A002562 Number of ways of placing n nonattacking queens on n X n board (symmetric solutions count only once). (Formerly M0180 N0068)

1, 0, 0, 1, 2, 1, 6, 12, 46, 92, 341, 1787, 9233, 45752, 285053, 1846955, 11977939, 83263591, 621012754, 4878666808, 39333324973, 336376244042, 3029242658210, 28439272956934, 275986683743434, 2789712466510289, 29363495934315694

Offset

1,5

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 238.

Links

Table of n, a(n) for n=1..27.

J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656. [Annotated scanned copy]

J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656.

P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]

V. Chvatal, All solutions to the problem of eight queens

V. Chvatal, All solutions to the problem of eight queens [Cached copy, pdf format, with permission]

Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-1. Illustrates a(8)=12.

Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-2.

Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-3.

Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-4.

Thomas Preusser, Queens%40TUD-Project

E. M. Reingold, Letter to N. J. A. Sloane, Dec 27 1973

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]

Eric Weisstein's World of Mathematics, Queens Problem.

M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]

WikiMili, Eight queens puzzle, (2019).

Formula

a(n) = (1/8) * (Q(n) + P(n) + 2 * R(n)), where Q(n) = A000170(n) [all solutions], P(n) = A032522(n) [point symmetric solutions] and R(n) = A033148(n) [rotationally symmetric solutions].

Example

a(4) = 1:
  +---------+
  | . . Q . |
  | Q . . . |
  | . . . Q |
  | . Q . . |
  +---------+
a(5) = 2:
  +-----------+ +-----------+
  | . . . Q . | | . . . Q . |
  | . Q . . . | | Q . . . . |
  | . . . . Q | | . . Q . . |
  | . . Q . . | | . . . . Q |
  | Q . . . . | | . Q . . . |
  +-----------+ +-----------+
a(6) = 1:
  +-------------+
  | . . . . Q . |
  | . . Q . . . |
  | Q . . . . . |
  | . . . . . Q |
  | . . . Q . . |
  | . Q . . . . | - Hugo Pfoertner, Mar 17 2019
  +-------------+
a(7) = 6:
  +---------------+  +---------------+  +---------------+
  | 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 . . |
  +---------------+  +---------------+  +---------------+
- Hugo Pfoertner, Mar 18 2019

Crossrefs

Cf. A000170, A032522, A033148.

Sequence in context: A350749 A347594 A128534 * A218492 A136456 A123968

Adjacent sequences: A002559 A002560 A002561 * A002563 A002564 A002565

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

a(17) and a(18) found by Ulrich Schimke in Goettingen, Germany (UlrSchimke(AT)aol.com)

Formula and a(19) to a(23) added by Matthias Engelhardt in Nuremberg, Germany, Jan 23 2000

Terms (calculated from formula) added by Thomas B. Preußer, Dec 15 2008

a(26) (derived from formula after recent extension of A000170) added by Thomas B. Preußer, Jul 12 2009

a(27) (derived from formula after recent extension of A000170) added by Thomas B. Preußer, Sep 23 2016

Status

approved