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A108576
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Number of 3 X 3 magic squares (with distinct positive entries) having all entries < n.
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8
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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; internal format)
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OFFSET
| 1,10
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COMMENTS
| Contribution from Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), 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)
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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. [From Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Jan 29 2010]
M. Beck and T. Zaslavsky, Six little squares and how their numbers grow, submitted. [From Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Jan 29 2010]
Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation. [From Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Mar 12 2010]
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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.
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EXAMPLE
| a(10) = 8 because there are 8 3 X 3 magic squares with distinct entries < 10 (they are the standard magic squares).
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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])
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CROSSREFS
| Cf. A108577, A108578, A108579.
Sequence in context: A063526 A156331 A024700 * A052207 A038578 A155110
Adjacent sequences: A108573 A108574 A108575 * A108577 A108578 A108579
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KEYWORD
| nonn
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AUTHOR
| Thomas Zaslavsky (zaslav(AT)math.binghamton.edu) and Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 11 2005
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EXTENSIONS
| Edited by N, J. A. Sloane, Feb 05 2010
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