12,2

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.

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.

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).

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.

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

Sequence in context: A364612 A176099 A160790 * A000376 A000375 A244488

Adjacent sequences: A173723 A173724 A173725 * A173727 A173728 A173729

Thomas Zaslavsky, Feb 23 2010

approved