A005650 Number of "magic squares" of order n (see comment line for exact definition).

1, 1, 3, 7, 47, 207, 2249, 14501, 216273, 1830449, 34662523, 362983263, 8330310559, 103938238111, 2801976629841, 40574514114061, 1256354802202337, 20708839411614753, 724160187180691379, 13386284447024070647

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice
• easy

Comments

This is the number of n X n square arrays with nonnegative integer entries in which every row and column add to 2 (A000681) normalized by dividing by n!/2^floor(n/2).

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 125, #25, A_n.
• 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..200
• Index entries for sequences related to magic squares

Programs

Mathematica

A000681[n_] := Sum[((2*i)!*n!^2)/(2^i*(i!^2*(n - i)!)), {i, 0, n}]/2^n;
a[n_] := A000681[n]*2^Floor[n/2]/n!;
Table[a[n], {n, 0, 19}]
(* Jean-François Alcover, Mar 14 2012, after Shanzhen Gao *)

Formula

a(n) = A000681(n)*2^floor(n/2)/n!.

a(n) ~ 2^(floor(n/2) + 1/2) * n^n * exp(1/2-n). - Vaclav Kotesovec, Aug 13 2013

Recurrence: a(n) = (2*n^2 - 4*n + 1)*a(n-2) - (n-3)*n*a(n-4). - Vaclav Kotesovec, Aug 13 2013

Extensions

More terms from Vladeta Jovovic, Feb 11 2001