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%I A007705 M4691 #59 Apr 10 2024 09:16:52
%S A007705 1,0,10,28,0,88,4524,0,140692,820496,0,128850048,1957725000,0,
%T A007705 605917055356,13404947681712,0
%N A007705 Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board.
%C A007705 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 A007705 W. Ahrens, Mathematische Unterhaltungen und Spiele, Vol. 1, B. G. Teubner, Leipzig, 1921, pp. 363-374.
%D A007705 R. K. Guy, Unsolved problems in Number Theory, 3rd Edn., Springer, 1994, p. 202 [with extensive bibliography]
%D A007705 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007705 Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesly, 1991, Chapter 6.
%H A007705 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 A007705 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, pp. 62-63.
%H A007705 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 A007705 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QueensProblem.html">Queens Problem</a>.
%H A007705 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 A007705 a(n) = A071607(n) * (2*n+1). - _Eduard I. Vatutin_, Jan 22 2024, corrected Mar 14 2024
%F A007705 a(n) = A342990(n) / (2n)!. - _Eduard I. Vatutin_, Apr 09 2024
%e A007705 From _Eduard I. Vatutin_, Jan 22 2024: (Start)
%e A007705 N=5=2*2+1 (all 10 solutions are shown below):
%e A007705 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A007705 | Q . . . . | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
%e A007705 | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | Q . . . . |
%e A007705 | . . . . Q | | . Q . . . | | Q . . . . | | . . Q . . | | . . . Q . |
%e A007705 | . Q . . . | | . . . . Q | | . . Q . . | | Q . . . . | | . Q . . . |
%e A007705 | . . . Q . | | . . Q . . | | . . . . Q | | . . . Q . | | . . . . Q |
%e A007705 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A007705 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A007705 | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | . . . . Q |
%e A007705 | . . . . Q | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
%e A007705 | . Q . . . | | . . Q . . | | . . . . Q | | . . . Q . | | Q . . . . |
%e A007705 | . . . Q . | | . . . . Q | | . . Q . . | | Q . . . . | | . . . Q . |
%e A007705 | Q . . . . | | . Q . . . | | Q . . . . | | . . Q . . | | . Q . . . |
%e A007705 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A007705 N=7=2*3+1:
%e A007705 +---------------+
%e A007705 | Q . . . . . . |
%e A007705 | . . . Q . . . |
%e A007705 | . . . . . . Q |
%e A007705 | . . Q . . . . |
%e A007705 | . . . . . Q . |
%e A007705 | . Q . . . . . |
%e A007705 | . . . . Q . . |
%e A007705 +---------------+
%e A007705 N=11=5*2+1:
%e A007705 +-----------------------+
%e A007705 | Q . . . . . . . . . . |
%e A007705 | . . Q . . . . . . . . |
%e A007705 | . . . . Q . . . . . . |
%e A007705 | . . . . . . Q . . . . |
%e A007705 | . . . . . . . . Q . . |
%e A007705 | . . . . . . . . . . Q |
%e A007705 | . Q . . . . . . . . . |
%e A007705 | . . . Q . . . . . . . |
%e A007705 | . . . . . Q . . . . . |
%e A007705 | . . . . . . . Q . . . |
%e A007705 | . . . . . . . . . Q . |
%e A007705 +-----------------------+
%e A007705 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 A007705 +---------------------------+ +---------------------------+
%e A007705 | Q . . . . . . . . . . . . | | Q . . . . . . . . . . . . |
%e A007705 | . . Q . . . . . . . . . . | | . . Q . . . . . . . . . . |
%e A007705 | . . . . Q . . . . . . . . | | . . . . Q . . . . . . . . |
%e A007705 | . . . . . . Q . . . . . . | | . . . . . . Q . . . . . . |
%e A007705 | . . . . . . . . Q . . . . | | . . . . . . . . . . . Q . |
%e A007705 | . . . . . . . . . . Q . . | | . . . . . . . . . Q . . . |
%e A007705 | . . . . . . . . . . . . Q | | . . . . . . . . . . . . Q |
%e A007705 | . Q . . . . . . . . . . . | | . . . . . Q . . . . . . . |
%e A007705 | . . . Q . . . . . . . . . | | . . . Q . . . . . . . . . |
%e A007705 | . . . . . Q . . . . . . . | | . Q . . . . . . . . . . . |
%e A007705 | . . . . . . . Q . . . . . | | . . . . . . . Q . . . . . |
%e A007705 | . . . . . . . . . Q . . . | | . . . . . . . . . . Q . . |
%e A007705 | . . . . . . . . . . . Q . | | . . . . . . . . Q . . . . |
%e A007705 +---------------------------+ +---------------------------+
%e A007705 (End)
%Y A007705 Cf. A051906, A071607, A342990.
%K A007705 nonn,nice,hard,more
%O A007705 0,3
%A A007705 _N. J. A. Sloane_
%E A007705 Two more terms from _Matthias Engelhardt_, Dec 17 1999 and Jan 11 2001
%E A007705 13404947681712 from _Matthias Engelhardt_, May 01 2005

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