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
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Sep 19 2021
EXTENSIONS
Terms a(6) and beyond from Andrew Howroyd, Jan 19 2022
STATUS
approved