A006331 a(n) = n*(n+1)*(2*n+1)/3.

0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770, 1012, 1300, 1638, 2030, 2480, 2992, 3570, 4218, 4940, 5740, 6622, 7590, 8648, 9800, 11050, 12402, 13860, 15428, 17110, 18910, 20832, 22880, 25058, 27370, 29820, 32412, 35150, 38038, 41080, 44280

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Triangles in rhombic matchstick arrangement of side n.

Maximum accumulated number of electrons at energy level n. - Scott A. Brown, Feb 28 2000

Let M_n denote the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002

Convolution of odds (A005408) and evens (A005843). - Graeme McRae, Jun 06 2006

a(n) is the number of non-monotonic functions with domain {0,1,2} and codomain {0,1,...,n}. - Dennis P. Walsh, Apr 25 2011

For any odd number 2n+1, find Sum_{aJ. M. Bergot, Jul 16 2011

a(n) gives the number of (n+1) X (n+1) symmetric (0,1)-matrices containing three ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012

a(n) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and |w - x| < y. - Clark Kimberling, Jun 02 2012

Partial sums of A001105. - Omar E. Pol, Jan 12 2013

Total number of square diagonals (of any size) in an n X n square grid. - Wesley Ivan Hurt, Mar 24 2015

Number of diagonal attacks of two queens on (n+1) X (n+1) chessboard. - Antal Pinter, Sep 20 2015

a(n) is the minimum value obtainable by partitioning either the set {x in the natural numbers | 1 <= x <= 2n} or the set {x in the natural numbers | 0 <= x <= 2n+1} into pairs, taking the product of all such pairs, and taking the sum of all such products. - Thomas Anton, Oct 21 2020

a(n) is the irregularity of the n-th power of a path of length at least 3*n. (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 16 2023

a(n) is the maximum possible total number of inversions in all rows and all columns of a Latin square of order n+1. - Ivaylo Kortezov, Jun 28 2025

Examples

			For n=2, a(2)=10 since there are 10 non-monotonic functions f from {0,1,2} to {0,1,2}, namely, functions f = <f(1),f(2),f(3)> given by <0,1,0>, <0,2,0>, <0,2,1>, <1,0,1>, <1,0,2>, <1,2,0>, <1,2,1>, <2,0,1>, <2,0,2>, and <2,1,2>. - _Dennis P. Walsh_, Apr 25 2011
Let n=4, 2*n+1 = 9. Since 9 = 1+8 = 3+6 = 5+4 = 7+2, a(4) = 1*8 + 3*6 + 5*4 + 7*2 = 60. - _Vladimir Shevelev_, May 11 2012
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359.
• J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
• Rowan Beckworth, Basic atomic information.
• Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
• P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
• Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 25.
• N. S. S. Gu, H. Prodinger and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat., Vol. 31 (2010), pp. 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2 at n=3.
• JBMO 2025, 29th Junior Balkan Mathematical Olympiad, Problem 4, author: Boris Mihov
• Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, Vol. 6 (1965), circa p. 82.
• 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
• Dennis Walsh, Notes on finite monotonic and non-monotonic functions.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

A row of A132339.

Cf. A002378, A048147, A098077, A163102.

Cf. A005900, A084980, A077415, A112524, A188475, A100178, A035597, A096000.

Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

Programs

Haskell

a006331 n = sum $ zipWith (*) [2*n-1, 2*n-3 .. 1] [2, 4 ..]
-- Reinhard Zumkeller, Feb 11 2012

Magma

[n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

Maple

A006331 := proc(n)
    n*(n+1)*(2*n+1)/3 ;
end proc:
seq(A006331(n),n=0..80) ; # R. J. Mathar, Sep 27 2013

Mathematica

Table[n(n+1)(2n+1)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,2,10,28},50] (* Harvey P. Dale, Apr 12 2013 *)

PARI

a(n)=if(n<0,0,n*(n+1)*(2*n+1)/3)

Formula

G.f.: 2*x*(1 + x)/(1 - x)^4. - Simon Plouffe (in his 1992 dissertation)

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

a(n) = 2*A000330(n) = A002492(n)/2.

a(n) = Sum_{i=0..n} T(i,n-i), array T as in A048147. - N. J. A. Sloane, Dec 11 1999

From the formula for the sum of squares of positive integers 1^2 + 2^2 + 3^2 + ... + n^2 = n*(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum_{k=0..n} 2*k^2 = n*(n+1)*(2*n+1)/3, which is an alternative formula for this sequence. - Mike Warburton, Sep 08 2007

10*a(n) = A016755(n) - A001845(n); since A016755 are odd cubes and A001845 centered octahedral numbers, 10*a(n) are the "odd cubes without their octahedral contents." - Damien Pras, Mar 19 2011

a(n) = sum(a*b), where the summing is over all unordered partitions 2*n+1=a+b. - Vladimir Shevelev, May 11 2012

a(n) = binomial(2*n+2, 3)/2. - Ronan Flatley, Dec 13 2012

a(n) = A000292(n) + A002411(n). - Omar E. Pol, Jan 11 2013

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, with a(0)=0, a(1)=2, a(2)=10, a(3)=28. - Harvey P. Dale, Apr 12 2013

a(n) = A208532(n+1,2). - Philippe Deléham, Dec 05 2013

Sum_{n>0} 1/a(n) = 9 - 12*log(2). - Enrique Pérez Herrero, Dec 03 2014

a(n) = A000292(n-1) + (n+1)*A000217(n). - J. M. Bergot, Sep 02 2015

a(n) = 2*(A000332(n+3) - A000332(n+1)). - Antal Pinter, Sep 20 2015

From Bruno Berselli, May 17 2018: (Start)

a(n) = n*A002378(n) - Sum_{k=0..n-1} A002378(k) for n>0, a(0)=0. Also:

A163102(n) = n*a(n) - Sum_{k=0..n-1} a(k) for n>0, A163102(0)=0. (End)

a(n) = A005900(n) - A000290(n) = A096000(n) - A000578(n+1) = A000578(n+1) - A084980(n+1) = A000578(n+1) - A077415(n)-1 = A112524(n) + 1 = A188475(n) - 1 = A061317(n) - A100178(n) = A035597(n+1) - A006331(n+1). - Bruce J. Nicholson, Jun 24 2018

E.g.f.: (1/3)*exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Jan 05 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi - 9. - Amiram Eldar, Jan 04 2022