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

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

Author

Eric W. Weisstein, Jan 16 2019

Keywords

• nonn

Links

• 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

Crossrefs

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

Cf. A286886, A289164, A303141.

Programs

Mathematica

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 *)

PARI

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

PARI

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

Formula

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)

Extensions

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