

A108579


Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having magic sum 3n.


10



0, 0, 0, 0, 1, 3, 4, 7, 10, 13, 17, 22, 26, 32, 38, 44, 51, 59, 66, 75, 84, 93, 103, 114, 124, 136, 148, 160, 173, 187, 200, 215, 230, 245, 261, 278, 294, 312, 330, 348, 367, 387, 406, 427, 448, 469, 491, 514, 536, 560, 584, 608, 633, 659, 684, 711, 738, 765, 793, 822, 850
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OFFSET

1,6


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 6.
It appears that A108579(n) is the number of ordered triples (w,x,y) with components all in {1,...,n} and w+n=2x+3y, as in the Mathematica section. For related sequences, see A211422.  Clark Kimberling, Apr 15 2012


LINKS

Yu. V. Chebrakov, Section 2.6.3 in "Theory of Magic Matrices. Issue TMM1.", 2008. (in Russian)


FORMULA

a(n) = floor((1/4)*(n2)^2)floor((1/3)*(n1)).  Mircea Merca, Oct 08 2013
G.f.: x^5*(1+2*x)/((1x)*(1x^2)*(1x^3)).


EXAMPLE

a(5) = 1 because there is a unique 3 X 3 magic square, up to symmetry, using the first 9 positive integers.


MATHEMATICA

(* This program generates a sequence described in the Comments section *)
t[n_] := t[n] = Flatten[Table[w^2 + x*y + n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 1, 80}] (* A211506 *)


CROSSREFS

Nonzero entries are the second differences of A055328.


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



