A027567 -id:A027567 - cp's OEIS frontend

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

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

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

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.

Cf. A027567, A014820, A111158, A053494.

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

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

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 *)

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

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]

More terms from Benoit Cloitre, May 12 2002

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

A160540 The number of distinct n X n panmagic = pandiagonal = diabolic = Nasik squares, not counting rotations, reflections, or row/column cyclings of others, with the additional property that each i X j rectangle, including each "wrap-around" i X j rectangle, where i and j are positive integers whose product is n, also sums to the magic constant A006003(n) = n(n^2+1)/2.

Original entry on oeis.org

1, 0, 0, 3

Offset: 1

Anonymous, May 18 2009

A "row/column cycling" of another square is a square that can be formed by moving any number of rows from the top of the other square to the bottom of it and keeping them in the same order, or by moving any number of columns from the left of it to the right of it and keeping them in the same order, or by doing both. Since these squares are panmagic, and so the "wrap-around" diagonals also must sum to the magic constant, row/column cyclings of a square are not essentially different from that square.

			a(4) = 3 because there are 3 distinct 4 X 4 panmagic squares, not counting rotations, reflections, or row/column cyclings of others, with the additional property that each 2 X 2 square, including each "wrap around" 2 X 2 square such as the one consisting of a11, a12, a41, and a42, and the one consisting of a11, a14, a41, and a44, also sums to A006003(4) = 4(4^2+1)/2 = 34:
     1  8 10 15       1  8 11 14       1  8 13 12
    12 13  3  6      12 13  2  7      14 11  2  7
     7  2 16  9       6  3 16  9       4  5 16  9
    14 11  5  4      15 10  5  4      15 10  3  6
The following panmagic square does not count because it can be formed from the third panmagic square given above by moving the first column on the left of it to the right of it and then reflecting it in the y-axis:
     1 12 13  8
    14  7  2 11
     4  9 16  5
    15  6  3 10

Cf. A027567.

cp's OEIS Frontend

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

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