A266237 Number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations, reflections, and M-transformations.

1, 0, 1, 220, 68826306, 739745383235859818

Offset: 1

Author

Max Alekseyev, Dec 25 2015

Keywords

• nonn
• hard
• more

Comments

Chebrakov (2008) defines M-transformations of a magic square to be simultaneous permutations of its rows/columns that preserve the content of each diagonal (i.e., M-transformations can only shuffle the diagonal elements). The number of M-transformations of a magic square of order n equals A000165(floor(n/2)) = 2*A002866(floor(n/2)). Half of the M-transformations can be obtained from the other half by rotations by 180 degrees (or by reflections about a diagonal).

Obviously, there is no magic square for n=2, although the MATLAB command magic(n) returns a non-magic square with determinant -10. - Altug Alkan, Dec 25 2015

Links

• Yu. V. Chebrakov, Section 3.1.2 and Section 3.2.2 in "Theory of Magic Matrices. Issue TMM-1.", 2008. (in Russian)
• Hidetoshi Mino, The number of magic squares of order 6.

Crossrefs

Cf. A006052.

Formula

a(n) = A006052(n) / A002866(floor(n/2)).

Extensions

a(6) from Hidetoshi Mino, Jul 22 2023

a(6) corrected by Hidetoshi Mino, May 31 2024