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

12,1

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).

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)] }.

Crossrefs

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

Sequence in context: A060661 A050495 A137883 * A173547 A173727 A342993

Adjacent sequences: A173725 A173726 A173727 * A173729 A173730 A173731

Keyword

nonn

Author

Thomas Zaslavsky, Mar 03 2010

Status

approved