A008911 a(n) = n^2*(n^2 - 1)/6.

0, 0, 2, 12, 40, 100, 210, 392, 672, 1080, 1650, 2420, 3432, 4732, 6370, 8400, 10880, 13872, 17442, 21660, 26600, 32340, 38962, 46552, 55200, 65000, 76050, 88452, 102312, 117740, 134850, 153760, 174592, 197472, 222530, 249900, 279720, 312132

Offset

0,3

Comments

Number of equilateral triangles in rhombic portion of side n+1 in hexagonal lattice.

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Sum of squared distances on n X n board between n queens each on its own row and column. - Zak Seidov, Sep 04 2002

For queens "each on its column and row" the sum of squared distances does not depend on configuration - while sum of distances does.

Number of cycles of length 3 in the bishop's graph associated with an (n+1) X (n+1) chessboard. - Anton Voropaev (anton.n.voropaev(AT)gmail.com), Feb 01 2009

a(n) is number of ways to place 3 queens on an (n+1) X (n+1) chessboard so that they diagonally attack each other exactly 3 times. The maximal possible attack number, p=binomial(k,2)=3 for k=3 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - Antal Pinter, Dec 27 2015

From a(1), convolution of the oblong numbers (A002378) with the odd numbers (A005408). - Bruno Berselli, Oct 24 2016

Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, p and |q-p|. - Wesley Ivan Hurt, Apr 15 2018

References

James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 6).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics, 1999.

James Propp, Updated article, 2009.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: 2*x^2*(1+x)/(1-x)^5.

a(n) = 2*A002415(n) = A047928(n-1)/6 = A083374(n-1)/3 = A006011(n)*2/3. - Zerinvary Lajos, May 09 2007

a(n) = n*binomial(n+1,3). - Martin Renner, Apr 03 2011

a(n+1) = (n+1)*A000292(n). - Tom Copeland, Sep 13 2011

From G. C. Greubel, Sep 13 2019: (Start)

a(n) = binomial(n^2,2)/3.

E.g.f.: x^2*(6 + 6*x + x^2)*exp(x)/6. (End)

From Amiram Eldar, Nov 02 2021: (Start)

Sum_{n>=2} 1/a(n) = 21/2 - Pi^2.

Sum_{n>=2} (-1)^n/a(n) = (Pi^2 - 9)/2. (End)

a(n) = Sum_{j=0..n-1} binomial(n,2) + binomial(n,3). - Detlef Meya, Jan 20 2024

Example

a(2)=2 because on 2 X 2 board queens "each on its column and row" may take only two angular cells, then squared distance is 1^2+1^2=2. a(3)=12 because on 3 X 3 board queens "each on its column and row" make only two essentially distinct configurations: {1,2,3}, {1,3,2} and in both cases the sum of three squared distances is 12.
G.f.: 2*x^2 + 12*x^3 + 40*x^4 + 100*x^5 + 210*x^6 + 392*x^7 + 672*x^8 + ...

Maple

A008911 := n->n^2*(n^2-1)/6; seq(A008911(n), n=0..40);

Mathematica

a[m_]:= m^2(m^2-1)/6;
Binomial[Range[0, 40]^2, 2]/3 (* G. C. Greubel, Sep 13 2019 *)

Prog

(PARI) {a(n) = n^2*(n^2-1)/6};
(Magma) [n^2*(n^2-1)/6: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011
(Sage) [n^2*(n^2-1)/6 for n in (0..40)] # G. C. Greubel, Sep 13 2019
(GAP) List([0..40], n-> n^2*(n^2-1)/6); # G. C. Greubel, Sep 13 2019

Crossrefs

Cf. A002415, A006011, A047928, A083374.

Cf. A002378, A005408.

Convolution of the oblong numbers with the even numbers: A033488.

Sequence in context: A363123 A168057 A290131 * A005719 A143126 A118417

Adjacent sequences: A008908 A008909 A008910 * A008912 A008913 A008914

Keyword

nonn,easy

Author

N. J. A. Sloane, R. K. Guy

Status

approved