## On the factors of the Thue-Morse word on three symbols.(English)Zbl 0746.68067

Summary: Let $$m$$ be the Thue-Morse word in a three-letter alphabet. A finite factor $$v$$ of $$m$$ is called special if there exist two distinct letters $$x$$, $$y$$ of the alphabet such that $$vx$$ and $$vy$$ are factors of $$m$$. Define, for each $$n>0$$, $$\psi(n)$$ (respectively $$g(n)$$) to be the number of special factors (respectively factors) of length $$n$$. By using some of our previous results Aldo De Luca and Stefano Varricchio [Theor. Comput. Sci. 63(3), 333-348 (1989; Zbl 0671.10050)] on the Thue- Morse word on two symbols we prove the following proposition: For each $$n>0$$, $$\psi(n)=4$$ if $$n\leq3\times 2^{[\log_ 2n]-1}-1$$, and $$\psi(n)=2$$ otherwise (where $$[x]$$ is the integer part of $$x$$). As a corollary one has: For each $$n>1$$, $$g(n)=4n-2^{[\log_ 2n]}$$ if $$n\leq 3\times 2^{[\log_ 2n]-1}$$ and $$g(n)=2n+2^{[\log_ 2n]+1}$$ otherwise.

### MSC:

 68R15 Combinatorics on words 68Q45 Formal languages and automata

### Keywords:

square-free word; special factor; structure-function

Zbl 0671.10050
Full Text:

### References:

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