A173725 Number of symmetry classes of 3 X 3 semimagic squares with distinct positive values and magic sum n.

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

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 395-413. MR 2007m:05010. Zbl 1116.05071.

Links

Thomas Zaslavsky, Table of n, a(n) for n = 15..10000.

Matthias Beck and Thomas 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.

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).

Formula

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

a(n) = n^4/576 - n^3/16 + O(n^2). - Charles R Greathouse IV, May 26 2026

a(n) = floor(n^4/576 - n^3/16 + 697*n^2/864 - 2493*n/560 + 8038/945 - (n mod 2)*(2*n^2-48*n+313)/64 + (((n+2) mod 3) - (n mod 3))*(n^2-27*n+154)/54 + ((n mod 4) - ((n+2) mod 4))*(n-16)/32 + ((2*n+4) mod 5)*2/5 + (((n+4) mod 6) - ((n+3) mod 6) + ((n+2) mod 6) - ((n+1) mod 6))/6 + ((5*n+6) mod 7)/7). - Hoang Xuan Thanh, Jun 04 2026

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 3 X 3 magic square). a(16)=2 because the values (1,...,6,8,9,10) and (1,...,7,9,11) give two normal forms.

Crossrefs

Cf. A173547, A173726. A173723 counts symmetry types by largest cell value.

Sequence in context: A263615 A347789 A303748 * A300414 A307732 A103258

Adjacent sequences: A173722 A173723 A173724 * A173726 A173727 A173728

Keyword

nonn,easy,changed

Author

Thomas Zaslavsky, Feb 23 2010

Status

approved