A289255 a(n) = 4^n - 2*n - 1.

1, 11, 57, 247, 1013, 4083, 16369, 65519, 262125, 1048555, 4194281, 16777191, 67108837, 268435427, 1073741793, 4294967263, 17179869149, 68719476699, 274877906905, 1099511627735, 4398046511061, 17592186044371, 70368744177617, 281474976710607, 1125899906842573

Offset: 1

Author

Eric W. Weisstein, Jun 29 2017

Keywords

• nonn
• easy

Comments

Number of dominating sets in the n-cocktail party graph.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Cocktail Party Graph.
• Eric Weisstein's World of Mathematics, Dominating Set.
• Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Crossrefs

Cf. A000302, A005408, A096046 (first differences).

Programs

Mathematica

Table[4^n - 2 n - 1, {n, 20}]
LinearRecurrence[{6, -9, 4}, {1, 11, 57}, 20]
CoefficientList[Series[(-1 - 5 x)/((-1 + x)^2 (-1 + 4 x)), {x, 0, 20}], x]

PARI

Vec(x*(1 + 5*x) / ((1 - x)^2*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jun 30 2017

Formula

a(n) = 4^n - 2*n - 1.

a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3).

G.f.: (-1 - 5*x)*x/((-1 + x)^2*(-1 + 4*x)).

From Elmo R. Oliveira, Apr 02 2025: (Start)

E.g.f.: exp(x)*(exp(3*x) - (2*x + 1)).

a(n) = A000302(n) - A005408(n). (End)