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A173546 Number of 3 X 3 semimagic squares with distinct positive values < n. In a semimagic squares the row and column sums must all be equal (the "magic sum").

%I A173546 #17 Jun 29 2023 08:13:46
%S A173546 72,288,936,2592,5760,11520,20952,35712,57168,88272,131112,189504,
%T A173546 265752,365760,492480,653040,851472,1096416,1392768,1751904,2178864,
%U A173546 2687184,3283632,3983760,4794984,5736528,6816456,8056224,9466128
%N A173546 Number of 3 X 3 semimagic squares with distinct positive values < n. In a semimagic squares the row and column sums must all be equal (the "magic sum").
%C A173546 a(n) is given by a quasipolynomial of degree 5 and period 60.
%D A173546 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 A173546 Thomas Zaslavsky, <a href="/A173546/b173546.txt">Table of n, a(n) for n = 10..10000</a>.
%H A173546 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 A173546 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 A173546 <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 A173546 G.f.: 72 * 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
%Y A173546 A173547 counts the same squares by magic sum.
%Y A173546 Cf. A108576, A108577, A108578, A108579, A173548, A173549.
%K A173546 nonn
%O A173546 10,1
%A A173546 _Thomas Zaslavsky_, Feb 21 2010