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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 (list; graph; refs; listen; history; text; internal format)
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
Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
Yu. V. Chebrakov, Section 2.6.3 in "Theory of Magic Matrices. Issue TMM-1.", 2008. (in Russian)
FORMULA
a(n) = floor((1/4)*(n-2)^2)-floor((1/3)*(n-1)). - Mircea Merca, Oct 08 2013
G.f.: x^5*(1+2*x)/((1-x)*(1-x^2)*(1-x^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 *)
(* Clark Kimberling, Apr 15 2012 *)
CROSSREFS
Nonzero entries are the second differences of A055328.
Sequence in context: A360254 A108855 A026488 * A332822 A287410 A228643
KEYWORD
nonn,easy
AUTHOR
Thomas Zaslavsky, Jun 11 2005
EXTENSIONS
Edited by N. J. A. Sloane, Oct 04 2010
STATUS
approved

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Last modified July 17 20:28 EDT 2024. Contains 374377 sequences. (Running on oeis4.)