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%I #16 May 27 2023 20:38:04
%S 1,4,13,36,80,160,291,496,794,1226,1821,2632,3691,5080,6840,9070,
%T 11826,15228,19344,24332,30262,37322,45606,55330,66597,79674,94673,
%U 111892,131474,153756,178891,207278,239074,274724,314427,358666,407649,461936
%N Number of symmetry classes of 3 X 3 semimagic squares with distinct positive values < n.
%C In a semimagic square the row and column sums must all be equal (the "magic sum"). Symmetry is up to 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="/A173723/b173723.txt">Table of n, a(n) for n = 10..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_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^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 a(10) the cells contain the nine integers from 1 to 9, which can be arranged in 1 way to make a magic square, up to symmetry. For a(11) the cells contain nine of the ten integers from 1 to 10. The omitted number can only be 1, 4, 7, or 10. Each selection of numbers can be arranged in 1 way, up to symmetry.
%Y Cf. A173546, A173724. A173725 counts symmetry types by magic sum.
%K nonn,easy
%O 10,2
%A _Thomas Zaslavsky_, Feb 22 2010, Mar 03 2010
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