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

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

Author

Eric W. Weisstein, Jan 16 2019

Keywords

• nonn

Links

• 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

Crossrefs

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

Cf. A287897, A289170, A303144.

Programs

Mathematica

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 *)

PARI

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

PARI

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

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)! 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)

Extensions

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