An exercise on Fibonacci representations
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 6, pp. 491-498.

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.

Classification: 68R15, 68R05
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     title = {An exercise on {Fibonacci} representations},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {491--498},
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Berstel, Jean. An exercise on Fibonacci representations. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 35 (2001) no. 6, pp. 491-498. http://archive.numdam.org/item/ITA_2001__35_6_491_0/

[1] T.C. Brown, Descriptions of the characteristic sequence of an irrational. Canad. Math. Bull. 36 (1993) 15-21. | MR | Zbl

[2] L. Carlitz, Fibonacci representations. Fibonacci Quarterly 6 (1968) 193-220. | MR | Zbl

[3] S. Eilenberg, Automata, Languages, and Machines, Vol. A. Academic Press (1974). | MR | Zbl

[4] A.S. Fraenkel, Systems of numeration. Amer. Math. Monthly 92 (1985) 105-114. | MR | Zbl

[5] C. Frougny and J. Sakarovitch, Automatic conversion from Fibonacci representation to representation in base ϕ and a generalization. Int. J. Algebra Comput. 9 (1999) 51-384. | MR | Zbl

[6] A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximation I. Abh. Math. Sem. Hamburg 1 (1922) 77-98. | JFM | MR

[7] J. Sakarovitch, Éléments de théorie des automates. Vuibert (to appear). | Zbl

[8] D. Simplot and A. Terlutte, Closure under union and composition of iterated rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 183-212. | EuDML | Numdam | MR | Zbl

[9] D. Simplot and A. Terlutte, Iteration of rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 99-129. | EuDML | Numdam | MR | Zbl

[10] E. Zeckendorff, 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. | MR | Zbl