A173730 Number of symmetry classes of 3 X 3 magilatin squares with positive values and magic sum n.

1, 1, 2, 4, 7, 10, 20, 22, 35, 50, 63, 78, 116, 131, 170, 215, 260, 306, 395, 440, 537, 640, 737, 841, 1025, 1125, 1310, 1507, 1700, 1898, 2213, 2404, 2729, 3071, 3391, 3725, 4242, 4566, 5075, 5612, 6127, 6656, 7418, 7931, 8703, 9499, 10254, 11038, 12140

Offset: 6

Author

Thomas Zaslavsky, Mar 04 2010, Apr 24 2010

Keywords

• nonn

Comments

A magilatin square has equal row and column sums and no number repeated in any row or column. The symmetries are row and column permutations and diagonal flip.

a(n) is given by a quasipolynomial of degree 4 and period 840.

Links

• T. Zaslavsky, Table of n, a(n) for n = 6..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 (-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

Cf. A173549 (all squares), A173548 (counted by upper bound), A173729 (symmetry types by upper bound).

Programs

Mathematica

LinearRecurrence[{-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}, {1, 1, 2, 4, 7, 10, 20, 22, 35, 50, 63, 78, 116, 131, 170, 215, 260, 306, 395, 440, 537, 640, 737, 841, 1025, 1125, 1310, 1507, 1700, 1898, 2213, 2404, 2729, 3071}, 50] (* Jean-François Alcover, Nov 17 2018 *)

Formula

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