)/24]; (* Michael Somos, Feb 19 2024 *)

PARI

{a(n) = (8 + 34/3*n + 5*n^2 + 2/3*n^3) \ 8}; /* Michael Somos, Sep 04 1999 */

PARI

x='x+O('x^50); Vec(1/((1 - x)^3 * (1 - x^2))) \\ Indranil Ghosh, Apr 04 2017

Python

def A002623(n): return ((n+2)*(n+4)*((n<<1)+3)>>3)//3 # Chai Wah Wu, Mar 25 2024

Formula

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

a(n)+a(n+1) = A000292(n+1).

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

Also: a(n) = C(n+3, 3) - a(n-1) with a(0)=1. - Labos Elemer, Apr 26 2003

From Paul Barry, Jul 01 2003: (Start)

a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+3,3).

The signed version 1, -3, 7, ... has the formula:

a(n) = (4*n^3 + 30*n^2 + 68*n + 45)*(-1)^n/48 + 1/16.

This is the partial sums of the signed version of A000292. (End)

From Paul Barry, Jul 21 2003: (Start)

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. (End)

a(n) = a(n - 2) + (n*(n - 1))/2, with n>2, a(1)=0, a(2)=1; a(n) = (4*n^3+6*n^2-4*n+3*(-1)^n-3)/48, with offset 2. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004 (formula simplified by Bruno Berselli, Aug 29 2013)

a(n) = ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4, with offset 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

A002620(n+3) = a(n+1) - a(n). - Michael Somos, Sep 04 1999

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

a(n) = -a(-5-n) for all n in Z. - Michael Somos, Sep 04 2006

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, Feb 18 2007

a(n) = (n+(3+(-1)^n)/2)*(n+(7+(-1)^n)/2)*(2*n+5-2*(-1)^n)/24. - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007 (corrected by Bruno Berselli, Aug 30 2013)

From Johannes W. Meijer, May 20 2011: (Start)

a(n) = A006918(n+1) + A006918(n).

a(n) = A058187(n-2) + 2*A058187(n-1) + A058187(n). (End)

a(0)=1, a(1)=3, a(2)=7, a(3)=13, a(4)=22; for n > 4, a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). - Harvey P. Dale, Jul 19 2011

a(n) = Sum_{i=0..n+2} floor(i/2)*ceiling(i/2). - Bruno Berselli, Aug 30 2013

a(n) = 15/16 + (1/16)*(-1)^n + (17/12)*n + (5/8)*n^2 + (1/12)*n^3. - Robert Israel, Jul 07 2014

a(n) = Sum_{i=0..n+2} (n+1-i)*floor(i/2+1). - Bruno Berselli, Apr 04 2017

a(n) = 1 + floor((2*n^3 + 15*n^2 + 34*n) / 24). - Allan Bickle, Aug 01 2020

E.g.f.: ((24 + 51*x + 21*x^2 + 2*x^3)*cosh(x) + (21 + 51*x + 21*x^2 + 2*x^3)*sinh(x))/24. - Stefano Spezia, Jun 02 2021