%I #39 Aug 27 2024 01:29:21

%S 1,8,48,200,675,1904,4736,10608,21925,42328,77328,134680,225351,

%T 364000,570368,869856,1295433,1888296,2700400,3795176,5250795,7160912,

%U 9638784,12818000,16857581,21942648,28290640,36151864

%N Number of 4 X 4 magic squares with line sum n.

%C 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.

%H M. M. Ahmed, <a href="https://arxiv.org/abs/math/0405476">Algebraic Combinatorics of Magic Squares</a>, arXiv:math/0405476 [math.CO], 2004.

%H M. Ahmed, J. De Loera and R. Hemmecke, <a href="http://arxiv.org/abs/math/0201108">Polyhedral Cones of Magic Cubes and Squares</a>, arXiv:math/0201108 [math.CO], 2002.

%H Maya Ahmed, Jesús De Loera and Raymond Hemmecke, <a href="https://doi.org/10.1007/978-3-642-55566-4_2">Polyhedral cones of magic cubes and squares</a>, Discrete and Computational Geometry, Volume 25, 2003, pp. 25-41.

%H Matthias Beck, <a href="https://arxiv.org/abs/math/0201013">The number of "magic" squares and hypercubes</a>, arXiv:math/0201013 [math.CO], 2002-2005.

%H V. Baldoni et al., <a href="https://www.math.ucdavis.edu/~latte/software.php">A User's Guide for LattE integrale</a>. Section 5.1 Counting Magic Squares.

%F 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

%F 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

%t 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);

%t Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Jan 18 2019 *)

%o (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)

%K nonn,easy,changed

%O 0,2

%A _Ralf Stephan_, Apr 22 2004