A182333 -id:A182333 - cp's OEIS frontend

A005635 Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).

1, 1, 1, 1, 3, 8, 36, 110, 666, 3250, 23436, 125198, 1037520, 7241272, 66360960, 500827928, 5080370400, 45926666984, 508032504000, 4919789029480, 59256857923200, 656763542278304, 8532986822438400, 100525959568386848, 1405335514253932800, 18431883489984091552

Offset: 0

N. J. A. Sloane

From Vaclav Kotesovec, Apr 26 2012: (Start)
This sequence gives (according to the article by Robinson) the number of inequivalent solutions.
For the total number of all arrangements of n non-attacking bishops such that every square of the board is controlled by at least one bishop, see A122749.
For the total number of all arrangements of n bishops (in any position) such that every square of the board is controlled by at least one bishop, see A182333.
(End)

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

N. J. A. Sloane, Table of n, a(n) for n = 0..250
Jean-François Alcover, Mathematica program.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Cf. A005631, A005632, A005633, A005634, A122749, A123071, A123072, A182333.

E:=proc(n) local k; if n mod 2 = 0 then k := n/2; if k mod 2 = 0 then RETURN( (k!*(k+2)/2)^2 ); else RETURN( ((k-1)!*(k+1)^2/2)^2 ); fi; else k := (n-1)/2; if k mod 2 = 0 then RETURN( ((k!)^2/12)*(3*k^3+16*k^2+18*k+8) ); else RETURN( ((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3) ); fi; fi; end; # Gives A122749
unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085
C:=proc(n) local k; if n mod 2 = 0 then RETURN(0); fi; k:=(n-1)/2; if k mod 2 = 0 then RETURN( k*2^(k-1)*((k/2)!)^2 ); else RETURN( 2^k*(((k+1)/2)!)^2 ); fi; end; # Gives A122693
Q:=proc(n) local m; if n mod 8 <> 1 then RETURN(0); fi; m:=(n-1)/8; ((2*m)!)^2/(m!)^2; end; # Gives A122747
M:=proc(n) local k; if n mod 2 = 0 then k:=n/2; if k mod 2 = 0 then RETURN( k!*(k+2)/2 ); else RETURN( (k-1)!*(k+1)^2/2 ); fi; else k:=(n-1)/2; RETURN(D(k)*D(k+1)); fi; end; # Gives A122748
a:=n-> if n <= 1 then RETURN(1) else E(n)/8 + C(n)/8 + Q(n)/4 + M(n)/4; fi; # Gives A005635
# The following additional Maple programs produce A123071, A005631, A123072, A005633, A005632, A005634
S:=proc(n) local k; if n mod 2 = 0 then RETURN(0) else k:=(n-1)/2; RETURN(B(k)*B(k+1)); fi; end; # Gives A123071
psi:=n->S(n)/2; # Gives A005631
zeta:=n->Q(n)/2; # Gives A123072
mu:=n->(M(n)-S(n))/2; # Gives A005633
chi:=n->(C(n)-S(n)-Q(n))/4; # Gives A005632
eps:=n->E(n)/8-C(n)/8+S(n)/4-M(n)/4; # Gives A005634

Entry revised by N. J. A. Sloane, Sep 25 2006

A122749 Number of arrangements of n non-attacking bishops on an n X n board such that every square of the board is controlled by at least one bishop.

Original entry on oeis.org

4, 2, 16, 44, 256, 768, 5184, 25344, 186624, 996480, 8294400, 57888000, 530841600, 4006195200, 40642560000, 367408742400, 4064256000000, 39358255104000, 474054819840000, 5254107586560000, 68263894056960000, 804207665479680000, 11242684107325440000

Offset: 2

N. J. A. Sloane, Sep 25 2006

nonn

Vincenzo Librandi, Table of n, a(n) for n = 2..200
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [E_n, n >= 2.]

Cf. A005635, A182333.

E:=proc(n) local k; if n mod 2 = 0 then k := n/2; if k mod 2 = 0 then RETURN( (k!*(k+2)/2)^2 ); else RETURN( ((k-1)!*(k+1)^2/2)^2 ); fi; else k := (n-1)/2; if k mod 2 = 0 then RETURN( ((k!)^2/12)*(3*k^3+16*k^2+18*k+8) ); else RETURN( ((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3) ); fi; fi; end;

