SYMMETRIC FIBONACCI WORDS *

In [1] the author studied Fibonacci words; the study was motivated by the consideration of Fibonacci strings and Fibonacci word patterns by Knuth [5] and Turner [6, 7], respectively. It was shown in [1] that all the /1 Fibonacci words can be obtained from any particular /2 Fibonacci word, for example w®, by shifting in a cyclic way the letters in it. Also it was shown that each of the Fibonacci words w° («>3) has a representation as a product of two symmetric words. In this paper, we show that every Fibonacci word has such a representation and that this representation is unique (Theorem 3). Furthermore, we prove that, for each positive integer n that is not a multiple of 3, there is precisely one symmetric Fibonacci word of length Fn, where Fn denotes the n^ Fibonacci number, while there are no symmetric Fibonacci words of length Fn if n is a multiple of 3 (Theorem 7). Let X be an alphabet and let X* be a free monoid of words over X with identity 1. Denote by £(w) the length of a word w. Define the reverse R and the shift T on X*l{ 1} by