

A000170


Number of ways of placing n nonattacking queens on an n X n board.
(Formerly M1958 N0775)


73



1, 1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 365596, 2279184, 14772512, 95815104, 666090624, 4968057848, 39029188884, 314666222712, 2691008701644, 24233937684440, 227514171973736, 2207893435808352, 22317699616364044, 234907967154122528
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OFFSET

0,5


COMMENTS

For n > 3, a(n) is the number of maximum independent vertex sets in the n X n queen graph.  Eric W. Weisstein, Jun 20 2017
Number of nodes on level n of the backtrack tree for the n queens problem (a(n) = A319284(n, n)).  Peter Luschny, Sep 18 2018


REFERENCES

M. Gardner, The Unexpected Hanging, pp. 1902, Simon & Shuster NY 1969
Jieh Hsiang, YuhPyng Shieh and YaoChiang Chen, The cyclic complete mappings counting problems, in Problems and Problem Sets for ATP, volume 0210 of DIKU technical reports, G. Sutcliffe, J. Pelletier and C. Suttner, eds., 2002.
D. E. Knuth, The Art of Computer Programming, Volume 4, Prefascicle 5B, Introduction to Backtracking, 7.2.2. Backtrack programming. 2018.
Massimo Nocentini, "An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation", PhD Thesis, University of Florence, 2019. See Ex. 67.
M. A. SainteLaguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, GauthierVillars, Paris, 1926, p. 47.
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).
R. J. Walker, An enumerative technique for a class of combinatorial problems, pp. 9194 of Proc. Sympos. Applied Math., vol. 10, Amer. Math. Soc., 1960.
M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 238.


LINKS

Table of n, a(n) for n=0..27.
Jordan Bell, Brett Stevens, A survey of known results and research areas for nqueens, Discrete Mathematics, Volume 309, Issue 1, Jan 06 2009, Pages 131.
D. Bill, Durango Bill's The NQueens Problem
J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651656.
J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651656. [Annotated scanned copy]
P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?)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]
Gheorghe Coserea, Solutions for n=8.
Gheorghe Coserea, Solutions for n=9.
Gheorghe Coserea, MiniZinc model for generating solutions.
Matteo Fischetti, Domenico Salvagnin, Chasing First Queens by Integer Programming, 2018.
J. Freeman, A neural network solution to the nqueens problem, The Mathematica J., 3 (No. 3, 1993), 5256.
Ian P. Gent, Christopher Jefferson, Peter Nightingale, Complexity of nQueens Completion, Journal of Artificial Intelligence Research 59 (2017), see p 816.
Eric Grigoryan, Investigation of the Regularities in the Formation of Solutions nQueens Problem, Modeling of Artificial Intelligence, 2018, 5(1), 321.
James Grime and Brady Haran, The 8 Queen Problem, Numberphile video (2015).
Kenji Kise, Takahiro Katagiri, Hiroki Honda and Toshitsugu Yuba, Solving the 24queens Problem using MPI on a PC Cluster, Technical Report UECIS20046, Graduate School of Information Systems, The University of ElectroCommunications (2004).
D. E. Knuth, Donald Knuth's 24th Annual Christmas Lecture: Dancing Links, Stanfordonline, Video published on YouTube, Dec 12, 2018.
W. Kosters, nQueens bibliography
V. Kotesovec, Nonattacking chess pieces, Sixth edition, 795 pages, Feb 02 2013 (minor update Mar 29 2016).
E. Masehian, H. Akbaripour, N. MohabbatiKalejahi, Solving the n Queens Problem using a Tuned Hybrid Imperialist Competitive Algorithm, 2013.
E. Masehian, H. Akbaripour, N. MohabbatiKalejahi, Landscape analysis and efficient metaheuristics for solving the nqueens problem, Computational Optimization and Applications, 2013; DOI 10.1007/s105890139578z.
Nasrin MohabbatiKalejahi, Hossein Akbaripour, Ellips Masehian, Basic and Hybrid Imperialist Competitive Algorithms for Solving the Nonattacking and Nondominating n Queens Problems, Studies in Computational Intelligence Volume 577, 2015, pp 7996. DOI 10.1007/9783319112718_6.
J. Pope, D. Sonnier, A linear solution to the nQueens problem using vector spaces, Journal of Computing Sciences in Colleges, Volume 29 Issue 5, May 2014 Pages 7783.
T. B. Preußer, M. R. Engelhardt, Putting Queens in Carry Chains, No. 27, Journal of Signal Processing Systems, Volume 88, Issue 2, August 2017. (The title refers to the fact that the article discusses the case n = 27.)
Thomas B. Preußer, Bernd Nägel, Rainer G. Spallek, Putting Queens in Carry Chains, Slides, HIPEAC WRC'09.
E. M. Reingold, Letter to N. J. A. Sloane, Dec 27 1973
I. Rivin, I. Vardi and P. Zimmermann, The nqueens problem, Amer. Math. Monthly, 101 (1994), 629639.
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set
Eric Weisstein's World of Mathematics, Queen Graph
Eric Weisstein's World of Mathematics, Queens Problem
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237240]
Cheng Zhang, Jianpeng Ma, Counting Solutions for the Nqueens and Latin Square Problems by Efficient Monte Carlo Simulations, arXiv:0808.4003 [condmat.statmech], 2008.


