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

1,10

Comments

From Thomas Zaslavsky, Mar 12 2010: (Start)

A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal.

a(n) is given by a quasipolynomial of period 12. (End)

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).

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.

Example

a(10) = 8 because there are 8 3 X 3 magic squares with distinct entries < 10 (they are the standard magic squares).

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 *)

Prog

(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])

Crossrefs

Cf. A108577, A108578, A108579.

Sequence in context: A156331 A269513 A024700 * A052207 A038578 A348925

Adjacent sequences: A108573 A108574 A108575 * A108577 A108578 A108579

Keyword

nonn,easy

Author

Thomas Zaslavsky and Ralf Stephan, Jun 11 2005

Extensions

Edited by N. J. A. Sloane, Feb 05 2010

Status

approved