A108579 Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having magic sum 3n.

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

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

T. Zaslavsky, Table of n, a(n) for n = 1..10000

Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.

Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow: Maple Worksheets and Supporting Documentation.

Yu. V. Chebrakov, Section 2.6.3 in "Theory of Magic Matrices. Issue TMM-1.", 2008. (in Russian)

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

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

Cf. A108576, A108577, A108578.

Nonzero entries are the second differences of A055328.

Sequence in context: A360254 A108855 A026488 * A332822 A287410 A228643

Adjacent sequences: A108576 A108577 A108578 * A108580 A108581 A108582

Keyword

nonn,easy

Author

Thomas Zaslavsky, Jun 11 2005

Extensions

Edited by N. J. A. Sloane, Oct 04 2010

Status

approved