A099484 A Fibonacci convolution.

1, 1, 2, 7, 19, 48, 125, 329, 862, 2255, 5903, 15456, 40465, 105937, 277346, 726103, 1900963, 4976784, 13029389, 34111385, 89304766, 233802911, 612103967, 1602508992, 4195423009, 10983760033, 28755857090, 75283811239, 197095576627

Offset

0,3

Comments

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

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

Formula

G.f.: (1-x)^2/((1+x^2)*(1-3*x+x^2)), convolution of A176742 and A001906.

a(n)=3a(n-1)-2a(n-2)+3a(n-3);

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

a(n)=sum{k=0..n, (0^k-2sin(pi*k/2))F(2(n-k)+2)}.

(1/3) [Fib(2n+2) + I^n + (-I)^n ]. - Ralf Stephan, Dec 04 2004

3*a(n) = A001906(n+1) +2*A056594(n). - R. J. Mathar, Jun 17 2020

Mathematica

LinearRecurrence[{3, -2, 3, -1}, {1, 1, 2, 7}, 40] (* Harvey P. Dale, Mar 25 2020 *)

Crossrefs

Cf. A000045, A099483, A099485.

Sequence in context: A238914 A227946 A328990 * A018030 A051354 A073799

Adjacent sequences: A099481 A099482 A099483 * A099485 A099486 A099487

Keyword

easy,nonn

Author

Paul Barry, Oct 18 2004

Status

approved