1,5

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.

a(n) is divisible by n.

Only terms indexed by odd numbers coprime to 3 are nonzero, thereforeA007705(n) = a(2n+1) is the main entry. -M. F. Hasler,Jul 01 2019

FromEduard I. Vatutin,Nov 27 2023: (Start)

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).

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).

(End)

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, p. 62-63.

Kevin Pratt,Closed-Form Expressions for the n-Queens Problem and Related Problems,arXiv:1609.09585 [cs.DM], 2016.

I. Rivin, I. Vardi and P. Zimmermann,The n-queens problem,Amer. Math. Monthly, 101 (1994), 629-639.

a(n) =A071607((n-1)/2) * n for odd n. -Eduard I. Vatutin,Nov 27 2023, corrected Apr 10 2024

FromEduard I. Vatutin,Nov 27 2023: (Start)

n=5 (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:

+---------------+

| Q...... |

|... Q... |

|...... Q |

|.. Q.... |

|..... Q. |

|. Q..... |

|.... Q.. |

+---------------+

n=11:

+-----------------------+

| Q.......... |

|.. Q........ |

|.... Q...... |

|...... Q.... |

|........ Q.. |

|.......... Q |

|. Q......... |

|... Q....... |

|..... Q..... |

|....... Q... |

|......... Q. |

+-----------------------+

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):

+---------------------------+ +---------------------------+

| 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)

SeeA007705,which is the main entry for this sequence.

Cf.A000170,A071607.

Sequence in context:A107781A107772A087474*A171809A167302A302576

Adjacent sequences:A051903A051904A051905*A051907A051908A051909

nonn,nice,hard,more

Matthias Engelhardt,Dec 17 1999

Term a(31) added fromA007705byVaclav Kotesovec,Aug 25 2012

approved