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A173727
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Number of reduced 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.
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1
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72, 144, 432, 1008, 1512, 2592, 3672, 5328, 6696, 9648, 11736, 15552, 17856, 23760, 26712, 33840, 37872, 46512, 51408, 62784, 67824, 81360, 88128, 103680, 111096, 130320, 138384, 159840, 170136, 194400, 205416, 234144, 245448, 277488, 291816
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OFFSET
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8,1
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COMMENTS
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In a semimagic square the row and column sums must all be equal (the "magic sum"). A reduced square has least entry 0.
a(n) is given by a quasipolynomial of degree 5 and period 60.
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REFERENCES
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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.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (-2, -1, 2, 5, 5, 2, -3, -7, -7, -3, 2, 5, 5, 2, -1, -2, -1).
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FORMULA
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G.f.: 72 * { x^5/[(x-1)*(x^4-1)] + x^5/[(x-1)^2*(x^3-1)] + x^5/[(x-1)^3*(x^2-1)] + 2*x^5/[(x-1)*(x^2-1)^2) + 2*x^5/[(x^2-1)*(x^3-1)] + x^5/(x^5-1) + 2*x^6/[(x-1)*(x^2-1)*(x^3-1)] + x^6/(x^2-1)^3 + 2*x^6/[(x^2-1)*(x^4-1)] + x^6/(x^3-1)^2 + x^7/[(x^2-1)*(x^5-1)] + x^7/[(x^2-1)^2*(x^3-1)] + x^7/[(x^3-1)*(x^4-1)] + x^8/[(x^3-1)*(x^5-1)] }
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EXAMPLE
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For n=8 the cells contain 0,...,8, which have one semimagic arrangement up to symmetry. All examples are obtained by symmetries from (by rows): 0, 5, 7; 4, 6, 2; 8, 1, 3.
For n=9 the cells contain all of 0,...,9 except 3 or 6, since 0 and 9 must be used; each selection has one semimagic arrangement up to symmetry.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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