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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• N. H. Bong, C. Dalfó, 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).

Crossrefs

Cf. A000032, A000045, A119457.

Sequences of the form m*Fibonacci listed in A022086.

Programs

Magma

A022089:= func< n | 6*Fibonacci(n) >;
[A022089(n): n in [0..50]]; // G. C. Greubel, Apr 13 2025

Maple

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

Mathematica

6*Fibonacci[Range[0,50]] (* G. C. Greubel, Apr 13 2025 *)
LinearRecurrence[{1,1},{0,6},50] (* Harvey P. Dale, Dec 05 2015 *)

SageMath

def A022089(n): return 6*fibonacci(n)
print([A022089(n) for n in range(51)]) # G. C. Greubel, Apr 13 2025

Formula

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

a(n) = 6*F(n) = F(n+3) + F(n+1) + F(n-4), n>3, where F=A000045.

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