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
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).
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
E.g.f.: 12*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Nov 09 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 *)
PROG
(Magma)
A022089:= func< n | 6*Fibonacci(n) >;
[A022089(n): n in [0..50]]; // G. C. Greubel, Apr 13 2025
(SageMath)
def A022089(n): return 6*fibonacci(n)
print([A022089(n) for n in range(51)]) # G. C. Greubel, Apr 13 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved