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A173725
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Number of symmetry classes of 3 X 3 semimagic squares with distinct positive values and magic sum n.
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4
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1, 2, 4, 8, 12, 20, 29, 42, 54, 82, 97, 131, 169, 207, 249, 331, 372, 459, 551, 647, 745, 911, 1007, 1184, 1374, 1553, 1739, 2049, 2231, 2539, 2867, 3183, 3509, 3999, 4316, 4820, 5340, 5835, 6350, 7104, 7607, 8352, 9132, 9882, 10651, 11724, 12472, 13551
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OFFSET
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15,2
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COMMENTS
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In a semimagic square the row and column sums must all be equal to the magic sum. The symmetries are permutation of rows and columns and reflection in a diagonal. a(n) is given by a quasipolynomial of degree 4 and period 840.
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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, -3, -2, 0, 3, 6, 8, 9, 7, 3, -4, -10, -15, -16, -14, -8, 0, 8, 14, 16, 15, 10, 4, -3, -7, -9, -8, -6, -3, 0, 2, 3, 2, 1).
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FORMULA
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G.f.: (x^3)/(1-x^3) * { x^7/[(x-1)*(x^2-1)^3] + 2x^7/[(x-1)*(x^2-1)*(x^4-1)] + x^7/[(x-1)*(x^6-1)] + x^7/[(x^2-1)^2*(x^3-1)] + x^7/[(x^2-1)*(x^5-1)] + x^7/[(x^3-1)*(x^4-1)] + x^7/(x^7-1) + x^9/[(x-1)*(x^4-1)^2] + 2*x^9/[(x^2-1)*(x^3-1)*(x^4-1)] + 2*x^9/[(x^3-1)*(x^6-1)] + x^9/[(x^4-1)*(x^5-1)] + x^11/[(x^3-1)*(x^4-1)^2] + x^11/[(x^3-1)*(x^8-1)] + x^11/[(x^5-1)*(x^6-1)] + x^13/[(x^5-1)*(x^8-1)] }. - Thomas Zaslavsky, Mar 03 2010
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EXAMPLE
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a(15) is the first term because the values 1,...,9 make magic sum 15. By symmetries one can assume a_{11} is smallest, and a_{11} < a_{12} < a_{21} < a_{31} < a_{13}. a(15)=1 because there is only one normal form with values 1,...,9 (equivalent to the classical 3 X 3 magic square). a(16)=2 because the values 1,...,8,10 give two normal forms.
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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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