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

%I A093199 #44 Mar 14 2025 04:29:36
%S A093199 1,8,48,200,675,1904,4736,10608,21925,42328,77328,134680,225351,
%T A093199 364000,570368,869856,1295433,1888296,2700400,3795176,5250795,7160912,
%U A093199 9638784,12818000,16857581,21942648,28290640,36151864
%N A093199 Number of 4 X 4 magic squares with line sum n.
%C A093199 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 A093199 M. M. Ahmed, <a href="https://arxiv.org/abs/math/0405476">Algebraic Combinatorics of Magic Squares</a>, arXiv:math/0405476 [math.CO], 2004.
%H A093199 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 A093199 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 A093199 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 A093199 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.
%H A093199 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-12,17,8,-28,8,17,-12,-2,4,-1).
%F A093199 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 A093199 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
%F A093199 E.g.f.: ((480 + 3450*x + 7710*x^2 + 6545*x^3 + 2480*x^4 + 439*x^5 + 35*x^6 + x^7)*cosh(x) + (390 + 3570*x + 7665*x^2 + 6550*x^3 + 2480*x^4 + 439*x^5 + 35*x^6 + x^7)*sinh(x))/480. - _Stefano Spezia_, Mar 12 2025
%t A093199 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 A093199 Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Jan 18 2019 *)
%o A093199 (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 A093199 nonn,easy
%O A093199 0,2
%A A093199 _Ralf Stephan_, Apr 22 2004