OFFSET
0,2
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).
FORMULA
a(n) = round( ((6*phi-3)/5) * phi^n ) for n>2. - Thomas Baruchel, Sep 08 2004
a(n) = 3*F(n). Also, a(n) = F(n-2) + F(n+2) for n>1, with F=A000045.
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). - Bruno Berselli, Dec 29 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 *)
PROG
(PARI) a(n)=3*fibonacci(n) \\ Charles R Greathouse IV, Nov 06 2014
(Magma) [3*Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Dec 31 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved