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}, volume = {35}, year = {2001}, pages = {491-498}, mrnumber = {1922290}, zbl = {1005.68119}, language = {en}, url = {http://dml.mathdoc.fr/item/ITA_2001__35_6_491_0} }
Berstel, Jean. An exercise on Fibonacci representations. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) pp. 491-498. http://gdmltest.u-ga.fr/item/ITA_2001__35_6_491_0/
[1] Descriptions of the characteristic sequence of an irrational. Canad. Math. Bull. 36 (1993) 15-21. | MR 1205889 | Zbl 0804.11021
,[2] Fibonacci representations. Fibonacci Quarterly 6 (1968) 193-220. | MR 236094 | Zbl 0167.03901
,[3] Automata, Languages, and Machines, Vol. A. Academic Press (1974). | MR 530382 | Zbl 0317.94045
,[4] Systems of numeration. Amer. Math. Monthly 92 (1985) 105-114. | MR 777556 | Zbl 0568.10005
,[5] Automatic conversion from Fibonacci representation to representation in base and a generalization. Int. J. Algebra Comput. 9 (1999) 51-384. | MR 1723473 | Zbl 1040.68061
and ,[6] Bemerkungen zur Theorie der Diophantischen Approximation I. Abh. Math. Sem. Hamburg 1 (1922) 77-98. | JFM 48.0197.04 | JFM 48.0185.01 | MR 3069389
,[7] Éléments de théorie des automates. Vuibert (to appear). | Zbl 1178.68002
,[8] Closure under union and composition of iterated rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 183-212. | Numdam | MR 1796268 | Zbl 0970.68085
and ,[9] Iteration of rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 99-129. | Numdam | MR 1774304 | Zbl 0962.68090
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 308032 | Zbl 0252.10011
,