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A108576 Number of 3 X 3 magic squares (with distinct positive entries) having all entries < n. 8
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 (list; graph; refs; listen; history; text; internal format)
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, 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.

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).

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 A155110

Adjacent sequences:  A108573 A108574 A108575 * A108577 A108578 A108579

KEYWORD

nonn

AUTHOR

Thomas Zaslavsky and Ralf Stephan, Jun 11 2005

EXTENSIONS

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

STATUS

approved

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Last modified May 23 06:42 EDT 2017. Contains 286909 sequences.