Hostname: page-component-7dc689bd49-6fmns Total loading time: 0 Render date: 2023-03-20T08:53:33.324Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

An Exercise on Fibonacci Representations

Published online by Cambridge University Press:  15 July 2002

Jean Berstel*
Institut Gaspard Monge (IGM), Université de Marne-la-Vallée, 5 boulevard Descartes, 77454 Marne-la-Vallée Cedex 2, France; (
Get access


We give a partial answer to a question of Carlitz asking for a closed formula for the number of distinct representations of an integer in the Fibonacci base.


Research Article
© EDP Sciences, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Brown, T.C., Descriptions of the characteristic sequence of an irrational. Canad. Math. Bull. 36 (1993) 15-21. CrossRef
Carlitz, L., Fibonacci representations. Fibonacci Quarterly 6 (1968) 193-220.
S. Eilenberg, Automata, Languages, and Machines, Vol. A. Academic Press (1974).
Fraenkel, A.S., Systems of numeration. Amer. Math. Monthly 92 (1985) 105-114. CrossRef
Frougny, C. and Sakarovitch, J., Automatic conversion from Fibonacci representation to representation in base φ and a generalization. Int. J. Algebra Comput. 9 (1999) 51-384.
Ostrowski, A., Bemerkungen zur Theorie der Diophantischen Approximation I. Abh. Math. Sem. Hamburg 1 (1922) 77-98. CrossRef
J. Sakarovitch, Éléments de théorie des automates. Vuibert (to appear).
Simplot, D. and Terlutte, A., Closure under union and composition of iterated rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 183-212.
Simplot, D. and Terlutte, A., Iteration of rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 99-129.
Zeckendorff, E., Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas. Bull. Soc. Royale Sci. Liège 42 (1972) 179-182.