0,3

Number of inequivalent ways to color vertices of a square using <= n colors, allowing rotations and reflections. Group is dihedral group D_8 of order 8 with cycle index (1/8)*(x1^4+2*x4+3*x2^2+2*x1^2*x2); setting all x_i = n gives the formula a(n) = (1/8)*(n^4+2*n+3*n^2+2*n^3).

Number of semi-magic 3 X 3 squares with a line sum of n-1. That is, 3 X 3 matrices of nonnegative integers such that row sums and column sums are all equal to n-1. - Peter Bertok (peter(AT)bertok.com), Jan 12 2002. See A005045 for another version.

Also the coefficient h_2 of x^{n-3} in the shelling polynomial h(x)=h_0*x^n-1 + h_1*x^n-2 + h_2*x^n-3 + ... + h_n-1 for the independence complex of the cycle matroid of the complete graph K_n on n vertices (n>=2) - Woong Kook (andrewk(AT)math.uri.edu), Nov 01 2006

If X is an n-set and Y a fixed 3-subset of X then a(n-4) is equal to the number of 5-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Jul 30 2007

A. Bj\"orner, The homology and shellability of matroids and geometric lattices, in Matroid Applications (ed. N. White), Encyclopedia of Mathematics and Its Applications, 40, Cambridge Univ. Press 1992.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(3,r).

I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables. Ann. Statist. 4 (1976), no. 6, 1159-1189.

D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.

R. P. Stanley, Enumerative Combinatorics I, p. 292.

Warburton, Henry. "On Self-Repeating Series." Transactions of the Cambridge Philosophical Society, Vol. 9, 471-486, 1856.

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

G. E. Andrews, P. Paule, A. Riese and V. Strehl, MacMahon's partition analysis V. Bijections, recursions, and magic squares, p. 37.

Matthias Beck, The number of "magic" squares and hypercubes

P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials

Milan Janjic, Two Enumerative Functions

Neven Juric, Illustration of the 55 3 X 3 matrices

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.

G.f.: x(1+x+x^2)/(1-x)^5. a(n) = 3*binomial(n+2, 4)+binomial(n+1, 2).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3 - Warut Roonguthai (warut822(AT)yahoo.com) Dec 13 1999

a(n) = Sum [ Sum ( 1 + Sum (3*n) ) ]. - Xavier Acloque, Jan 21 2003

a(n) = (n+3 choose 4) + (n+2 choose 4) + (n+1 choose 4) - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Oct 17 2006

A002817 := n->n*(n+1)*(n^2+n+2)/8;

A002817:=-(1+z+z**2)/(z-1)**5; [S. Plouffe in his 1992 dissertation.]

a[n_] := n(n+1)(n^2+n+2)/8

(PARI) a(n)=n*(n+1)*(n^2+n+2)/8

Cf. A000217, A064322, A066370, A001496.

Adjacent sequences: A002814 A002815 A002816 this_sequence A002818 A002819 A002820

Sequence in context: A067680 A115052 A025203 this_sequence A132366 A015641 A050190

nonn,easy,nice,new

njas

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

Plouffe Maple line edited by njas, May 13 2008