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A022087 Fibonacci sequence beginning 0, 4. 12
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 (list; graph; refs; listen; history; text; internal format)
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 sequences related to linear recurrences with constant coefficients, signature (1,1).

FORMULA

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

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)-17F(n+3), n>=0, F(n)=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

MAPLE

a := n-> (Matrix([[4, 0]]). Matrix([[1, 1], [1, 0]])^n)[1, 2]: seq (a(n), n=0..32); # 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 Orlovsky_, Jul 22 2008 *)

PROG

(PARI) a(n)=4*fibonacci(n) \\ Charles R Greathouse IV, Jun 05, 2011

(MAGMA) [4*Fibonacci(n): n in [0..50]]; // Vincenzo Librandi, Oct 12 2013

CROSSREFS

Sequence in context: A194697 A194696 A002368 * A095294 A190100 A244421

Adjacent sequences:  A022084 A022085 A022086 * A022088 A022089 A022090

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified September 16 17:40 EDT 2014. Contains 246822 sequences.