A323501 -id:A323501 - cp's OEIS frontend

A182333 Number of arrangements of n bishops such that every square of the board is controlled by at least one bishop.

1, 4, 6, 25, 104, 484, 2136, 11664, 71136, 451584, 3006720, 21902400, 176774400, 1456185600, 12758860800, 117456998400, 1181072793600, 12023694950400, 130072449024000, 1451792885760000, 17487355576320000, 212389727477760000, 2729844680048640000

Offset: 1

Vaclav Kotesovec, Apr 25 2012

nonn

Number of minimum dominating sets in the n X n bishop graph. - Eric W. Weisstein, Jun 04 2017

A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, vol.1, 1987, p.11 and p.83-88.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Dominating Set

Cf. A005635, A122749, A323500, A323501.

Table[If[n==1,1,((2*Floor[n/4])!)^2/128*(n^5+3*n^4+n^3+35*n^2+38*n+2-(n^5-n^4-7*n^3-n^2-10*n-30)*(-1)^n-4*(n^3+2*n^2+n-4)*n*Cos[Pi*n/2]-2*(n^5+n^4-11*n^3-7*n^2-2*n+2)*Sin[Pi*n/2])],{n,1,25}]

a(n)={if(n==1, 1, (n\4*2)!^2*if(n%4<2, if(n%2==0, (n+1)^2, (n^3 + 3*n^2 + 2*n - 2)/2), if(n%2==0, (n^2+n+2)^2/4, (n+1)*(n-1)*(n^3 + n^2 - 6*n + 6)/8))/4)} \\ Andrew Howroyd, Sep 09 2019

a(n) = (((2*floor(n/4))!)^2/128)*(n^5 + 3*n^4 + n^3 + 35*n^2 + 38*n + 2 - (n^5 - n^4 - 7*n^3 - n^2 - 10*n - 30)*(-1)^n -4*(n^3 + 2*n^2 + n - 4)*n*cos(Pi*n/2) - 2*(n^5 + n^4 - 11*n^3 - 7*n^2 - 2*n + 2)*sin(Pi*n/2)), for n > 1.

a(n) = A323500(n) * A323501(n) for n > 1. - Andrew Howroyd, Sep 08 2019

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

A381727 Number of minimum connected dominating sets in the n X n white bishop graph.

Original entry on oeis.org

2, 4, 1, 4, 13, 64, 513, 4480, 41197, 444416, 5597201, 77253632, 1153902701, 18870222848, 336018968449, 6428081455104, 131386321421901, 2865273888571392, 66426533670738769, 1629643279560867840, 42175861619149917325, 1148845693539400548352, 32856688248674995989889

Offset: 2

Eric W. Weisstein, Mar 05 2025

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..200
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, White Bishop Graph.

Cf. A381726 (black bishop).

Cf. A072590, A132609, A289169, A303144, A323501.

Join[{2, 4}, Table[Sum[(2 k)^(n - 2 k - 2) (n - 2 k - 1)^(2 k - 1), {k, Floor[n/2] - 1}], {n, 4, 20}]] (* Eric W. Weisstein, Mar 22 2025 *)

\\ B(n, k) is A072590.
B(n,k) = n^(k-1) * k^(n-1)
a(n) = if(n <= 3, 2*n-2, sum(k=1, n\2-1, B(n-1-2*k, 2*k))) \\ Andrew Howroyd, Mar 20 2025

a(n) = Sum_{k=1..floor(n\2)-1} A072590(n-1-2*k, 2*k) for n >= 4. - Andrew Howroyd, Mar 20 2025

a(10) onwards from Andrew Howroyd, Mar 20 2025

cp's OEIS Frontend

A182333 Number of arrangements of n bishops such that every square of the board is controlled by at least one bishop.

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

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A381727 Number of minimum connected dominating sets in the n X n white bishop graph.

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