%I #51 Nov 09 2025 11:37:46

%S 0,6,6,12,18,30,48,78,126,204,330,534,864,1398,2262,3660,5922,9582,

%T 15504,25086,40590,65676,106266,171942,278208,450150,728358,1178508,

%U 1906866,3085374,4992240,8077614,13069854,21147468,34217322,55364790,89582112,144946902

%N Fibonacci sequence beginning 0, 6.

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

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

%H G. C. Greubel, <a href="/A022089/b022089.txt">Table of n, a(n) for n = 0..1000</a>

%H N. H. Bong, C. Dalfó, M. À. Fiol, and D. Závacká, <a href="https://arxiv.org/abs/2409.02125">Some inner metric parameters of a digraph: Iterated line digraphs and integer sequences</a>, arXiv:2409.02125 [math.CO], 2024. See p. 17.

%H Cristina Dalfó and Miquel Àngel Fiol, <a href="https://arxiv.org/abs/1607.08832">A Note on the Order of Iterated Line Digraphs</a>, 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.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Christoph Richard and Uwe Grimm, <a href="https://arxiv.org/abs/math/0302302">On the entropy and letter frequencies of ternary squarefree words</a>, arXiv:math/0302302 [math.CO], 2003.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).

%F a(n) = round( (12*phi-6)/5 * phi^n) for n>3. - _Thomas Baruchel_, Sep 08 2004

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

%F a(n) = A119457(n+4,n-1) for n>1. - _Reinhard Zumkeller_, May 20 2006

%F G.f.: 6*x/(1-x-x^2). - _Philippe Deléham_, Nov 20 2008

%F a(n) = 6 * A000045(n). - _Alois P. Heinz_, Jan 18 2019

%F E.g.f.: 12*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - _Stefano Spezia_, Nov 09 2025

%p a:= n-> 6*(<<0|1>, <1|1>>^n)[1,2]:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jan 18 2019

%t 6*Fibonacci[Range[0,50]] (* _G. C. Greubel_, Apr 13 2025 *)

%t LinearRecurrence[{1,1},{0,6},50] (* _Harvey P. Dale_, Dec 05 2015 *)

%o (Magma)

%o A022089:= func< n | 6*Fibonacci(n) >;

%o [A022089(n): n in [0..50]]; // _G. C. Greubel_, Apr 13 2025

%o (SageMath)

%o def A022089(n): return 6*fibonacci(n)

%o print([A022089(n) for n in range(51)]) # _G. C. Greubel_, Apr 13 2025

%Y Cf. A000032, A000045, A119457.

%Y Sequences of the form m*Fibonacci listed in A022086.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_