A022087 Fibonacci sequence beginning 0, 4.

0, 4, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356, 576, 932, 1508, 2440, 3948, 6388, 10336, 16724, 27060, 43784, 70844, 114628, 185472, 300100, 485572, 785672, 1271244, 2056916, 3328160, 5385076, 8713236, 14098312, 22811548, 36909860, 59721408, 96631268

Offset

0,2

Comments

For n > 1, this sequence gives the number of binary strings of length n that do not contain 0000, 0101, 1010, or 1111 as contiguous substrings (see A230127). - Nathaniel Johnston, Oct 11 2013

References

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 18.

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) = 4*F(n) = F(n-2) + F(n) + F(n+2), where F = A000045.

a(n) = round( phi^n*(8*phi-4)/5 ) for n>2. - Thomas Baruchel, Sep 08 2004

a(n) = A119457(n+2,n-1) for n>1. - Reinhard Zumkeller, May 20 2006

G.f.: 4*x/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = F(n+9) - 17*F(n+3), where F=A000045. - Manuel Valdivia, Dec 15 2009

G.f.: Q(0) -1, where Q(k) = 1 + x^2 + (4*k+5)*x - x*(4*k+1 + x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013

a(n) = Fibonacci(n+3) - Fibonacci(n-3), where Fibonacci(-3..-1) = 2,-1,1. [Bruno Berselli, May 22 2015]

Maple

a:= n-> (Matrix([[4, 0]]). Matrix([[1, 1], [1, 0]])^n)[1, 2]: seq(a(n), n=0..40); # Alois P. Heinz, Aug 17 2008

Mathematica

a={}; b=0; c=4; AppendTo[a, b]; AppendTo[a, c]; Do[b=b+c; AppendTo[a, b]; c=b+c; AppendTo[a, c], {n, 1, 9, 1}]; a (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
Table[4 Fibonacci(n), {n, 0, 40}] (* Bruno Berselli, May 22 2015 *)

Prog

(PARI) a(n)=4*fibonacci(n) \\ Charles R Greathouse IV, Jun 05 2011
(Magma) [4*Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Oct 12 2013

Crossrefs

Cf. A000045.

Cf. similar sequences listed in A258160.

Sequence in context: A302681 A002368 A299474 * A333149 A095294 A190100

Adjacent sequences: A022084 A022085 A022086 * A022088 A022089 A022090

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved