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%I #17 Jun 29 2023 09:19:53
%S 1,2,6,14,21,36,51,74,93,134,163,216,248,330,371,470,526,646,714,872,
%T 942,1130,1224,1440,1543,1810,1922,2220,2363,2700,2853,3252,3409,3854,
%U 4053,4536,4744,5304,5525,6134,6396,7056,7330,8080,8364,9170,9508,10366
%N Number of reduced, normalized 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.
%C In a semimagic square the row and column sums must all be equal (the "magic sum"). A "reduced" square has least entry 0. There is one normalized square for each symmetry class of reduced squares (symmetry under permutation of rows and columns and reflection in a diagonal). a(n) is given by a quasipolynomial of degree 5 and period 60.
%D Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Annals of Combinatorics, 10 (2006), no. 4, pages 395-413. MR 2007m:05010. Zbl 1116.05071.
%H Thomas Zaslavsky, <a href="/A173724/b173724.txt">Table of n, a(n) for n = 8..10000</a>.
%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_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 G.f.: x^5/[(1-x)^3*(1-x^2)] - 2x^5/[(1-x)*(1-x^2)^2] - x^5/[(1-x)^2*(1-x^3)] - 2x^6/[(1-x)*(1-x^2)*(1-x^3)] - x^6/(1-x^2)^3 - x^7/[(1-x^2)^2*(1-x^3)] + x^5/[(1-x)*(1-x^4)] + 2x^5/[(1-x^2)*(1-x^3)] + 2x^6/[(1-x^2)*(1-x^4)] + x^6/(1-x^3)^2 + x^7/[(1-x^2)*(1-x^5)] + x^7/[(1-x^3)*(1-x^4)] + x^8/[(1-x^3)*(1-x^5)] - x^5/(1-x^5). - _Thomas Zaslavsky_, Mar 03 2010
%e For n=8 the cells contain 0,...,8, which have one semimagic arrangement up to symmetry. 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 Cf. A173546, A173723. A173726 counts symmetry types by magic sum.
%K nonn
%O 8,2
%A _Thomas Zaslavsky_, Feb 22 2010, Mar 03 2010
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