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December 2012 Circular words and three applications: factors of the Fibonacci word, $\mathcal F$-adic numbers, and the sequence 1, 5, 16, 45, 121, 320,...
Benoît Rittaud, Laurent Vivier
Funct. Approx. Comment. Math. 47(2): 207-231 (December 2012). DOI: 10.7169/facm/2012.47.2.6

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 establish a new result on factors of the Fibonacci word $abaababaabaab\ldots$. Second, we present an expression of the sequence A004146 of [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 particular that it contains exactly $q$ ${\mathcal F}$-adic elements.

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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,...." Funct. Approx. Comment. Math. 47 (2) 207 - 231, December 2012. https://doi.org/10.7169/facm/2012.47.2.6

Information

Published: December 2012
First available in Project Euclid: 20 December 2012

zbMATH: 1260.68313
MathSciNet: MR3051449
Digital Object Identifier: 10.7169/facm/2012.47.2.6

Subjects:
Primary: 68R15
Secondary: 11A63, 11E95, 37B10

Rights: Copyright © 2012 Adam Mickiewicz University

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Vol.47 • No. 2 • December 2012
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