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

Abstract : We introduce the notion of {\em circular words} with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we give a new result on factors of the Fibonacci word $abaababaabaab\ldots$. Second, we present an expression of the sequence A004146 of \cite{Sloane} 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 {\em ${\mathcal F}$-adic numbers}, we give a characterization of the subset of those which are {\em rational} (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 $q$ ${\mathcal F}$-adic elements.
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Pré-publication, Document de travail
2011

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https://hal.archives-ouvertes.fr/hal-00566314
Contributeur : Benoît Rittaud <>
Soumis le : mardi 15 février 2011 - 23:09:47
Dernière modification le : dimanche 17 mars 2019 - 18:16:01
Document(s) archivé(s) le : lundi 16 mai 2011 - 03:28:31

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• HAL Id : hal-00566314, version 1

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Benoît Rittaud, Laurent Vivier. Circular words and three applications: factors of the Fibonacci word, ${\mathcal F}$-adic numbers, and the sequence $1$, $5$, $16$, $45$, $121$, $320$,\ldots. 2011. 〈hal-00566314〉

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