A173727 Number of reduced 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.

72, 144, 432, 1008, 1512, 2592, 3672, 5328, 6696, 9648, 11736, 15552, 17856, 23760, 26712, 33840, 37872, 46512, 51408, 62784, 67824, 81360, 88128, 103680, 111096, 130320, 138384, 159840, 170136, 194400, 205416, 234144, 245448, 277488, 291816

Offset: 8

Author

Thomas Zaslavsky, Mar 03 2010

Keywords

• nonn
• easy

Comments

In a semimagic square the row and column sums must all be equal (the "magic sum"). A reduced square has least entry 0.

a(n) is given by a quasipolynomial of degree 5 and period 60.

Examples

			For n=8 the cells contain 0,...,8, which have one semimagic arrangement up to symmetry. All examples are obtained by symmetries from (by rows): 0, 5, 7; 4, 6, 2; 8, 1, 3.
For n=9 the cells contain all of 0,...,9 except 3 or 6, since 0 and 9 must be used; each selection has one semimagic arrangement up to symmetry.
		

Links

• Thomas Zaslavsky, Table of n, a(n) for n = 8..10000.
• Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, arXiv:math/0506315 [math.CO], 2005.
• 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.
• 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,-1,2,5,5,2,-3,-7,-7,-3,2,5,5,2,-1,-2,-1).

Crossrefs

Cf. A173546, A173723, A173724. A173728 counts reduced squares by magic sum.

Formula

G.f.: 72 * ( x^5/((x-1)*(x^4-1)) + x^5/((x-1)^2*(x^3-1)) + x^5/((x-1)^3*(x^2-1)) + 2*x^5/((x-1)*(x^2-1)^2) + 2*x^5/((x^2-1)*(x^3-1)) + x^5/(x^5-1) + 2*x^6/((x-1)*(x^2-1)*(x^3-1)) + x^6/(x^2-1)^3 + 2*x^6/((x^2-1)*(x^4-1)) + x^6/(x^3-1)^2 + x^7/((x^2-1)*(x^5-1)) + x^7/((x^2-1)^2*(x^3-1)) + x^7/((x^3-1)*(x^4-1)) + x^8/((x^3-1)*(x^5-1)) ).