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

4,1

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).

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)] }.

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 *)

Crossrefs

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

Sequence in context: A165280 A371419 A280058 * A006564 A239352 A292022

Adjacent sequences: A173545 A173546 A173547 * A173549 A173550 A173551

Keyword

nonn

Author

Thomas Zaslavsky, Mar 03 2010, Apr 24 2010

Status

approved