%I #16 Jun 29 2023 09:27:23

%S 1,1,2,4,7,10,20,22,35,50,63,78,116,131,170,215,260,306,395,440,537,

%T 640,737,841,1025,1125,1310,1507,1700,1898,2213,2404,2729,3071,3391,

%U 3725,4242,4566,5075,5612,6127,6656,7418,7931,8703,9499,10254,11038,12140

%N Number of symmetry classes of 3 X 3 magilatin squares with positive values and magic sum 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 4 and period 840.

%H T. Zaslavsky, <a href="/A173730/b173730.txt">Table of n, a(n) for n = 6..10000</a>.

%H Matthias Beck and Thomas Zaslavsky, <a href="http://arxiv.org/abs/math/0506315">An enumerative geometry for magic and magilatin labellings</a>, arXiv:math/0506315 [math.CO], 2005.

%H Matthias Beck and Thomas Zaslavsky, <a href="https://doi.org/10.1007/s00026-006-0296-4">An enumerative geometry for magic and magilatin labellings</a>, Annals of Combinatorics, 10 (2006), no. 4, pages 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>, Journal of Integer Sequences, 13 (2010), Article 10.6.2.

%H <a href="/index/Rec#order_34">Index entries for linear recurrences with constant coefficients</a>, 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).

%F 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)) ).

%t 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 *)

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

%K nonn

%O 6,3

%A _Thomas Zaslavsky_, Mar 04 2010, Apr 24 2010