A002567 -id:A002567 - cp's OEIS frontend

A075324 Independent domination number for queens' graph Q(n).

1, 1, 1, 3, 3, 4, 4, 5, 5, 5, 5, 7, 7, 8, 9, 9, 9, 10, 11, 11, 11, 12, 13, 13, 13, 14, 15, 15, 16, 16, 17

Offset: 1

N. J. A. Sloane, Oct 16 2002

			a(8) = 5 queens attacking all squares of standard chessboard:
  . . . . . . . .
  . . . . . Q . .
  . . Q . . . . .
  . . . . Q . . .
  . . . . . . Q .
  . . . Q . . . .
  . . . . . . . .
  . . . . . . . .

W. W. R. Ball and H. S. M. Coxeter, Math'l Rec. and Essays, 13th Ed. Dover, p. 173.
C. Berge, Graphs and Hypergraphs, North-Holland, 1973; p. 304, Example 2.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 49.

William Herbert Bird, Computational methods for domination problems, University of Victoria, 2017. See Table 5.1 on p. 114.
Matthew D. Kearse and Peter B. Gibbons, Computational Methods and New Results for Chessboard Problems, Australasian Journal of Combinatorics 23 (2001), 253-284.
Alexis Langlois-Rémillard, Christoph Müßig, and Érika Róldan, Complexity of Chess Domination Problems, arXiv:2211.05651 [math.CO], 2022.
Alexis Langlois-Rémillard, Christoph Müßig, and Érika Róldan, Solution a(26)-a(31) and Julia code to compute the sequence, 2022.

A002567 gives the number of solutions.

Cf. A075458 (not necessarily independent).

a(19)-a(24) from Bird and a(25) from Kearse & Gibbons added by Andrey Zabolotskiy, Sep 03 2021
a(26) from Alexis Langlois-Rémillard, Christoph Müßig and Érika Roldán added by Christoph Muessig, Aug 25 2022
a(27)-a(31) from Alexis Langlois-Rémillard, Christoph Müßig and Érika Roldán added by Christoph Muessig, Sep 19 2022

A376732 Triangle read by rows: T(n,k) is the maximum number of squares covered (i.e., attacked) by k independent (i.e., non-attacking) queens on an n X n chessboard.

Original entry on oeis.org

1, 4, 0, 9, 9, 0, 12, 15, 16, 16, 17, 23, 25, 25, 25, 20, 30, 35, 36, 36, 36, 25, 37, 45, 49, 49, 49, 49, 28, 44, 55, 62, 64, 64, 64, 64, 33, 52, 66, 76, 81, 81, 81, 81, 81, 36, 60, 77, 92, 100, 100, 100, 100, 100, 100, 41, 68, 88, 104, 121, 121, 121, 121, 121, 121, 121

Offset: 1

John King, Oct 03 2024

T(2,2) = T(3,3) = 0 indicate that there are no solutions to the n-queens problem when n is 2 or 3.

			Triangle begins:
  n\k|  1    2    3    4    5    6    7    8    9   10   11   12
 ----+-----------------------------------------------------------
   1 |  1;
   2 |  4,   0;
   3 |  9,   9,   0;
   4 | 12,  15,  16,  16;
   5 | 17,  23,  25,  25,  25;
   6 | 20,  30,  35,  36,  36,  36;
   7 | 25,  37,  45,  49,  49,  49,  49;
   8 | 28,  44,  55,  62,  64,  64,  64,  64;
   9 | 33,  52,  66,  76,  81,  81,  81,  81,  81;
  10 | 36,  60,  77,  92, 100, 100, 100, 100, 100, 100;
  11 | 41,  68,  88, 104, 121, 121, 121, 121, 121, 121, 121;
  12 | 44,  76, 101, 120, 134, 142, 144, 144, 144, 144, 144, 144;
  13 | 49,  84, 112, 136, 153, 165, 169, 169, 169, 169, 169, ...;
  14 | 52,  92, 125, 152, 172, 186, 194, 196, 196, 196, 196, ...;
  15 | 57, 100, 136, 168, 193, 209, 221, 224, 225, 225, 225, ...;
  16 | 60, 108, 149, 184, 212, 231, 242, 251, 256, 256, 256, ...;
  17 | 65, 116, 160, 200, 233, 255, 269, 281, 289, 289, 289, ...;
  18 | 68, 124, 173, 216, 252, 277, 294, 310, 322, 324, 324, ...;
  ...

Mia Muessig, Table of n, a(n) for n = 1..240
John King, Examples for Queens 1,2,3,4,5 up to 11x11.
John King, Examples for Queens 6,7,8,9 up to 15x15.
Mia Muessig, Julia code to compute the sequence

Columns 1..8 are A047461, A374933, A375116, A374934, A374935, A374936, A374937, A374938.

Cf. A075324, A002567, A075458, A274947.

T(n,k) = n^2 for k >= A075324(n), n >= 4.

Initial terms by John King and Mia Müßig added by Mia Muessig, Oct 05 2024

A002568 Number of different ways one can attack all squares on an n X n chessboard with the smallest number of non-attacking queens needed.

Original entry on oeis.org

1, 4, 1, 16, 16, 120, 8, 728, 92, 8, 2, 840, 24, 436, 10188, 128, 12, 224, 8424, 312, 72, 192, 8784, 368, 56, 224, 14500, 280, 10880, 240

Offset: 1

N. J. A. Sloane

For same problem, but with queens in general position (without condition "non-attacking"), see A002564. - Vaclav Kotesovec, Sep 07 2012

			a(5) = 16 because it is impossible to attack all squares with 2 queens but with 3 queens you can do it in 16 different ways (with mirroring and rotation).

W. Ahrens, Mathematische Unterhaltungen und Spiele, second edition (1910), Vol. 1, p. 301.
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).

Mia Müßig, Julia code to compute the sequence
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), 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]

See A002567 for the number of non-isomorphic solutions.

Cf. A002564, A122749, A103315.

a(9)-a(12) from Johan Särnbratt, Mar 28 2008
Name of the sequence corrected by Vaclav Kotesovec, Sep 07 2012
a(13)-a(15) from Andrew Howroyd, Dec 07 2021
a(16)-a(30) from Mia Muessig, Oct 04 2024

A002565 Number of non-isomorphic ways to attack all squares on an n X n chessboard using the smallest possible number of queens with each queen attacking at least one other.

Original entry on oeis.org

0, 2, 5, 3, 15, 150, 5, 56, 3, 39, 681

Offset: 1

N. J. A. Sloane

Differs from A002563 and A002567 in that each queen is attacking at least one other queen. - Sean A. Irvine, Apr 05 2014

The Sainte-Laguë paper has "a(6)=140?". - Sean A. Irvine, Apr 05 2014

W. Ahrens, Mathematische Unterhaltungen und Spiele, second edition (1910), Vol. 1, p. 301.
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. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), 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]

Cf. A002566 (all solutions for attacking queens). - Sean A. Irvine, Apr 05 2014

a(6) corrected and a(9)-a(11) from Sean A. Irvine, Apr 05 2014

Better name from Sean A. Irvine, Apr 05 2014

cp's OEIS Frontend

A075324 Independent domination number for queens' graph Q(n).

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A376732 Triangle read by rows: T(n,k) is the maximum number of squares covered (i.e., attacked) by k independent (i.e., non-attacking) queens on an n X n chessboard.

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A002568 Number of different ways one can attack all squares on an n X n chessboard with the smallest number of non-attacking queens needed.

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A002565 Number of non-isomorphic ways to attack all squares on an n X n chessboard using the smallest possible number of queens with each queen attacking at least one other.

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