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A173728
Number of reduced 3 X 3 semimagic squares with magic sum n.
2
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
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
KEYWORD
nonn
AUTHOR
Thomas Zaslavsky, Mar 03 2010
STATUS
approved