A260318 Number of doubly symmetric characteristic solutions to the n-queens problem.

1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 32, 64, 0, 0, 240, 352, 0, 0, 1664, 1632, 0, 0, 16448, 21888, 0, 0, 203392, 333952, 0, 0, 2922752, 4325376, 0, 0, 38592000, 50746368, 0, 0, 630794240, 897616896, 0, 0, 10758713344, 17514086400, 0, 0, 203437559808, 326022221824, 0, 0, 4306790547456, 6265275064320, 0, 0, 97204813266944, 145913049251840, 0, 0

Offset: 1

Author

N. J. A. Sloane, Jul 22 2015

Keywords

• nonn
• more

Comments

The problem of placing eight queens on a chessboard so that no one of them can take any other in a single move is a particular case of the more general problem: On a square array of n X n cells place n objects, one on each of n different cells, in such a way that no two of them lie on the same row, column, or diagonal.

There are no (interesting) doubly centrosymmetric solutions for n < 4, and there is just one complete set for n = 4: 2413, 3142 and one for n = 5: 25314, 41352.

On the ordinary chessboard of 8 X 8 cells there are a total of 92 solutions, consisting of 11 sets of equivalent ordinary solutions and one set of equivalent symmetric solutions. There are no doubly symmetric solutions in this case.

References

• Maurice Kraitchik: Mathematical Recreations. Mineola, NY: Dover, 2nd ed. 1953, pp. 247-255 (The Problem of the Queens).

Links

• P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]
• M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47.
• M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47. [Incomplete annotated scan of title page and pages 18-51]

Crossrefs

A002562 = A260318 + A260319 + A260320, A000170 = 2*A260318 + 4*A260319 + 8*A260320 (n>1).

Formula

a(n) = A033148(n) / 2 for n >= 2. - Don Knuth, Jun 20 2017

Extensions

More terms, due to Don Knuth, added by Colin Barker, Jun 20 2017