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

0,2

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).

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

Mathematica

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

Crossrefs

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

Sequence in context: A062197 A030016 A248733 * A038990 A355387 A077938

Adjacent sequences: A099482 A099483 A099484 * A099486 A099487 A099488

Keyword

easy,nonn

Author

Paul Barry, Oct 18 2004

Status

approved