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

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

Author

Eric W. Weisstein, Mar 05 2025

Keywords

• nonn

Links

• 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.

Crossrefs

Cf. A381726 (black bishop).

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

Programs

Mathematica

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 *)

PARI

\\ 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

Formula

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

Extensions

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