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A173729 Number of symmetry classes of 3 X 3 magilatin squares with positive values < n. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
4,2
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
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).
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
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 *)
CROSSREFS
Cf. A173548 (total number of squares, A173549 (squares counted by magic sum), A173730 (symmetry types by magic sum).
Sequence in context: A162588 A280541 A080615 * A340569 A097976 A279851
KEYWORD
nonn
AUTHOR
Thomas Zaslavsky, Mar 04 2010, Apr 24 2010
STATUS
approved

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Last modified March 19 07:36 EDT 2024. Contains 370957 sequences. (Running on oeis4.)