

A173723


Number of symmetry classes of 3x3 semimagic squares with distinct positive values < n.


4



1, 4, 13, 36, 80, 160, 291, 496, 794, 1226, 1821, 2632, 3691, 5080, 6840, 9070, 11826, 15228, 19344, 24332, 30262, 37322, 45606, 55330, 66597, 79674, 94673, 111892, 131474, 153756, 178891, 207278, 239074, 274724, 314427, 358666, 407649, 461936
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OFFSET

10,2


COMMENTS

In a semimagic square the row and column sums must all be equal (the "magic sum"). Symmetry is up to permutation of rows and columns and reflection in a diagonal. 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

T. Zaslavsky, Table of n, a(n) for n=10..10000.
M. Beck, T. Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.
Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.


FORMULA

G.f.: x^2/(1x)^2 * { x^5/[(1x)^3*(1x^2)]  2x^5/[(1x)*(1x^2)^2]  x^5/[(1x)^2*(1x^3)]  2x^6/[(1x)*(1x^2)*(1x^3)]  x^6/(1x^2)^3  x^7/[(1x^2)^2*(1x^3)] + x^5/[(1x)*(1x^4)] + 2x^5/[(1x^2)*(1x^3)] + 2x^6/[(1x^2)*(1x^4)] + x^6/(1x^3)^2 + x^7/[(1x^2)*(1x^5)] + x^7/[(1x^3)*(1x^4)] + x^8/[(1x^3)*(1x^5)]  x^5/(1x^5) } [From Thomas Zaslavsky, Mar 03 2010]


EXAMPLE

For a(10) the cells contain the nine integers from 1 to 9, which can be arranged in 1 way to make a magic square, up to symmetry. For a(11) the cells contain nine of the ten integers from 1 to 10. The omitted number can only be 1, 4, 7, or 10. Each selection of numbers can be arranged in 1 way, up to symmetry.


CROSSREFS

Cf. A173546, A173724. A173725 counts symmetry types by magic sum.
Sequence in context: A268996 A270988 A272556 * A002727 A320589 A328542
Adjacent sequences: A173720 A173721 A173722 * A173724 A173725 A173726


KEYWORD

nonn


AUTHOR

Thomas Zaslavsky, Feb 22 2010, Mar 03 2010


STATUS

approved



