A070212 Number of 5 X 5 pandiagonal magic squares with sum n.

1, 10, 55, 220, 715, 2001, 4995, 11385, 24090, 47905, 90376, 162955, 282490, 473110, 768570, 1215126, 1875015, 2830620, 4189405, 6089710, 8707501, 12264175, 17035525, 23361975, 31660200, 42436251, 56300310, 73983205, 96354820, 124444540, 159463876, 202831420, 256200285, 321488190

Offset: 0

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 07 2002

Keywords

• nonn
• easy

Comments

In contrast to other definitions, a magic square may contain here any nonnegative integers, not necessarily distinct. For example, the 10 solutions for n = 1 are the 10 permutation matrices of size 5 X 5 which are pandiagonal in the sense that any of the 10 (principal or broken) diagonals has exactly one 1 and four 0's. - M. F. Hasler, Oct 23 2018

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..5000
• M. Ahmed, J. De Loera, and R. Hemmecke, Polyhedral Cones of Magic Cubes and Squares, arXiv:math/0201108 [math.CO], 2002.
• Maya Ahmed, Jesús De Loera and Raymond Hemmecke, Polyhedral cones of magic cubes and squares, in Discrete and Computational Geometry, Springer, Berlin, 2003, pp. 25-41.
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
• Index entries related to magic squares.

Crossrefs

Cf. A027567, A014820, A111158, A053494.

Programs

GAP

a:=[1, 10, 55, 220, 715, 2001, 4995, 11385, 24090];;  for n in [10..36] do a[n]:=9*a[n-1]-36*a[n-2]+84*a[n-3]-126*a[n-4]+126*a[n-5]-84*a[n-6]+36*a[n-7]-9*a[n-8]+a[n-9]; od; a; # Muniru A Asiru, Oct 23 2018

Maple

seq(coeff(series(-(x^4+x^3+x^2+x+1)/(x-1)^9,x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 23 2018

Mathematica

LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,10,55,220,715,2001,4995,11385,24090},40] (* Harvey P. Dale, Mar 13 2018 *)

PARI

apply( A070212(n)=1/8064*(n+4)*(n+3)*(n+2)*(n+1)*(n^2+5*n+8)*(n^2+5*n+42), [0..20]) \\ Edited by M. F. Hasler, Oct 23 2018

Formula

a(n) = (1/8064) * (n+4)*(n+3)*(n+2)*(n+1)*(n^2+5n+8)*(n^2+5n+42).

G.f.: -(x^4+x^3+x^2+x+1) / (x-1)^9. [Colin Barker, Dec 10 2012]

Extensions

More terms from Benoit Cloitre, May 12 2002

More terms from M. F. Hasler, Oct 23 2018