A007705 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board. (Formerly M4691)

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

Offset

0,3

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

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

Table of n, a(n) for n=0..16.

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).

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

Example

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)

Crossrefs

Cf. A051906, A071607, A342990.

Sequence in context: A362737 A262919 A370672 * A196356 A196359 A184686

Adjacent sequences: A007702 A007703 A007704 * A007706 A007707 A007708

Keyword

nonn,nice,hard,more

Author

N. J. A. Sloane

Extensions

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

13404947681712 from Matthias Engelhardt, May 01 2005

Status

approved