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Number of 3 X 3 magilatin squares with positive values < n.
8

%I #16 Jun 29 2023 08:17:31

%S 12,48,120,384,1068,2472,4896,9072,15516,25608,40296,61608,91068,

%T 131640,185136,255960,346860,463248,608088,789240,1010316,1280544,

%U 1604832,1994064,2454012,2998656,3633912,4376064,5232972,6223080,7354896

%N Number of 3 X 3 magilatin squares with positive values < n.

%C A magilatin squares has equal row and column sums and no number repeated in any row or column.

%C a(n) is given by a quasipolynomial of degree 5 and period 60.

%H T. Zaslavsky, <a href="/A173548/b173548.txt">Table of n, a(n) for n=4..10000</a>.

%H Matthias Beck and Thomas Zaslavsky, <a href="https://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_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 * { 12x^2/(x-1)^2 - 36x^3/(x-1)^3 - 72x^3/[(x-1)*(x^2-1)] - 36x^3/(x^3-1) - 72x^4/[(x-1)^2*(x^2-1)] - 36x^4/[(x-1)*(x^3-1)] - 72x^4/(x^2-1)^2 + 72x^5/[(x-1)^3*(x^2-1)] + 72x^5/[(x-1)^2*(x^3-1)] + 144x^5/[(x-1)*(x^2-1)^2] + 72x^5/[(x-1)*(x^4-1)] + 108x^5/[(x^2-1)*(x^3-1)] + 72x^5/(x^5-1) + 144x^6/[(x-1)*(x^2-1)*(x^3-1)] + 72x^6/(x^2-1)^3 + 144x^6/[(x^2-1)*(x^4-1)] + 72x^6/(x^3-1)^2 + 72x^7/[(x^2-1)^2*(x^3-1)] + 72x^7/[(x^2-1)*(x^5-1)] + 72x^7/[(x^3-1)*(x^4-1)] + 72x^8/[(x^3-1)*(x^5-1)] }.

%t LinearRecurrence[{0, 2, 2, 0, -3, -3, -2, 1, 4, 4, 1, -2, -3, -3, 0, 2, 2, 0, -1}, {12, 48, 120, 384, 1068, 2472, 4896, 9072, 15516, 25608, 40296, 61608, 91068, 131640, 185136, 255960, 346860, 463248, 608088}, 31] (* _Jean-François Alcover_, Nov 05 2018 *)

%Y Cf. A173729 (symmetry types), A173549 (counted by magic sum), A173730 (symmetry types by magic sum).

%K nonn

%O 4,1

%A _Thomas Zaslavsky_, Mar 03 2010, Apr 24 2010