Table[If[n==1,1,1/768*(2*(3*n^3+23*n^2+17*n+21)*(((n-1)/2)!)^2*(1-(-1)^n+2*Sin[(Pi*n)/2])-2*(3*n^3+17*n^2-47*n+3)*((n-3)/2)!*((n+1)/2)!*((-1)^n+2*Sin[(Pi*n)/2]-1)+3*(n+2)^4*((n/2-1)!)^2*((-1)^n-2*Cos[(Pi*n)/2]+1)+12*(n+4)^2*((n/2)!)^2*((-1)^n+2*Cos[(Pi*n)/2]+1))],{n,2,25}] (* Vaclav Kotesovec, Apr 26 2012 *)
a[n_] := Module[{k}, If[Mod[n, 2]==0, k = n/2; If[Mod[k, 2]==0, (k!*(k+2) /2)^2, ((k-1)!*(k+1)^2/2)^2], k = (n-1)/2; If[Mod[k, 2]==0, ((k!)^2/12)* (3*k^3+16*k^2+18*k+8), ((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3)]]];
Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Jul 23 2022, after Maple code *)

From Andy Huchala, Mar 22 2024 (based on Mathematica code): (Start)
a(4*n)   = (n+1)^2*((2*n)!)^2.
a(4*n+1) = (1/3)*(n+2)*(1+4*n+6*n^2)*((2*n)!)^2.
a(4*n+2) = 4*(n+1)^4*((2*n)!)^2.
a(4*n+3) = (1/3)*(3+17*n+22*n^2+6*n^3)*(2*n)!*(2*n+2)!.  (End)

New name from Vaclav Kotesovec, Apr 26 2012

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

Original entry on oeis.org

1, 4, 1, 12, 186, 4, 86, 4860, 114, 8, 2, 8, 288, 4632, 205832, 2968, 124, 16, 84

Offset: 1

N. J. A. Sloane

Number of distinct solutions to minimum dominating set on queens' graph Q(n). See A002563 for non-isomorphic solutions.
For same problem, but with non-attacking queens, see A002568. - Vaclav Kotesovec, Sep 07 2012
In other words, number of minimum dominating sets in the n X n queen graph. - Eric W. Weisstein, Dec 31 2017
For n > 2, also the number of minimal edge covers in the n X n queen graph. - Eric W. Weisstein, Dec 09 2024
a(20) >= 4152. - Eric W. Weisstein, Jul 28 2025

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).

Andy Huchala, Python program.
Matthew D. Kearse and Peter B. Gibbons, Computational Methods and New Results for Chessboard Problems, Australasian Journal of Combinatorics 23 (2001), 253-284.
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]
Eric Weisstein's World of Mathematics, Minimal Edge Cover.
Eric Weisstein's World of Mathematics, Minimum Dominating Set.
Eric Weisstein's World of Mathematics, Queen Graph.
Eric W. Weisstein, Symmetrically inequivalent configurations for n = 1 to 7
Eric W. Weisstein, Symmetrically inequivalent configurations for n = 8
Eric W. Weisstein, Symmetrically inequivalent configurations for n = 9 to 13
Eric W. Weisstein, Symmetrically inequivalent configurations for n = 14
Eric W. Weisstein, Symmetrically inequivalent configurations for n = 16
Eric W. Weisstein, Symmetrically inequivalent configurations for n = 17 to 19

Cf. A002563, A002568, A182333, A286883.

A075458 gives number of queens required. - Sean A. Irvine, Apr 05 2014

New name of the sequence from Vaclav Kotesovec, Sep 07 2012
a(9)-a(10) from Vaclav Kotesovec, Sep 07 2012
a(11) from Svyatoslav Starkov, Sep 16 2013
a(12)-a(13) from Sean A. Irvine, Apr 07 2014
Definition edited by N. J. A. Sloane, Dec 25 2017 at the suggestion of Brendan McKay.
a(14) from Andy Huchala, Mar 13 2024
a(15)-a(19) from Mia Muessig, Oct 04 2024

A323500 Number of minimum dominating sets in the n X n black bishop graph.

Original entry on oeis.org

1, 2, 1, 5, 52, 22, 6, 108, 2964, 672, 120, 4680, 245520, 38160, 5040, 342720, 29292480, 3467520, 362880, 38102400, 4819046400, 460857600, 39916800, 5987520000, 1050690009600, 84304281600, 6227020800, 1264085222400, 293878019635200, 20312541849600

Offset: 1

Eric W. Weisstein, Jan 16 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Black Bishop Graph
Eric Weisstein's World of Mathematics, Minimum Dominating Set

Cf. A182333 (bishop graph), A323501 (white bishop graph).

Cf. A286886, A289164, A303141.

