A007705 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board.

1, 0, 10, 28, 0, 88, 4524, 0, 140692, 820496, 0, 128850048, 1957725000, 0, 605917055356, 13404947681712, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more

Comments

Polya proved (see Ahrens) that the number of solutions to this problem for an m X m board is > 0 iff m is coprime to 6. - Jonathan Vos Post, Feb 20 2005

Examples

			From _Eduard I. Vatutin_, Jan 22 2024: (Start)
N=5=2*2+1 (all 10 solutions are shown below):
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
| 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 . . . |
+-----------+ +-----------+ +-----------+ +-----------+ +-----------+
N=7=2*3+1:
+---------------+
| Q . . . . . . |
| . . . Q . . . |
| . . . . . . Q |
| . . Q . . . . |
| . . . . . Q . |
| . Q . . . . . |
| . . . . Q . . |
+---------------+
N=11=5*2+1:
+-----------------------+
| Q . . . . . . . . . . |
| . . Q . . . . . . . . |
| . . . . Q . . . . . . |
| . . . . . . Q . . . . |
| . . . . . . . . Q . . |
| . . . . . . . . . . Q |
| . Q . . . . . . . . . |
| . . . Q . . . . . . . |
| . . . . . Q . . . . . |
| . . . . . . . Q . . . |
| . . . . . . . . . Q . |
+-----------------------+
N=13=6*2+1 (first example can be found using a knight moving from cell (1,1) with dx=1 and dy=2, second example can't be obtained in the same way):
+---------------------------+ +---------------------------+
| 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 . . . . |
+---------------------------+ +---------------------------+
(End)
		

References

• W. Ahrens, Mathematische Unterhaltungen und Spiele, Vol. 1, B. G. Teubner, Leipzig, 1921, pp. 363-374.
• R. K. Guy, Unsolved problems in Number Theory, 3rd Edn., Springer, 1994, p. 202 [with extensive bibliography]
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesly, 1991, Chapter 6.

Links

• M. R. Engelhardt, A group-based search for solutions of the n-queens problem, Discr. Math., 307 (2007), 2535-2551.
• Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 62-63.
• I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
• Eric Weisstein's World of Mathematics, Queens Problem.
• Eduard I. Vatutin, Numerical formula between number of horizontally or vertically semicyclic diagonal Latin squares and number of toroidal n-queens problem solutions (in Russian).

Crossrefs

Cf. A051906, A071607, A342990.

Formula

a(n) = A071607(n) * (2*n+1). - Eduard I. Vatutin, Jan 22 2024, corrected Mar 14 2024

a(n) = A342990(n) / (2n)!. - Eduard I. Vatutin, Apr 09 2024

Extensions

Two more terms from Matthias Engelhardt, Dec 17 1999 and Jan 11 2001

13404947681712 from Matthias Engelhardt, May 01 2005