A347922 Number of minimal total dominating sets in the n X n rook complement graph.

0, 1, 51, 492, 2500, 8925, 25431, 61936, 134352, 266625, 493075, 861036, 1433796, 2293837, 3546375, 5323200, 7786816, 11134881, 15604947, 21479500, 29091300, 38829021, 51143191, 66552432, 85650000, 109110625, 137697651, 172270476, 213792292, 263338125

Offset

1,3

Comments

From Andrew Howroyd, Jan 19 2022: (Start)

The vertex sets which are not totally dominating are just those that are contained in the union of a single row and column. Minimal total dominating sets are:

- any three vertices such that no two are in the same row or column,

- two vertices in each of two rows/columns. (End)

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Minimal Total Dominating Set

Eric Weisstein's World of Mathematics, Rook Complement Graph

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

From Andrew Howroyd, Jan 19 2022: (Start)

a(n) = 6*binomial(n,3)^2 + 2*binomial(n,2)^3 - binomial(n,2)^2.

a(n) = (5*n^2 - 11*n + 5)*n^2*(n-1)^2/12.

G.f.: x*(1 + 44*x + 156*x^2 + 92*x^3 + 7*x^4)/(1 - x)^7.

(End)

Mathematica

Table[(n - 1)^2 n^2 (5 n^2 - 11 n + 5)/12, {n, 20}] (* Eric W. Weisstein, May 11 2024 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 1, 51, 492, 2500, 8925, 25431}, 20] (* Eric W. Weisstein, May 11 2024 *)
CoefficientList[Series[-x (1 + 44 x + 156 x^2 + 92 x^3 + 7 x^4)/(-1 + x)^7, {x, 0, 20}], x] (* Eric W. Weisstein, May 11 2024 *)

Prog

(PARI) a(n) = (5*n^2 - 11*n + 5)*n^2*(n-1)^2/12 \\ Andrew Howroyd, Jan 19 2022

Crossrefs

Cf. A292074, A303209, A303212.

Sequence in context: A204215 A164646 A128511 * A355418 A142994 A251932

Adjacent sequences: A347919 A347920 A347921 * A347923 A347924 A347925

Keyword

nonn,easy

Author

Eric W. Weisstein, Sep 19 2021

Extensions

Terms a(6) and beyond from Andrew Howroyd, Jan 19 2022

Status

approved