A173548 Number of 3 X 3 magilatin squares with positive values < n.

12, 48, 120, 384, 1068, 2472, 4896, 9072, 15516, 25608, 40296, 61608, 91068, 131640, 185136, 255960, 346860, 463248, 608088, 789240, 1010316, 1280544, 1604832, 1994064, 2454012, 2998656, 3633912, 4376064, 5232972, 6223080, 7354896

Offset: 4

Author

Thomas Zaslavsky, Mar 03 2010, Apr 24 2010

Keywords

• nonn

Comments

A magilatin squares has equal row and column sums and no number repeated in any row or column.

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

Links

• T. Zaslavsky, Table of n, a(n) for n=4..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, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
• Index entries for linear recurrences with constant coefficients, signature (0, 2, 2, 0, -3, -3, -2, 1, 4, 4, 1, -2, -3, -3, 0, 2, 2, 0, -1).

Crossrefs

Cf. A173729 (symmetry types), A173549 (counted by magic sum), A173730 (symmetry types by magic sum).

Programs

Mathematica

LinearRecurrence[{0, 2, 2, 0, -3, -3, -2, 1, 4, 4, 1, -2, -3, -3, 0, 2, 2, 0, -1}, {12, 48, 120, 384, 1068, 2472, 4896, 9072, 15516, 25608, 40296, 61608, 91068, 131640, 185136, 255960, 346860, 463248, 608088}, 31] (* Jean-François Alcover, Nov 05 2018 *)

Formula

G.f.: x^2/(1-x)^2 * { 12x^2/(x-1)^2 - 36x^3/(x-1)^3 - 72x^3/[(x-1)*(x^2-1)] - 36x^3/(x^3-1) - 72x^4/[(x-1)^2*(x^2-1)] - 36x^4/[(x-1)*(x^3-1)] - 72x^4/(x^2-1)^2 + 72x^5/[(x-1)^3*(x^2-1)] + 72x^5/[(x-1)^2*(x^3-1)] + 144x^5/[(x-1)*(x^2-1)^2] + 72x^5/[(x-1)*(x^4-1)] + 108x^5/[(x^2-1)*(x^3-1)] + 72x^5/(x^5-1) + 144x^6/[(x-1)*(x^2-1)*(x^3-1)] + 72x^6/(x^2-1)^3 + 144x^6/[(x^2-1)*(x^4-1)] + 72x^6/(x^3-1)^2 + 72x^7/[(x^2-1)^2*(x^3-1)] + 72x^7/[(x^2-1)*(x^5-1)] + 72x^7/[(x^3-1)*(x^4-1)] + 72x^8/[(x^3-1)*(x^5-1)] }.