cp's OEIS Frontend

A002564 Number of different ways one can attack all squares on an n X n chessboard using the minimum number of queens.

%I A002564 M3199 N1293 #106 Jul 28 2025 08:44:54
%S A002564 1,4,1,12,186,4,86,4860,114,8,2,8,288,4632,205832,2968,124,16,84
%N A002564 Number of different ways one can attack all squares on an n X n chessboard using the minimum number of queens.
%C A002564 Number of distinct solutions to minimum dominating set on queens' graph Q(n). See A002563 for non-isomorphic solutions.
%C A002564 For same problem, but with non-attacking queens, see A002568. - _Vaclav Kotesovec_, Sep 07 2012
%C A002564 In other words, number of minimum dominating sets in the n X n queen graph. - _Eric W. Weisstein_, Dec 31 2017
%C A002564 For n > 2, also the number of minimal edge covers in the n X n queen graph. - _Eric W. Weisstein_, Dec 09 2024
%C A002564 a(20) >= 4152. - _Eric W. Weisstein_, Jul 28 2025
%D A002564 W. Ahrens, Mathematische Unterhaltungen und Spiele, second edition (1910), Vol. 1, p. 301.
%D A002564 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002564 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002564 Andy Huchala, <a href="/A002564/a002564.py.txt">Python program</a>.
%H A002564 Matthew D. Kearse and Peter B. Gibbons, <a href="http://ajc.maths.uq.edu.au/pdf/23/ocr-ajc-v23-p253.pdf">Computational Methods and New Results for Chessboard Problems</a>, Australasian Journal of Combinatorics 23 (2001), 253-284.
%H A002564 Mia Müßig, <a href="https://gist.github.com/PhoenixSmaug/98f30dbd66e17f51b05e5d5729500841">Julia code to compute the sequence</a>
%H A002564 M. A. Sainte-Laguë, <a href="https://eudml.org/doc/192551">Les Réseaux (ou Graphes)</a>, Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49.
%H A002564 M. A. Sainte-Laguë, <a href="/A002560/a002560.pdf">Les Réseaux (ou Graphes)</a>, Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49. [Incomplete annotated scan of title page and pages 18-51]
%H A002564 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalEdgeCover.html">Minimal Edge Cover</a>.
%H A002564 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</a>.
%H A002564 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QueenGraph.html">Queen Graph</a>.
%H A002564 Eric W. Weisstein, <a href="/A002564/a002564.png">Symmetrically inequivalent configurations for n = 1 to 7</a>
%H A002564 Eric W. Weisstein, <a href="/A002564/a002564_1.png">Symmetrically inequivalent configurations for n = 8</a>
%H A002564 Eric W. Weisstein, <a href="/A002564/a002564_2.png">Symmetrically inequivalent configurations for n = 9 to 13</a>
%H A002564 Eric W. Weisstein, <a href="/A002564/a002564_6.png">Symmetrically inequivalent configurations for n = 14</a>
%H A002564 Eric W. Weisstein, <a href="/A002564/a002564_5.png">Symmetrically inequivalent configurations for n = 16</a>
%H A002564 Eric W. Weisstein, <a href="/A002564/a002564_4.png">Symmetrically inequivalent configurations for n = 17 to 19</a>
%Y A002564 Cf. A002563, A002568, A182333, A286883.
%Y A002564 A075458 gives number of queens required. - _Sean A. Irvine_, Apr 05 2014
%K A002564 nonn,more
%O A002564 1,2
%A A002564 _N. J. A. Sloane_
%E A002564 New name of the sequence from _Vaclav Kotesovec_, Sep 07 2012
%E A002564 a(9)-a(10) from _Vaclav Kotesovec_, Sep 07 2012
%E A002564 a(11) from _Svyatoslav Starkov_, Sep 16 2013
%E A002564 a(12)-a(13) from _Sean A. Irvine_, Apr 07 2014
%E A002564 Definition edited by _N. J. A. Sloane_, Dec 25 2017 at the suggestion of _Brendan McKay_.
%E A002564 a(14) from _Andy Huchala_, Mar 13 2024
%E A002564 a(15)-a(19) from _Mia Muessig_, Oct 04 2024