A099483 A Fibonacci convolution.

0, 1, 3, 7, 18, 48, 126, 329, 861, 2255, 5904, 15456, 40464, 105937, 277347, 726103, 1900962, 4976784, 13029390, 34111385, 89304765, 233802911, 612103968, 1602508992, 4195423008, 10983760033, 28755857091, 75283811239, 197095576626

Offset: 0

Author

Paul Barry, Oct 18 2004

Keywords

• easy
• nonn

Comments

A Chebyshev transform of the sequence 0,1,3,9,27 with g.f. x/(1-3x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Links

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

Crossrefs

Cf. A000045, A099484, A099485.

Programs

Mathematica

LinearRecurrence[{3,-2,3,-1},{0,1,3,7},30] (* Harvey P. Dale, May 23 2016 *)

Formula

G.f.: x/((1+x^2)(1-3x+x^2)); a(n)=3a(n-1)-2a(n-2)+3a(n-3); a(n)=sum{k=0..n, cos(pi*k/2)F(2(n-k))}. a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^n*(3^(n-2k)-0^(n-2k))/3}.

(1/6) [2Fib(2n+2) - I^n - (-I)^n ]. - Ralf Stephan, Dec 04 2004