FORMULA

Strong conjecture: there is a constant c around 2.54 such that a(n) is asymptotic to n!/c^n; weak conjecture: lim_{n > infinity} (1/n) * log(n!/a(n)) = constant = 0.90....  Benoit Cloitre, Nov 10 2002


EXAMPLE

a(2) = a(3) = 0, since on 2 X 2 and 3 X 3 chessboards there are no solutions.
For n=4 the a(4)=2 solutions are:
oxoo ooxo
ooox xooo
xooo ooox
ooxo oxoo


CROSSREFS

See A140393 for another version. Cf. A002562, A065256.
Cf. A036464 (2Q), A047659 (3Q), A061994 (4Q), A108792 (5Q), A176186 (6Q).
Cf. A099152, A006717, A051906, A319284 (backtrack trees).
Sequence in context: A189869 A054790 A140393 * A038216 A213603 A145911
Adjacent sequences: A000167 A000168 A000169 * A000171 A000172 A000173


KEYWORD

nonn,hard,nice,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

Terms for n=2123 computed by Sylvain PION (Sylvain.Pion(AT)sophia.inria.fr) and JoelYann FOURRE (JoelYann.Fourre(AT)ens.fr).
a(24) from Kenji KISE (kis(AT)is.uec.ac.jp), Sep 01 2004
a(25) from Objectweb ProActive INRIA Team (proactive(AT)objectweb.org), Jun 11 2005 [Communicated by Alexandre Di Costanzo (Alexandre.Di_Costanzo(AT)sophia.inria.fr)]. This calculation took about 53 years of CPU time.
a(25) has been confirmed by the NTU 25Queen Project at National Taiwan University and Ming Chuan University, led by YuhPyng (Arping) Shieh, Jul 26 2005. This computation took 26613 days CPU time.
The NQueensatHome web site gives a different value for a(24), 226732487925864. Thanks to Goran Fagerstrom for pointing this out. I do not know which value is correct. I have therefore created a new entry, A140393, which gives the NQueensathome version of the sequence.  N. J. A. Sloane, Jun 18 2008
It now appears that this sequence (A000170) is correct and A140393 is wrong.  N. J. A. Sloane, Nov 08 2008
Added a(26) as calculated by Queens(AT)TUD [http://queens.inf.tudresden.de/].  Thomas B. Preußer, Jul 11 2009
Added a(27) as calculated by the Q27 Project [https://github.com/preusser/q27].  Thomas B. Preußer, Sep 23 2016
a(0) = 1 prepended by Joerg Arndt, Sep 16 2018


STATUS

approved



