A085279 Expansion of (1 - 2*x - 2*x^2)/((1 - 2*x)*(1 - 3*x)).

1, 3, 7, 17, 43, 113, 307, 857, 2443, 7073, 20707, 61097, 181243, 539633, 1610707, 4815737, 14414443, 43177793, 129402307, 387944777, 1163310043, 3488881553, 10464547507, 31389448217, 94159956043, 282463090913, 847355718307

Offset: 0

Author

Paul Barry, Jun 25 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A001045(n)+1.

For n > 1, also the number of independent vertex sets in the (n-1)-book graph. - Eric W. Weisstein, Aug 16 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Book Graph
• Eric Weisstein's World of Mathematics, Independent Vertex Set
• Index entries for linear recurrences with constant coefficients, signature (5,-6).

Crossrefs

Cf. A001045.

Programs

Magma

[1] cat [2^n+3^(n-1): n in [1..30]]; // Vincenzo Librandi, Sep 12 2014

Maple

seq(2^n + (3^n - charfcn[0](n))/3, n=0..100); # Robert Israel, Sep 12 2014

Mathematica

CoefficientList[Series[(1 - 2 x - 2 x^2)/((1 - 2 x) (1 - 3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 12 2014 *)
Join[{1}, LinearRecurrence[{5, -6}, {3, 7}, 20]] (* Eric W. Weisstein, Aug 16 2017 *)
Join[{1}, Table[2^n + 3^(n - 1), {n, 20}]] (* Eric W. Weisstein, Aug 16 2017 *)

PARI

Vec((1-2*x-2*x*x)/((1-2*x)*(1-3*x)) + O(x^50)) \\ Michel Marcus, Sep 12 2014

Formula

a(n) = (3*2^n + 3^n - 0^n)/3.

a(n) = 2^n + 3^(n-1) for n >= 1.

G.f.: (1 - 2*x - 2*x*x)/((1 - 2*x)*(1 - 3*x)).

a(n) = 5*a(n-1) - 6*a(n-2) for n > 1. - Vincenzo Librandi, Sep 12 2014

E.g.f.: (1/3)*(exp(3*x) + 3*exp(2*x) -1). - G. C. Greubel, Aug 17 2017