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Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board.
(Formerly M4691)
26

%I M4691 #59 Apr 10 2024 09:16:52

%S 1,0,10,28,0,88,4524,0,140692,820496,0,128850048,1957725000,0,

%T 605917055356,13404947681712,0

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

%C 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

%D W. Ahrens, Mathematische Unterhaltungen und Spiele, Vol. 1, B. G. Teubner, Leipzig, 1921, pp. 363-374.

%D R. K. Guy, Unsolved problems in Number Theory, 3rd Edn., Springer, 1994, p. 202 [with extensive bibliography]

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesly, 1991, Chapter 6.

%H M. R. Engelhardt, <a href="http://dx.doi.org/10.1016/j.disc.2007.01.007">A group-based search for solutions of the n-queens problem</a>, Discr. Math., 307 (2007), 2535-2551.

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, pp. 62-63.

%H I. Rivin, I. Vardi and P. Zimmermann, <a href="https://www.jstor.org/stable/2974691">The n-queens problem</a>, Amer. Math. Monthly, 101 (1994), 629-639.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QueensProblem.html">Queens Problem</a>.

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_2691">Numerical formula between number of horizontally or vertically semicyclic diagonal Latin squares and number of toroidal n-queens problem solutions</a> (in Russian).

%F a(n) = A071607(n) * (2*n+1). - _Eduard I. Vatutin_, Jan 22 2024, corrected Mar 14 2024

%F a(n) = A342990(n) / (2n)!. - _Eduard I. Vatutin_, Apr 09 2024

%e From _Eduard I. Vatutin_, Jan 22 2024: (Start)

%e N=5=2*2+1 (all 10 solutions are shown below):

%e +-----------+ +-----------+ +-----------+ +-----------+ +-----------+

%e | Q . . . . | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |

%e | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | Q . . . . |

%e | . . . . Q | | . Q . . . | | Q . . . . | | . . Q . . | | . . . Q . |

%e | . Q . . . | | . . . . Q | | . . Q . . | | Q . . . . | | . Q . . . |

%e | . . . Q . | | . . Q . . | | . . . . Q | | . . . Q . | | . . . . Q |

%e +-----------+ +-----------+ +-----------+ +-----------+ +-----------+

%e +-----------+ +-----------+ +-----------+ +-----------+ +-----------+

%e | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | . . . . Q |

%e | . . . . Q | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |

%e | . Q . . . | | . . Q . . | | . . . . Q | | . . . Q . | | Q . . . . |

%e | . . . Q . | | . . . . Q | | . . Q . . | | Q . . . . | | . . . Q . |

%e | Q . . . . | | . Q . . . | | Q . . . . | | . . Q . . | | . Q . . . |

%e +-----------+ +-----------+ +-----------+ +-----------+ +-----------+

%e N=7=2*3+1:

%e +---------------+

%e | Q . . . . . . |

%e | . . . Q . . . |

%e | . . . . . . Q |

%e | . . Q . . . . |

%e | . . . . . Q . |

%e | . Q . . . . . |

%e | . . . . Q . . |

%e +---------------+

%e N=11=5*2+1:

%e +-----------------------+

%e | Q . . . . . . . . . . |

%e | . . Q . . . . . . . . |

%e | . . . . Q . . . . . . |

%e | . . . . . . Q . . . . |

%e | . . . . . . . . Q . . |

%e | . . . . . . . . . . Q |

%e | . Q . . . . . . . . . |

%e | . . . Q . . . . . . . |

%e | . . . . . Q . . . . . |

%e | . . . . . . . Q . . . |

%e | . . . . . . . . . Q . |

%e +-----------------------+

%e 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):

%e +---------------------------+ +---------------------------+

%e | Q . . . . . . . . . . . . | | Q . . . . . . . . . . . . |

%e | . . Q . . . . . . . . . . | | . . Q . . . . . . . . . . |

%e | . . . . Q . . . . . . . . | | . . . . Q . . . . . . . . |

%e | . . . . . . Q . . . . . . | | . . . . . . Q . . . . . . |

%e | . . . . . . . . Q . . . . | | . . . . . . . . . . . Q . |

%e | . . . . . . . . . . Q . . | | . . . . . . . . . Q . . . |

%e | . . . . . . . . . . . . Q | | . . . . . . . . . . . . Q |

%e | . Q . . . . . . . . . . . | | . . . . . Q . . . . . . . |

%e | . . . Q . . . . . . . . . | | . . . Q . . . . . . . . . |

%e | . . . . . Q . . . . . . . | | . Q . . . . . . . . . . . |

%e | . . . . . . . Q . . . . . | | . . . . . . . Q . . . . . |

%e | . . . . . . . . . Q . . . | | . . . . . . . . . . Q . . |

%e | . . . . . . . . . . . Q . | | . . . . . . . . Q . . . . |

%e +---------------------------+ +---------------------------+

%e (End)

%Y Cf. A051906, A071607, A342990.

%K nonn,nice,hard,more

%O 0,3

%A _N. J. A. Sloane_

%E Two more terms from _Matthias Engelhardt_, Dec 17 1999 and Jan 11 2001

%E 13404947681712 from _Matthias Engelhardt_, May 01 2005