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A173727 Number of reduced 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.

%I A173727 #22 Feb 08 2025 03:44:04
%S A173727 72,144,432,1008,1512,2592,3672,5328,6696,9648,11736,15552,17856,
%T A173727 23760,26712,33840,37872,46512,51408,62784,67824,81360,88128,103680,
%U A173727 111096,130320,138384,159840,170136,194400,205416,234144,245448,277488,291816
%N A173727 Number of reduced 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.
%C A173727 In a semimagic square the row and column sums must all be equal (the "magic sum"). A reduced square has least entry 0.
%C A173727 a(n) is given by a quasipolynomial of degree 5 and period 60.
%H A173727 Thomas Zaslavsky, <a href="/A173727/b173727.txt">Table of n, a(n) for n = 8..10000</a>.
%H A173727 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 A173727 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 A173727 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 A173727 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 A173727 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-1,2,5,5,2,-3,-7,-7,-3,2,5,5,2,-1,-2,-1).
%F A173727 G.f.: 72 * ( x^5/((x-1)*(x^4-1)) + x^5/((x-1)^2*(x^3-1)) + x^5/((x-1)^3*(x^2-1)) + 2*x^5/((x-1)*(x^2-1)^2) + 2*x^5/((x^2-1)*(x^3-1)) + x^5/(x^5-1) + 2*x^6/((x-1)*(x^2-1)*(x^3-1)) + x^6/(x^2-1)^3 + 2*x^6/((x^2-1)*(x^4-1)) + x^6/(x^3-1)^2 + x^7/((x^2-1)*(x^5-1)) + x^7/((x^2-1)^2*(x^3-1)) + x^7/((x^3-1)*(x^4-1)) + x^8/((x^3-1)*(x^5-1)) ).
%e A173727 For n=8 the cells contain 0,...,8, which have one semimagic arrangement up to symmetry. All examples are obtained by symmetries from (by rows): 0, 5, 7; 4, 6, 2; 8, 1, 3.
%e A173727 For n=9 the cells contain all of 0,...,9 except 3 or 6, since 0 and 9 must be used; each selection has one semimagic arrangement up to symmetry.
%Y A173727 Cf. A173546, A173723, A173724. A173728 counts reduced squares by magic sum.
%K A173727 nonn,easy
%O A173727 8,1
%A A173727 _Thomas Zaslavsky_, Mar 03 2010