We describe a technique for mechanically proving certain kinds of theorems in combinatorics on words, using finite automata and a software package for manipulating them. We illustrate our technique by applying it to (a) solve an open problem of Currie and Saari on the lengths of unbordered factors in the Thue-Morse sequence; (b) verify an old result of Prodinger and Urbanek on the regular paperfolding sequence; (c) find an explicit expression for the recurrence function for the Rudin-Shapiro sequence; and (d) improve the avoidance bound in Leech's squarefree sequence. We also introduce a new measure of infinite words called condensation and compute it for some famous sequences. We follow up on the study of Currie and Saari of least periods of infinite words. We show that the characteristic sequence of least periods of a k-automatic sequence is (effectively) k-automatic. We compute the least periods for several famous sequences. Many of our results were obtained by machine computations.
We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f (n) of unbordered factors of the Thue-Morse sequence. We show that f (n) ≤ n for n ≥ 4 and that f (n) = n infinitely often. We also give examples of automatic sequences having exactly 2 unbordered factors of every length.
We investigate questions related to the presence of primitive words and Lyndon words in automatic and linearly recurrent sequences. We show that the Lyndon factorization of a k-automatic sequence is itself k-automatic. We also show that the function counting the number of primitive factors (resp., Lyndon factors) of length n in a k-automatic sequence is k-regular. Finally, we show that the number of Lyndon factors of a linearly recurrent sequence is bounded.
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