Circular words and three applications: factors of the Fibonacci word, $\mathcal F$-adic numbers, and the sequence 1, 5, 16, 45, 121, 320,...

Abstract

We introduce the notion of circular words with a combi- natorial constraint derived from the Zeckendorf (Fibonacci) numera- tion system, and get explicit group structures for these words. As a first application, we give a new result on factors of the Fibonacci word abaababaabaab.... Second, we present an expression of the sequence A004146 of (S) in terms of a product of expressions involving roots of unity. Third, we consider the equivalent of p-adic numbers that arise by the use of the numeration system defined by the Fibonacci sequence instead of the usual numeration system in base p. Among such F-adic numbers, we give a characterization of the subset of those which are ra- tional (that is: a root of an equation of the form qX = p, for integral values of p and q) by a periodicity property. Eventually, with the help of circular words, we give a complete description of the set of roots of qX = p, showing in particuler that it contains exactly q F-adic elements. Classically, a (finite) word is a finite sequence of elements (or letters) of a given set, the alphabet. Here, we mean by circular word a finite word w0 ...wn in which the last letter, wn, is assumed to be followed by the first one, w0. This definition gives rise to interesting properties when circular words are assumed to be admissible, that is, made of letters in the alphabet {0,1} without any two successive letters equal to 1. These properties derive from an underlying algebraic structure: the set of admissible circular words of fixed even length is an abelian group, which can be explicitely written as a product of finite monogenetic groups.