

A173727


Number of reduced 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.


1



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

8,1


COMMENTS

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.


REFERENCES

Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Annals of Combinatorics, 10 (2006), no. 4, pages 395413. MR 2007m:05010. Zbl 1116.05071.


LINKS

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).


FORMULA

G.f.: 72 * { x^5/[(x1)*(x^41)] + x^5/[(x1)^2*(x^31)] + x^5/[(x1)^3*(x^21)] + 2*x^5/[(x1)*(x^21)^2) + 2*x^5/[(x^21)*(x^31)] + x^5/(x^51) + 2*x^6/[(x1)*(x^21)*(x^31)] + x^6/(x^21)^3 + 2*x^6/[(x^21)*(x^41)] + x^6/(x^31)^2 + x^7/[(x^21)*(x^51)] + x^7/[(x^21)^2*(x^31)] + x^7/[(x^31)*(x^41)] + x^8/[(x^31)*(x^51)] }


EXAMPLE

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.


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



