Enumeration of factors in the Thue-Morse word

Abstract

We give a new combinatorial property of square factors in the infinite Thue-Morse word M2: every nontrivial factorization M2=w1w2 implies that w1 has a square suffix, or w2 has a square prefix. The proof is based on a coding which yields also a new demonstration of the characterization of square factors. Finally, we present results on the enumeration of factors: first explicit formulas, then the asymptotic values as well as a recurrence formula are given. © 1992.