A370672 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board using knight moves.

1, 0, 10, 28, 0, 88, 130, 0, 238, 304, 0, 460, 250, 0, 754, 868, 0, 280, 1258, 0, 1558, 1720, 0, 2068, 1372, 0, 2650, 880, 0, 3304, 3538, 0, 1300, 4288, 0, 4828, 5110, 0, 2464, 6004, 0, 6640, 2380, 0, 7654, 3640, 0

Offset: 0

Author

Eduard I. Vatutin, Feb 25 2024

Keywords

• nonn

Comments

All solutions of this type can be found using a knight moving with some displacements dx and dy starting from some cell with coordinates (x,y): (x,y) -> (x+dx,y+dy) -> (x+2*dx,y+2*dy) -> ... -> (x,y) (all operations modulo n). For n <= 11 all solutions of n nonattacking queens on n X n a toroidal board problem are solutions of this type, for n >= 13 some solutions are not of this type (see A051906 for examples).

Examples

			For n=2*2+1=5 there are 10 solutions:
.
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
| Q . . . . | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
| . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | Q . . . . |
| . . . . Q | | . Q . . . | | Q . . . . | | . . Q . . | | . . . Q . |
| . Q . . . | | . . . . Q | | . . Q . . | | Q . . . . | | . Q . . . |
| . . . Q . | | . . Q . . | | . . . . Q | | . . . Q . | | . . . . Q |
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
.
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
| . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | . . . . Q |
| . . . . Q | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
| . Q . . . | | . . Q . . | | . . . . Q | | . . . Q . | | Q . . . . |
| . . . Q . | | . . . . Q | | . . Q . . | | Q . . . . | | . . . Q . |
| Q . . . . | | . Q . . . | | Q . . . . | | . . Q . . | | . Q . . . |
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
.
so a(2)=10.
		

Links

• Eduard I. Vatutin, Arranging of N queens on toroidal board and generating pandiagonal Latin squares using them (in Russian).
• Eduard I. Vatutin, Numerical formula between number of cyclic diagonal Latin squares and number of toroidal n-queens problem solutions getting by knight movement (in Russian).
• Index entries for sequences related to Latin squares and rectangles.

Crossrefs

Cf. A007705, A051906, A123565, A338562.

Formula

a(n) = A123565(2*n+1) * (2*n+1).

a(n) = A338562(n) / (2n)!. - Eduard I. Vatutin, Mar 13 2024