A173729 Number of symmetry classes of 3 X 3 magilatin squares with positive values < n.

1, 4, 10, 24, 53, 106, 191, 328, 528, 822, 1230, 1794, 2542, 3534, 4802, 6428, 8460, 10996, 14087, 17870, 22405, 27850, 34286, 41896, 50773, 61148, 73116, 86942, 102751, 120840, 141343, 164618, 190808, 220306, 253292, 290202, 331226, 376872

Offset: 4

Author

Thomas Zaslavsky, Mar 04 2010, Apr 24 2010

Keywords

• nonn
• easy

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 5 and period 60.

Links

• Thomas Zaslavsky, Table of n, a(n) for n = 4..10000.
• Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Ann. Combinatorics, 10 (2006), no. 4, 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 (0, 2, 2, 0, -3, -3, -2, 1, 4, 4, 1, -2, -3, -3, 0, 2, 2, 0, -1).

Crossrefs

Cf. A173548 (total number of squares), A173549 (squares counted by magic sum), A173730 (symmetry types by magic sum).

Programs

Mathematica

CoefficientList[Series[x^4*(1 + 4*x + 8*x^2 + 14*x^3 + 25*x^4 + 41*x^5 + 52*x^6 + 54*x^7 + 43*x^8 + 27*x^9 + 13*x^10 + 10*x^11 + 16*x^12 + 23*x^13 + 20*x^14 + 9*x^15)/((1 + x^2)*(1 + x)^3*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)*(1 - x)^6), {x, 0, 41}], x] (* L. Edson Jeffery, Sep 10 2017 *)

Formula

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

G.f.: x^4*(1 + 4*x + 8*x^2 + 14*x^3 + 25*x^4 + 41*x^5 + 52*x^6 + 54*x^7 + 43*x^8 + 27*x^9 + 13*x^10 + 10*x^11 + 16*x^12 + 23*x^13 + 20*x^14 + 9*x^15)/((1 + x^2)*(1 + x)^3*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)*(1 - x)^6). - L. Edson Jeffery, Sep 10 2017