A173726 Number of reduced, normalized 3x3 semimagic squares with magic sum n.

1, 2, 4, 7, 10, 16, 21, 30, 34, 53, 55, 77, 87, 110, 118, 162, 165, 210, 220, 275, 286, 360, 360, 439, 463, 546, 555, 675, 678, 800, 818, 952, 970, 1132, 1133, 1311, 1341, 1519, 1530, 1764, 1772, 2002, 2028, 2275, 2299, 2592, 2590, 2900, 2939, 3250, 3265, 3644

Offset: 12

Author

Thomas Zaslavsky, Feb 23 2010

Keywords

• nonn

Comments

In a semimagic square the row and column sums must all equal the magic sum. The symmetries are permutation of rows and columns and reflection in a diagonal. A "reduced" square has least entry 0. There is one normalized square for each symmetry class of reduced squares. See A173725 for a general normal form. a(n) is given by a quasipolynomial of degree 4 and period 840.

Examples

			a(12) is the first term because the values 0,...,8 make magic sum 12. a(12)=1 because there is only one normal form with values 0 to 8: (by rows) 0,4,8;5,6,1;7,2,3. a(13)=2 because the values 0,...,5,7,8,9 give two normal forms: 0,4,9;5,7,1;8,2,3 and 0,4,9;5,7,1;8,2,3.
		

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=12..10000.
• Matthias Beck, 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, -3, -2, 0, 3, 6, 9, 10, 9, 5, 0, -6, -11, -14, -14, -11, -6, 0, 5, 9, 10, 9, 6, 3, 0, -2, -3, -3, -2, -1).

Crossrefs

Cf. A173547, A173725. A173724 counts squares by largest cell value.