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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.
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References
[1]
Brown, T.C., Descriptions of the characteristic sequence of an irrational. Canad. Math. Bull.36 (1993) 15-21.
CrossRef
S. Eilenberg, Automata, Languages, and Machines, Vol. A. Academic Press (1974).
[4]
Fraenkel, A.S., Systems of numeration. Amer. Math. Monthly92 (1985) 105-114.
CrossRef
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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.
[6]
Ostrowski, A., Bemerkungen zur Theorie der Diophantischen Approximation I. Abh. Math. Sem. Hamburg1 (1922) 77-98.
CrossRef
[7]
J. Sakarovitch, Éléments de théorie des automates. Vuibert (to appear).
[8]
Simplot, D. and Terlutte, A., Closure under union and composition of iterated rational transductions. RAIRO: Theoret. Informatics Appl.34 (2000) 183-212.
[9]
Simplot, D. and Terlutte, A., Iteration of rational transductions. RAIRO: Theoret. Informatics Appl.34 (2000) 99-129.
[10]
Zeckendorff, E., Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas. Bull. Soc. Royale Sci. Liège42 (1972) 179-182.