A022088 Fibonacci sequence beginning 0, 5.

0, 5, 5, 10, 15, 25, 40, 65, 105, 170, 275, 445, 720, 1165, 1885, 3050, 4935, 7985, 12920, 20905, 33825, 54730, 88555, 143285, 231840, 375125, 606965, 982090, 1589055, 2571145, 4160200, 6731345, 10891545, 17622890, 28514435, 46137325, 74651760, 120789085

Offset

0,2

References

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, pp. 15, 34, 52.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

Kristina Lund, Steven Schlicker and Patrick Sigmon, Fibonacci sequences and the space of compact sets, Involve, 1:2 (2008), pp. 159-165.

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

Formula

a(n) = round( (2*phi-1)*phi^n ) for n>3. - Thomas Baruchel, Sep 08 2004

a(n) = 5*Fibonacci(n).

a(n) = A119457(n+3,n-1) for n>1. - Reinhard Zumkeller, May 20 2006

G.f.: 5*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008

a(n+2) = A014217(n+4) - A014217(n). - Paul Curtz, Dec 22 2008

a(n) = sqrt(5*(A000032(n)^2 - 4*(-1)^n)). - Alexander Samokrutov, Sep 02 2015

From Tom Copeland, Jan 25 2016: (Start)

The o.g.f. for the shifted series b(0)=0 and b(n) = a(n+1) is G(x) = 5*x*(1+x)/(1-x*(1+x)) = 5 L(-Cinv(-x)), where L(x) = x/(1-x) with inverse Linv(x) = x/(1+x) and Cinv(x) = x*(1-x), the inverse of the o.g.f. for the shifted Catalan numbers of A000108, C(x) = (1-sqrt(1-4*x))/2. Then Ginv(x) = -C(-Linv(x/5)) = (-1 + sqrt(1+4*x/(5+x)))/2.

a(n+1) = 5*Sum_{k=0..n} binomial(n-k,k) = 5 * A000045(n+1), from A267633, with the convention for zeros of the binomial assumed there.

(End)

For n > 0, 1/a(n) = Sum_{k>=1} F(n*k)/(L(n+1)^(k+1)), where F(n) = A000045(n) and L(n) = A000032(n). - Diego Rattaggi, Oct 26 2022

Mathematica

LinearRecurrence[{1, 1}, {0, 5}, 40] (* Harvey P. Dale, Jan 13 2012 *)
5*Fibonacci[Range[0, 50]] (* G. C. Greubel, Feb 10 2023 *)

Prog

(Magma) [5*Fibonacci(n): n in [1..40]]; // Vincenzo Librandi, Sep 03 2015
(PARI) a(n) = 5*fibonacci(n); \\ Michel Marcus, Sep 03 2015
(SageMath) [5*fibonacci(n) for n in range(51)] # G. C. Greubel, Feb 10 2023

Crossrefs

Cf. A000032, A000045, A000108, A014217, A267633.

Sequence in context: A000728 A242895 A242129 * A245418 A082450 A304266

Adjacent sequences: A022085 A022086 A022087 * A022089 A022090 A022091

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved