A093199 Number of 4 X 4 magic squares with line sum n.

1, 8, 48, 200, 675, 1904, 4736, 10608, 21925, 42328, 77328, 134680, 225351, 364000, 570368, 869856, 1295433, 1888296, 2700400, 3795176, 5250795, 7160912, 9638784, 12818000, 16857581, 21942648, 28290640, 36151864

Offset

0,2

Comments

A magic square is defined here as a square matrix whose entries are nonnegative integers and whose rows, columns and main diagonals add up to the same number.

Links

Table of n, a(n) for n=0..27.

M. M. Ahmed, Algebraic Combinatorics of Magic Squares, arXiv:math/0405476 [math.CO], 2004.

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, Discrete and Computational Geometry, Volume 25, 2003, pp. 25-41.

Matthias Beck, The number of "magic" squares and hypercubes, arXiv:math/0201013 [math.CO], 2002-2005.

V. Baldoni et al., A User's Guide for LattE integrale. Section 5.1 Counting Magic Squares.

Formula

G.f.: (x^8+4x^7+18x^6+36x^5+50x^4+36x^3+18x^2+4x+1)/(1-x)^4/(1-x^2)^4 [Ahmed]. - sent by R. J. Mathar, Jan 25 2007

a(n) = 4*a(n-1) - 2*a(n-2) - 12*a(n-3) + 17*a(n-4) + 8*a(n-5) - 28*a(n-6) + 8*a(n-7) + 17*a(n-8) - 12*a(n-9) - 2*a(n-10) + 4*a(n-11) - a(n-12) for n > 11. - Chai Wah Wu, Jan 15 2019

Mathematica

a[n_] := (1/960)(n + 2)(2 n^6 + 24 n^5 + 130 n^4 + 400 n^3 + 5 (-1)^n n^2 + 763 n^2 + 20 (-1)^n n + 876 n + 45 (-1)^n + 435);
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jan 18 2019 *)

Prog

(PARI) a(n)=if(n%2==0, 1/480*n^7+7/240*n^6+89/480*n^5+11/16*n^4+49/30*n^3+38/15*n^2+71/30*n+1, 1/480*n^7+7/240*n^6+89/480*n^5+11/16*n^4+779/480*n^3+593/240*n^2+1051/480*n+13/16))

Crossrefs

Sequence in context: A247727 A187174 A128796 * A263507 A261975 A087914

Adjacent sequences: A093196 A093197 A093198 * A093200 A093201 A093202

Keyword

nonn

Author

Ralf Stephan, Apr 22 2004

Status

approved