

A108576


Number of 3 X 3 magic squares (with distinct positive entries) having all entries < n.


8



0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 40, 64, 96, 128, 184, 240, 320, 400, 504, 608, 744, 880, 1056, 1232, 1440, 1648, 1904, 2160, 2464, 2768, 3120, 3472, 3880, 4288, 4760, 5232, 5760, 6288, 6888, 7488, 8160, 8832, 9576, 10320, 11144, 11968, 12880, 13792, 14784, 15776
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OFFSET

1,10


COMMENTS

A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal.
a(n) is given by a quasipolynomial of period 12. (End)


LINKS

Index entries for linear recurrences with constant coefficients, signature (2,1,0,1,2,2,2,1,0,1,2,1).


FORMULA

G.f.: (8*x^10*(2*x^2+1)) / ((1x^6)*(1x^4)*(1x)^2).
a(n) is given by a quasipolynomial of period 12.


EXAMPLE

a(10) = 8 because there are 8 3 X 3 magic squares with distinct entries < 10 (they are the standard magic squares).


MATHEMATICA

LinearRecurrence[{2, 1, 0, 1, 2, 2, 2, 1, 0, 1, 2, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 40}, 60] (* JeanFrançois Alcover, Nov 12 2018 *)
CoefficientList[Series[(8 x^10 (2 x^2 + 1)) / ((1  x^6) (1  x^4) (1  x)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Nov 12 2018 *)


PROG

(PARI) a(n)=1/6*(n^316*n^2+(763*(n%2))*n [96, 58, 96, 102, 112, 90, 96, 70, 96, 90, 112, 102][(n%12)+1])


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



