A173728 Number of reduced 3 X 3 semimagic squares with magic sum n.

72, 144, 288, 504, 720, 1152, 1512, 2160, 2448, 3816, 3960, 5544, 6264, 7920, 8496, 11664, 11880, 15120, 15840, 19800, 20592, 25920, 25920, 31608, 33336, 39312, 39960, 48600, 48816, 57600, 58896, 68544, 69840, 81504, 81576, 94392, 96552

Offset: 12

Author

Thomas Zaslavsky, Mar 03 2010

Keywords

• nonn

Comments

In a semimagic square the row and column sums must all equal the magic sum. The symmetries are permutation of rows and columns and reflection in a diagonal. A "reduced" square has least entry 0.

a(n) is given by a quasipolynomial of degree 4 and period 840.

References

• Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Annals of Combinatorics, 10 (2006), no. 4, pages 395-413. MR 2007m:05010. Zbl 1116.05071.

Links

• Thomas Zaslavsky, Table of n, a(n) for n = 12..10000.
• Matthias Beck and Thomas Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.
• Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
• Index entries for linear recurrences with constant coefficients, signature (-2, -3, -3, -2, 0, 3, 6, 9, 10, 9, 5, 0, -6, -11, -14, -14, -11, -6, 0, 5, 9, 10, 9, 6, 3, 0, -2, -3, -3, -2, -1).

Crossrefs

Cf. A173547, A173725, A173726. A173724 counts squares by largest cell value.

Formula

G.f.: 72 * { x^7/[(x-1)*(x^2-1)^3] + 2x^7/[(x-1)*(x^2-1)*(x^4-1)] + x^7/[(x-1)*(x^6-1)] + x^7/[(x^2-1)^2*(x^3-1)] + x^7/[(x^2-1)*(x^5-1)] + x^7/[(x^3-1)*(x^4-1)] + x^7/(x^7-1) + x^9/[(x-1)*(x^4-1)^2] + 2*x^9/[(x^2-1)*(x^3-1)*(x^4-1)] + 2*x^9/[(x^3-1)*(x^6-1)] + x^9/[(x^4-1)*(x^5-1)] + x^11/[(x^3-1)*(x^4-1)^2] + x^11/[(x^3-1)*(x^8-1)] + x^11/[(x^5-1)*(x^6-1)] + x^13/[(x^5-1)*(x^8-1)] }.