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%I #22 Jun 29 2023 09:26:09
%S 1,4,10,24,53,106,191,328,528,822,1230,1794,2542,3534,4802,6428,8460,
%T 10996,14087,17870,22405,27850,34286,41896,50773,61148,73116,86942,
%U 102751,120840,141343,164618,190808,220306,253292,290202,331226,376872
%N Number of symmetry classes of 3 X 3 magilatin squares with positive values < n.
%C 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.
%C a(n) is given by a quasipolynomial of degree 5 and period 60.
%H Thomas Zaslavsky, <a href="/A173729/b173729.txt">Table of n, a(n) for n = 4..10000</a>.
%H Matthias Beck and Thomas Zaslavsky, <a href="http://arXiv.org/abs/math.CO/0506315">An enumerative geometry for magic and magilatin labellings</a>, Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071.
%H Matthias Beck and Thomas Zaslavsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Zaslavsky/sls.html">Six Little Squares and How Their Numbers Grow </a>, J. Int. Seq. 13 (2010), 10.6.2.
%H Matthias Beck and Thomas Zaslavsky, <a href="https://people.math.binghamton.edu/zaslav/Tpapers/SLSfiles/">"Six Little Squares and How their Numbers Grow" Web Site</a>: Maple worksheets and supporting documentation.
%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (0, 2, 2, 0, -3, -3, -2, 1, 4, 4, 1, -2, -3, -3, 0, 2, 2, 0, -1).
%F 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)] }.
%F 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
%t 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 *)
%Y Cf. A173548 (total number of squares, A173549 (squares counted by magic sum), A173730 (symmetry types by magic sum).
%K nonn
%O 4,2
%A _Thomas Zaslavsky_, Mar 04 2010, Apr 24 2010
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