A049602 a(n) = (Fibonacci(2*n)-(-1)^n*Fibonacci(n))/2.

0, 1, 1, 5, 9, 30, 68, 195, 483, 1309, 3355, 8900, 23112, 60813, 158717, 416325, 1088661, 2852242, 7463884, 19546175, 51163695, 133962621, 350695511, 918170280, 2403740304, 6293172025, 16475579353, 43133883845, 112925557953

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

A049602 gives the coefficients of x in the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1. For the constant terms, see A192352. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. - Clark Kimberling, Jun 29 2011

Links

• Index entries for linear recurrences with constant coefficients, signature (2,3,-4,1).

Crossrefs

Cf. A049601.

Programs

Mathematica

LinearRecurrence[{2,3,-4,1},{0,1,1,5},30] (* Harvey P. Dale, Jul 07 2017 *)

PARI

a(n)=(fibonacci(2*n)-(-1)^n*fibonacci(n))/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n)=Sum{T(2i+1, n-2i-1): i=0, 1, ..., [ (n+1)/2 ]}, array T as in A049600.

Cosh transform of Fibonacci numbers A000045 (or mean of binomial and inverse binomial transforms of A000045). E.g.f.: cosh(x)(2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - Paul Barry, May 10 2003

a(n)=sum{k=0..floor(n/2), C(n, 2k)Fib(n-2k)}; - Paul Barry, May 01 2005

a(n)=2a(n-1)+3a(n-2)-4a(n-3)+a(n-4). - Paul Curtz, Jun 16 2008

G.f.: x(1-x)/((1+x-x^2)(1-3x+x^2)); a(n)=sum{k=0..n-1, (-1)^(n-k+1)*F(2k+2)*F(n-k+1)}; - Paul Barry, Jul 11 2008

Extensions

Simpler description from Vladeta Jovovic and Thomas Baruchel, Aug 24 2004

More terms from Paul Curtz, Jun 16 2008