A290721 a(n) = 4^n - n - 1.

2, 13, 60, 251, 1018, 4089, 16376, 65527, 262134, 1048565, 4194292, 16777203, 67108850, 268435441, 1073741808, 4294967279, 17179869166, 68719476717, 274877906924, 1099511627755, 4398046511082, 17592186044393, 70368744177640, 281474976710631, 1125899906842598

Offset

1,1

Comments

For n > 1, also the number of (non-null) connected induced subgraphs in the n-cocktail party graph.

Links

Table of n, a(n) for n=1..25.

Eric Weisstein's World of Mathematics, Cocktail Party Graph.

Eric Weisstein's World of Mathematics, Connected Graph.

Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph.

Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Formula

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

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

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

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

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

a(n) = A000302(n) - n - 1. (End)

Mathematica

Table[4^n - n - 1, {n, 20}]
LinearRecurrence[{6, -9, 4}, {2, 13, 60}, 20]
CoefficientList[Series[(-2 - x)/((-1 + x)^2 (-1 + 4 x)), {x, 0, 20}], x]

Crossrefs

Cf. A000302.

Sequence in context: A382990 A187560 A338818 * A354942 A205532 A294052

Adjacent sequences: A290718 A290719 A290720 * A290722 A290723 A290724

Keyword

nonn,easy

Author

Eric W. Weisstein, Aug 09 2017

Status

approved