A173547 Number of 3 X 3 semimagic squares with distinct positive values and magic sum n.

72, 144, 288, 576, 864, 1440, 2088, 3024, 3888, 5904, 6984, 9432, 12168, 14904, 17928, 23832, 26784, 33048, 39672, 46584, 53640, 65592, 72504, 85248, 98928, 111816, 125208, 147528, 160632, 182808, 206424, 229176, 252648, 287928, 310752

Offset: 15

Author

Thomas Zaslavsky, Feb 21 2010, Feb 24 2010, Mar 03 2010

Keywords

• nonn

Comments

In a semimagic squares the row and column sums must all be equal to the magic sum. a(n) is given by a quasipolynomial of degree 4 and period 840.

a(15) is the first term because the values 1,...,9 make magic sum 15. [From Thomas Zaslavsky, Mar 03 2010]

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=15..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, -2, 0, 3, 6, 8, 9, 7, 3, -4, -10, -15, -16, -14, -8, 0, 8, 14, 16, 15, 10, 4, -3, -7, -9, -8, -6, -3, 0, 2, 3, 2, 1).

Crossrefs

A173546 counts the same squares by upper bound on the entries. Cf. A108576, A108577, A108578, A108579, A173548, A173549.

Formula

G.f.: 72 * x^3/(1-x)^3 * { 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)] }