A108576 Number of 3 X 3 magic squares (with distinct positive entries) having all entries < n.
0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 40, 64, 96, 128, 184, 240, 320, 400, 504, 608, 744, 880, 1056, 1232, 1440, 1648, 1904, 2160, 2464, 2768, 3120, 3472, 3880, 4288, 4760, 5232, 5760, 6288, 6888, 7488, 8160, 8832, 9576, 10320, 11144, 11968, 12880, 13792, 14784, 15776
Offset: 1
Examples
a(10) = 8 because there are 8 3 X 3 magic squares with distinct entries < 10 (they are the standard magic squares).
Links
- T. Zaslavsky, Table of n, a(n) for n = 1..10000.
- M. Beck and T. Zaslavsky, An enumerative geometry for magic and magilatin labellings, arXiv:math/0506315 [math.CO], 2005; Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071. - _Thomas Zaslavsky_, Jan 29 2010
- M. Beck and T. Zaslavsky, Six little squares and how their numbers grow, submitted. - _Thomas Zaslavsky_, Jan 29 2010
- Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
- Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,2,-2,1,0,-1,2,-1).
Programs
-
Mathematica
LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 1, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 40}, 60] (* Jean-François Alcover, Nov 12 2018 *) CoefficientList[Series[(8 x^10 (2 x^2 + 1)) / ((1 - x^6) (1 - x^4) (1 - x)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Nov 12 2018 *) -
PARI
a(n)=1/6*(n^3-16*n^2+(76-3*(n%2))*n -[96,58,96,102,112,90,96,70,96,90,112,102][(n%12)+1])
Formula
G.f.: (8*x^10*(2*x^2+1)) / ((1-x^6)*(1-x^4)*(1-x)^2).
a(n) is given by a quasipolynomial of period 12.
Extensions
Edited by N. J. A. Sloane, Feb 05 2010
Comments