A173728 - OEIS

%I #16 Jun 29 2023 09:24:33

%S 72,144,288,504,720,1152,1512,2160,2448,3816,3960,5544,6264,7920,8496,

%T 11664,11880,15120,15840,19800,20592,25920,25920,31608,33336,39312,

%U 39960,48600,48816,57600,58896,68544,69840,81504,81576,94392,96552

%N Number of reduced 3 X 3 semimagic squares with magic sum n.

%C In a semimagic square the row and column sums must all equal the magic sum. The symmetries are permutation of rows and columns and reflection in a diagonal. A "reduced" square has least entry 0.

%C a(n) is given by a quasipolynomial of degree 4 and period 840.

%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="/A173728/b173728.txt">Table of n, a(n) for n = 12..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_31">Index entries for linear recurrences with constant coefficients</a>, signature (-2, -3, -3, -2, 0, 3, 6, 9, 10, 9, 5, 0, -6, -11, -14, -14, -11, -6, 0, 5, 9, 10, 9, 6, 3, 0, -2, -3, -3, -2, -1).

%F G.f.: 72 * { x^7/[(x-1)*(x^2-1)^3] + 2x^7/[(x-1)*(x^2-1)*(x^4-1)] + x^7/[(x-1)*(x^6-1)] + x^7/[(x^2-1)^2*(x^3-1)] + x^7/[(x^2-1)*(x^5-1)] + x^7/[(x^3-1)*(x^4-1)] + x^7/(x^7-1) + x^9/[(x-1)*(x^4-1)^2] + 2*x^9/[(x^2-1)*(x^3-1)*(x^4-1)] + 2*x^9/[(x^3-1)*(x^6-1)] + x^9/[(x^4-1)*(x^5-1)] + x^11/[(x^3-1)*(x^4-1)^2] + x^11/[(x^3-1)*(x^8-1)] + x^11/[(x^5-1)*(x^6-1)] + x^13/[(x^5-1)*(x^8-1)] }.

%Y Cf. A173547, A173725, A173726. A173724 counts squares by largest cell value.

%K nonn

%O 12,1

%A _Thomas Zaslavsky_, Mar 03 2010