A121364 Convolution of A066983 with the double Fibonacci sequence A103609.

0, 0, 1, 2, 3, 6, 10, 18, 29, 50, 81, 136, 220, 364, 589, 966, 1563, 2550, 4126, 6710, 10857, 17622, 28513, 46224, 74792, 121160, 196041, 317434, 513619, 831430, 1345282, 2177322, 3522981, 5701290, 9224881, 14927768, 24153636, 39083988, 63239221, 102327390

Offset: 1

Author

Graeme McRae, Jul 23 2006

Keywords

• nonn
• easy

Comments

The convolution of 1,0,1,1,1,3,3,7,9,17,25,... (A066983 with 1,0 added to the front) with "A double Fibonacci sequence" (A103609) is the Fibonacci sequence (A000045), with an extra initial 0.

Examples

			a(7)=10 because F(7)=13 and D(8)=3 and a(7)=F(7)-D(8).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,0,-1,-1).

Crossrefs

Cf. A000045, A066983, A103609.

Programs

Magma

A121364:= func< n | Fibonacci(n) - Fibonacci(Floor((n+1)/2)) >;
[A121364(n): n in [1..70]]; // G. C. Greubel, Oct 23 2024

Mathematica

LinearRecurrence[{1,2,-1,0,-1,-1},{0, 0, 1, 2, 3, 6},40] (* James C. McMahon, Oct 17 2024 *)

PARI

concat([0,0], Vec(-x^3*(x^2-x-1)/((x^2+x-1)*(x^4+x^2-1)) + O(x^100))) \\ Colin Barker, Oct 13 2014

SageMath

def A121364(n): return fibonacci(n) - fibonacci((n+1)//2)
[A121364(n) for n in range(1,71)] # G. C. Greubel, Oct 23 2024

Formula

a(n) = F(n) - D(n+1), where F is the Fibonacci sequence (A000045) and D is "A double Fibonacci sequence" (A103609).

G.f.: x^3*(1+x-x^2) / ((1-x-x^2)*(1-x^2-x^4)). - Colin Barker, Oct 13 2014

Extensions

More terms from Colin Barker, Oct 13 2014