A292073 Number of dominating sets in the n X n rook complement graph.

1, 9, 421, 64727, 33548731, 68719441881, 562949953225997, 18446744073708516927, 2417851639229258344138819, 1267650600228229401496677076985, 2658455991569831745807614120434020301, 22300745198530623141535718272648360902500919

Offset: 1

Author

Eric W. Weisstein, Sep 12 2017

Keywords

• nonn

Comments

Non-dominating sets are just those that are contained in the union of a single row and column minus the intersecting vertex. - Andrew Howroyd, Sep 13 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Dominating Set
• Eric Weisstein's World of Mathematics, Rook Complement Graph

Crossrefs

Cf. A291623, A292074.

Programs

Magma

[1] cat [2^(n^2)-2*n*(2^n-2)+n^2-n^2*(2^(n-1)-1)^2+ n^2*(n-1)^2-2*Binomial(n,2)^2-1: n in [2..15]]; // Vincenzo Librandi, Mar 17 2018

Mathematica

Table[If[n == 1, 1, 2^n^2 + (2^n (n - 2) - 4^(n - 1) n + (n - 1)^2 n/2 + 4) n - 1], {n, 20}]

PARI

a(n) = if(n == 1, 1, 2^(n^2) - 2*n*(2^n - 2) + n^2 - n^2*(2^(n-1)-1)^2 + n^2*(n-1)^2 - 2*binomial(n,2)^2 - 1); \\ Andrew Howroyd, Sep 13 2017

Formula

a(n) = 2^(n^2) - 2*n*(2^n - 2) + n^2 - n^2*(2^(n-1)-1)^2 + n^2*(n-1)^2 - 2*binomial(n,2)^2 - 1 for n > 1. - Andrew Howroyd, Sep 13 2017

Extensions

a(6)-a(12) from Andrew Howroyd, Sep 13 2017