Table[Piecewise[{{1, n == 1}, {(n/2)! (n + 1)/2, Mod[n, 4] == 0}, {((n - 1)/2)! (n^3 + 3 n^2 + 2 n - 2)/8, Mod[n, 4] == 1}, {(n/2 - 1)! (n^2 + n + 2)/4, Mod[n, 4] == 2}, {((n - 1)/2)!, Mod[n, 4] == 3}}], {n, 20}] (* Eric W. Weisstein, Feb 27 2025 *)

\\ See A286886 for DomSetCount, Bishop.
a(n)={Vec(DomSetCount(Bishop(n, 0), x + O(x^((n+3)\2))))[1]} \\ Andrew Howroyd, Sep 08 2019

a(n)=if(n==1, 1, (n\4*2)!*if(n%4<2, if(n%2==0, (n+1)/2, (n^3 + 3*n^2 + 2*n - 2)/8), if(n%2==0, (n^2+n+2)/4, (n-1)/2))); \\ Andrew Howroyd, Sep 09 2019

From Andrew Howroyd, Sep 09 2019: (Start)
a(n) = (n/2)! * (n + 1)/2 for n mod 4 = 0;
a(n) = ((n-1)/2)! * (n^3 + 3*n^2 + 2*n - 2)/8 for n mod 4 = 1, n > 1;
a(n) = (n/2-1)! * (n^2 + n + 2)/4 for n mod 4 = 2;
a(n) = ((n-1)/2)! for n mod 4 = 3.
(End)

Terms a(11) and beyond from Andrew Howroyd, Sep 08 2019

A323501 Number of minimum dominating sets in the n X n white bishop graph.

Original entry on oeis.org

2, 6, 5, 2, 22, 356, 108, 24, 672, 25056, 4680, 720, 38160, 2531520, 342720, 40320, 3467520, 358444800, 38102400, 3628800, 460857600, 68388364800, 5987520000, 479001600, 84304281600, 16979648716800, 1264085222400, 87178291200, 20312541849600

Offset: 2

Eric W. Weisstein, Jan 16 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..50
Eric Weisstein's World of Mathematics, Minimum Dominating Set
Eric Weisstein's World of Mathematics, White Bishop Graph

Cf. A182333 (bishop graph), A323500 (black bishop graph).

Cf. A287897, A289170, A303144.

Table[Piecewise[{{(n/2)! (n + 1)/2, Mod[n, 4] == 0}, {((n - 1)/2)!, Mod[n, 4] == 1}, {(n/2 - 1)! (n^2 + n + 2)/4, Mod[n, 4] == 2}, {((n - 3)/2)! (n + 1) (n^3 + n^2 - 6 n + 6)/16, Mod[n, 4] == 3}}], {n, 2, 20}] (* Eric W. Weisstein, Feb 27 2025 *)

\\ See A289170 for DomSetCount, Bishop.
a(n)={Vec(DomSetCount(Bishop(n, 1), x + O(x^((n+3)\2))))[1]} \\ Andrew Howroyd, Sep 08 2019

a(n)=(n\4*2)!*if(n%4<2, if(n%2==0, (n + 1)/2, 1), if(n%2==0, (n^2 + n + 2)/4, (n + 1)*(n^3 + n^2 - 6*n + 6)/16)); \\ Andrew Howroyd, Sep 09 2019

From Andrew Howroyd, Sep 09 2019: (Start)
a(n) = (n/2)! * (n + 1)/2 for n mod 4 = 0;
a(n) = ((n-1)/2)! for n mod 4 = 1;
a(n) = (n/2-1)! * (n^2 + n + 2)/4 for n mod 4 = 2;
a(n) = ((n-3)/2)! * (n + 1)*(n^3 + n^2 - 6*n + 6)/16 for n mod 4 = 3.
(End)

Offset corrected and terms a(11) and beyond from Andrew Howroyd, Sep 08 2019

A295899 Number of minimal dominating sets in the n X n bishop graph.

Original entry on oeis.org

1, 4, 30, 361, 6552, 180625, 6768900, 330039889, 20267217352, 1535068962441

Offset: 1

Eric W. Weisstein, Nov 29 2017

Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Minimal Dominating Set

Cf. A182333, A286886, A287897, A290714, A295898.

a(n) = A286886(n) * A287897(n). - Andrew Howroyd, Nov 30 2017

a(6)-a(10) from Andrew Howroyd, Nov 30 2017

cp's OEIS Frontend

A005635 Number of ways of placing n non-attacking bishops on an n X n board so that every square is attacked (or occupied).

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A122749 Number of arrangements of n non-attacking bishops on an n X n board such that every square of the board is controlled by at least one bishop.

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

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A323500 Number of minimum dominating sets in the n X n black bishop graph.

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A323501 Number of minimum dominating sets in the n X n white bishop graph.

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A295899 Number of minimal dominating sets in the n X n bishop graph.

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