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.
@article{ITA_2001__35_6_491_0, author = {Berstel, Jean}, title = {An exercise on {Fibonacci} representations}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, pages = {491--498}, publisher = {EDP-Sciences}, volume = {35}, number = {6}, year = {2001}, mrnumber = {1922290}, zbl = {1005.68119}, language = {en}, url = {http://archive.numdam.org/item/ITA_2001__35_6_491_0/} }
TY - JOUR AU - Berstel, Jean TI - An exercise on Fibonacci representations JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2001 SP - 491 EP - 498 VL - 35 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/item/ITA_2001__35_6_491_0/ LA - en ID - ITA_2001__35_6_491_0 ER -
%0 Journal Article %A Berstel, Jean %T An exercise on Fibonacci representations %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2001 %P 491-498 %V 35 %N 6 %I EDP-Sciences %U http://archive.numdam.org/item/ITA_2001__35_6_491_0/ %G en %F ITA_2001__35_6_491_0
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] Descriptions of the characteristic sequence of an irrational. Canad. Math. Bull. 36 (1993) 15-21. | MR | Zbl
,[2] Fibonacci representations. Fibonacci Quarterly 6 (1968) 193-220. | MR | Zbl
,[3] Automata, Languages, and Machines, Vol. A. Academic Press (1974). | MR | Zbl
,[4] Systems of numeration. Amer. Math. Monthly 92 (1985) 105-114. | MR | Zbl
,[5] Automatic conversion from Fibonacci representation to representation in base and a generalization. Int. J. Algebra Comput. 9 (1999) 51-384. | MR | Zbl
and ,[6] Bemerkungen zur Theorie der Diophantischen Approximation I. Abh. Math. Sem. Hamburg 1 (1922) 77-98. | JFM | MR
,[7] Éléments de théorie des automates. Vuibert (to appear). | Zbl
,[8] Closure under union and composition of iterated rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 183-212. | EuDML | Numdam | MR | Zbl
and ,[9] Iteration of rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 99-129. | EuDML | Numdam | MR | Zbl
and ,[10] 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
,