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
Links
- 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.
- Index entries for linear recurrences with constant coefficients, signature (4,-2,-12,17,8,-28,8,17,-12,-2,4,-1).
Programs
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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 *) -
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)
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
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
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