A022086 Fibonacci sequence beginning 0, 3.

0, 3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432, 699, 1131, 1830, 2961, 4791, 7752, 12543, 20295, 32838, 53133, 85971, 139104, 225075, 364179, 589254, 953433, 1542687, 2496120, 4038807, 6534927, 10573734, 17108661, 27682395, 44791056, 72473451, 117264507

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

First differences of A111314. - Ross La Haye, May 31 2006

Pisano period lengths: 1, 3, 1, 6, 20, 3, 16, 12, 8, 60, 10, 6, 28, 48, 20, 24, 36, 24, 18, 60, ... . - R. J. Mathar, Aug 10 2012

For n>=6, a(n) is the number of edge covers of the union of two cycles C_r and C_s, r+s=n, with a single common vertex. - Feryal Alayont, Oct 17 2024

References

• A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 7,17.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Essentially the same as A097135.

Cf. A026390, A036999, A111314, A119457, A187893.

Sequences of the form Fibonacci(n+k) + Fibonacci(n-k) are listed in A280154.

Sequences of the form m*Fibonacci: A000045 (m=1), A006355 (m=2), this sequence (m=3), A022087 (m=4), A022088 (m=5), A022089 (m=6), A022090 (m=7), A022091 (m=8), A022092 (m=8), A022093 (m=10), A022345...A022366 (m=11...32).

Programs

Magma

[3*Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Dec 31 2016

Maple

BB := n->if n=0 then 3; > elif n=1 then 0; > else BB(n-2)+BB(n-1); > fi: > L:=[]: for k from 1 to 34 do L:=[op(L),BB(k)]: od: L; # Zerinvary Lajos, Mar 19 2007
with (combinat):seq(sum((fibonacci(n,1)),m=1..3),n=0..32); # Zerinvary Lajos, Jun 19 2008

Mathematica

LinearRecurrence[{1, 1}, {0, 3}, 40] (* Arkadiusz Wesolowski, Aug 17 2012 *)
Table[Fibonacci[n + 4] + Fibonacci[n - 4] - 4 Fibonacci[n], {n, 0, 40}] (* Bruno Berselli, Dec 30 2016 *)
Table[3 Fibonacci[n], {n, 0, 40}] (* Vincenzo Librandi, Dec 31 2016 *)

PARI

a(n)=3*fibonacci(n) \\ Charles R Greathouse IV, Nov 06 2014

SageMath

def A022086(n): return 3*fibonacci(n)
print([A022086(n) for n in range(41)]) # G. C. Greubel, Apr 10 2025

Formula

a(n) = 3*Fibonacci(n).

a(n) = F(n-2) + F(n+2) for n>1, with F=A000045.

a(n) = round( ((6*phi-3)/5) * phi^n ) for n>2. - Thomas Baruchel, Sep 08 2004

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

G.f.: 3*x/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = A187893(n) - 1. - Filip Zaludek, Oct 29 2016

E.g.f.: 6*sinh(sqrt(5)*x/2)*exp(x/2)/sqrt(5). - Ilya Gutkovskiy, Oct 29 2016

a(n) = F(n+4) + F(n-4) - 4*F(n), F = A000045. - Bruno Berselli, Dec 29 2016