A022089 Fibonacci sequence beginning 0, 6.

0, 6, 6, 12, 18, 30, 48, 78, 126, 204, 330, 534, 864, 1398, 2262, 3660, 5922, 9582, 15504, 25086, 40590, 65676, 106266, 171942, 278208, 450150, 728358, 1178508, 1906866, 3085374, 4992240, 8077614, 13069854, 21147468, 34217322, 55364790, 89582112, 144946902

Offset

0,2

Comments

Starting with a(0)=1, a(1)=3, a(n) = the number of ternary length-2 squarefree words of length n.

References

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

Links

Table of n, a(n) for n=0..37.

N. H. Bong, C. Dalfó, and M. À. Fiol, and D. Závacká, Some inner metric parameters of a digraph: Iterated line digraphs and integer sequences, arXiv:2409.02125 [math.CO], 2024. See p. 17.

Cristina Dalfó and Miquel Àngel Fiol, A Note on the Order of Iterated Line Digraphs, Journal of Graph Theory, Volume 85, Issue 2, June 2017, Pages 395-39, 2016; DOI: 10.1002/jgt.22068; arXiv:1607.08832 [math.CO], 2016.

Tanya Khovanova, Recursive Sequences

Christoph Richard and Uwe Grimm, On the entropy and letter frequencies of ternary squarefree words, arXiv:math/0302302 [math.CO], 2003.

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

Formula

a(n) = round( (12*phi-6)/5 * phi^n) for n>3. - Thomas Baruchel, Sep 08 2004

a(n) = 6F(n) = F(n+3) + F(n+1) + F(n-4), n>3.

a(n) = A119457(n+4,n-1) for n>1. - Reinhard Zumkeller, May 20 2006

G.f.: 6*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008

a(n) = 6 * A000045(n). - Alois P. Heinz, Jan 18 2019

Maple

a:= n-> 6*(<<0|1>, <1|1>>^n)[1, 2]:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 18 2019

Mathematica

a={}; b=0; c=6; AppendTo[a, b]; AppendTo[a, c]; Do[b=b+c; AppendTo[a, b]; c=b+c; AppendTo[a, c], {n, 1, 12, 1}]; a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
LinearRecurrence[{1, 1}, {0, 6}, 50] (* Harvey P. Dale, Dec 05 2015 *)

Crossrefs

Cf. A000032, A000045.

Sequence in context: A315795 A315796 A242951 * A275288 A110357 A091827

Adjacent sequences: A022086 A022087 A022088 * A022090 A022091 A022092

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved