A173723 Number of symmetry classes of 3 X 3 semimagic squares with distinct positive values < n.
1, 4, 13, 36, 80, 160, 291, 496, 794, 1226, 1821, 2632, 3691, 5080, 6840, 9070, 11826, 15228, 19344, 24332, 30262, 37322, 45606, 55330, 66597, 79674, 94673, 111892, 131474, 153756, 178891, 207278, 239074, 274724, 314427, 358666, 407649, 461936
Offset: 10
Examples
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.
References
- 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.
Links
- Thomas Zaslavsky, Table of n, a(n) for n = 10..10000.
- Matthias Beck and Thomas Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.
- Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
- Index entries for linear recurrences with constant coefficients, signature (0,2,2,0,-3,-3,-2,1,4,4,1,-2,-3,-3,0,2,2,0,-1).
Formula
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
Comments