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A173724
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Number of reduced, normalized 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.
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5
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1, 2, 6, 14, 21, 36, 51, 74, 93, 134, 163, 216, 248, 330, 371, 470, 526, 646, 714, 872, 942, 1130, 1224, 1440, 1543, 1810, 1922, 2220, 2363, 2700, 2853, 3252, 3409, 3854, 4053, 4536, 4744, 5304, 5525, 6134, 6396, 7056, 7330, 8080, 8364, 9170, 9508, 10366
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OFFSET
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8,2
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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. There is one normalized square for each symmetry class of reduced squares (symmetry under permutation of rows and columns and reflection in a diagonal). 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.: x^5/[(1-x)^3*(1-x^2)] - 2x^5/[(1-x)*(1-x^2)^2] - x^5/[(1-x)^2*(1-x^3)] - 2x^6/[(1-x)*(1-x^2)*(1-x^3)] - x^6/(1-x^2)^3 - x^7/[(1-x^2)^2*(1-x^3)] + x^5/[(1-x)*(1-x^4)] + 2x^5/[(1-x^2)*(1-x^3)] + 2x^6/[(1-x^2)*(1-x^4)] + x^6/(1-x^3)^2 + x^7/[(1-x^2)*(1-x^5)] + x^7/[(1-x^3)*(1-x^4)] + x^8/[(1-x^3)*(1-x^5)] - x^5/(1-x^5). - Thomas Zaslavsky, Mar 03 2010
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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. 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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