A099485 A Fibonacci convolution.

1, 2, 5, 14, 37, 96, 251, 658, 1723, 4510, 11807, 30912, 80929, 211874, 554693, 1452206, 3801925, 9953568, 26058779, 68222770, 178609531, 467605822, 1224207935, 3205017984, 8390846017, 21967520066, 57511714181, 150567622478

Offset: 0

Author

Paul Barry, Oct 18 2004

Keywords

• easy
• nonn

Comments

A Chebyshev transform of A025192 with g.f. (1-x)/(1-3*x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))*G(x/(1+x^2)).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2392
• Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics. 2017, Vol. 41 Issue 6, pp. 849-853.
• Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-1).

Crossrefs

Cf. A000045, A099483, A099484, A001045, A128834.

Programs

Mathematica

LinearRecurrence[{3,-2,3,-1},{1,2,5,14},30] (* Harvey P. Dale, Jul 06 2017 *)

Formula

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

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

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^n*(2*3^(n-2*k)+0^(n-2*k))/3.

a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*Fibonacci(2*(n-k)+2).

a(n) = (1/6) * (4*Fibonacci(2*n+2) + I^n + (-I)^n). - Ralf Stephan, Dec 04 2004

Also a transformation of the Jacobsthal numbers A001045(n+1) under the mapping G(x)-> (1/(1-x+x^2))*G(x/(1-x+x^2)). - Paul Barry, Dec 11 2004

G.f.: g(f(x))/x, where g is g.f. of A001045 and f is g.f. of A128834. - Oboifeng Dira, Jun 21 2020