

A173725


Number of symmetry classes of 3x3 semimagic squares with distinct positive values and magic sum n.


4



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

15,2


COMMENTS

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.


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=15..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^3)/(1x^3) * { x^7/[(x1)*(x^21)^3] + 2x^7/[(x1)*(x^21)*(x^41)] + x^7/[(x1)*(x^61)] + x^7/[(x^21)^2*(x^31)] + x^7/[(x^21)*(x^51)] + x^7/[(x^31)*(x^41)] + x^7/(x^71) + x^9/[(x1)*(x^41)^2] + 2*x^9/[(x^21)*(x^31)*(x^41)] + 2*x^9/[(x^31)*(x^61)] + x^9/[(x^41)*(x^51)] + x^11/[(x^31)*(x^41)^2] + x^11/[(x^31)*(x^81)] + x^11/[(x^51)*(x^61)] + x^13/[(x^51)*(x^81)] } [From Thomas Zaslavsky, Mar 03 2010]


EXAMPLE

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 3x3 magic square). a(16)=2 because the values 1,...,8,10 give two normal forms.


CROSSREFS

Cf. A173547, A173726. A173723 counts symmetry types by largest cell value.
Sequence in context: A023598 A263615 A303748 * A300414 A307732 A103258
Adjacent sequences: A173722 A173723 A173724 * A173726 A173727 A173728


KEYWORD

nonn


AUTHOR

Thomas Zaslavsky, Feb 23 2010


STATUS

approved



