cp's OEIS Frontend

A051906 Number of ways of placing n nonattacking queens on an n X n toroidal chessboard.

%I A051906 #76 Apr 13 2024 22:59:37
%S A051906 1,0,0,0,10,0,28,0,0,0,88,0,4524,0,0,0,140692,0,820496,0,0,0,
%T A051906 128850048,0,1957725000,0,0,0,605917055356,0,13404947681712,0,0,0
%N A051906 Number of ways of placing n nonattacking queens on an n X n toroidal chessboard.
%C A051906 The sequence has been computed up to n = 23 by Rivin, Vardi & Zimmermann, see p. 637 of their paper from 1994. Further terms were calculated by the submitter, Dec 17 1999 and Jan 11 2001.
%C A051906 a(n) is divisible by n.
%C A051906 Only terms indexed by odd numbers coprime to 3 are nonzero, therefore A007705(n) = a(2n+1) is the main entry. - _M. F. Hasler_, Jul 01 2019
%C A051906 From _Eduard I. Vatutin_, Nov 27 2023: (Start)
%C A051906 For n <= 11 all solutions 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 >= 13 some solutions are same, but not all (see examples).
%C A051906 All solutions of n-queens problem on toroidal chessboard are also solutions of n-queens problem on classical chessboard; the converse is not true, so a(n) <= A000170(n).
%C A051906 (End)
%H A051906 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 A051906 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 62-63.
%H A051906 Kevin Pratt, <a href="https://arxiv.org/abs/1609.09585">Closed-Form Expressions for the n-Queens Problem and Related Problems</a>, arXiv:1609.09585 [cs.DM], 2016.
%H A051906 I. Rivin, I. Vardi and P. Zimmermann, <a href="http://www.jstor.org/stable/2974691">The n-queens problem</a>, Amer. Math. Monthly, 101 (1994), 629-639.
%F A051906 a(n) = A071607((n-1)/2) * n for odd n. - _Eduard I. Vatutin_, Nov 27 2023, corrected Apr 10 2024
%e A051906 From _Eduard I. Vatutin_, Nov 27 2023: (Start)
%e A051906 n=5 (all 10 solutions are shown below):
%e A051906 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A051906 | Q . . . . | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
%e A051906 | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | Q . . . . |
%e A051906 | . . . . Q | | . Q . . . | | Q . . . . | | . . Q . . | | . . . Q . |
%e A051906 | . Q . . . | | . . . . Q | | . . Q . . | | Q . . . . | | . Q . . . |
%e A051906 | . . . Q . | | . . Q . . | | . . . . Q | | . . . Q . | | . . . . Q |
%e A051906 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A051906 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A051906 | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | . . . . Q |
%e A051906 | . . . . Q | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
%e A051906 | . Q . . . | | . . Q . . | | . . . . Q | | . . . Q . | | Q . . . . |
%e A051906 | . . . Q . | | . . . . Q | | . . Q . . | | Q . . . . | | . . . Q . |
%e A051906 | Q . . . . | | . Q . . . | | Q . . . . | | . . Q . . | | . Q . . . |
%e A051906 +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
%e A051906 n=7:
%e A051906 +---------------+
%e A051906 | Q . . . . . . |
%e A051906 | . . . Q . . . |
%e A051906 | . . . . . . Q |
%e A051906 | . . Q . . . . |
%e A051906 | . . . . . Q . |
%e A051906 | . Q . . . . . |
%e A051906 | . . . . Q . . |
%e A051906 +---------------+
%e A051906 n=11:
%e A051906 +-----------------------+
%e A051906 | Q . . . . . . . . . . |
%e A051906 | . . Q . . . . . . . . |
%e A051906 | . . . . Q . . . . . . |
%e A051906 | . . . . . . Q . . . . |
%e A051906 | . . . . . . . . Q . . |
%e A051906 | . . . . . . . . . . Q |
%e A051906 | . Q . . . . . . . . . |
%e A051906 | . . . Q . . . . . . . |
%e A051906 | . . . . . Q . . . . . |
%e A051906 | . . . . . . . Q . . . |
%e A051906 | . . . . . . . . . Q . |
%e A051906 +-----------------------+
%e A051906 n=13 (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 A051906 +---------------------------+ +---------------------------+
%e A051906 | Q . . . . . . . . . . . . | | Q . . . . . . . . . . . . |
%e A051906 | . . Q . . . . . . . . . . | | . . Q . . . . . . . . . . |
%e A051906 | . . . . Q . . . . . . . . | | . . . . Q . . . . . . . . |
%e A051906 | . . . . . . Q . . . . . . | | . . . . . . Q . . . . . . |
%e A051906 | . . . . . . . . Q . . . . | | . . . . . . . . . . . Q . |
%e A051906 | . . . . . . . . . . Q . . | | . . . . . . . . . Q . . . |
%e A051906 | . . . . . . . . . . . . Q | | . . . . . . . . . . . . Q |
%e A051906 | . Q . . . . . . . . . . . | | . . . . . Q . . . . . . . |
%e A051906 | . . . Q . . . . . . . . . | | . . . Q . . . . . . . . . |
%e A051906 | . . . . . Q . . . . . . . | | . Q . . . . . . . . . . . |
%e A051906 | . . . . . . . Q . . . . . | | . . . . . . . Q . . . . . |
%e A051906 | . . . . . . . . . Q . . . | | . . . . . . . . . . Q . . |
%e A051906 | . . . . . . . . . . . Q . | | . . . . . . . . Q . . . . |
%e A051906 +---------------------------+ +---------------------------+
%e A051906 (End)
%Y A051906 See A007705, which is the main entry for this sequence.
%Y A051906 Cf. A000170, A071607.
%K A051906 nonn,nice,hard,more
%O A051906 1,5
%A A051906 _Matthias Engelhardt_, Dec 17 1999
%E A051906 Term a(31) added from A007705 by _Vaclav Kotesovec_, Aug 25 2012