An exercise on Fibonacci representations
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 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.

@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},
     zbl = {1005.68119},
     mrnumber = {1922290},
     language = {en},
     url = {http://www.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
DA  - 2001///
SP  - 491
EP  - 498
VL  - 35
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/ITA_2001__35_6_491_0/
UR  - https://zbmath.org/?q=an%3A1005.68119
UR  - https://www.ams.org/mathscinet-getitem?mr=1922290
LA  - en
ID  - ITA_2001__35_6_491_0
ER  - 
Berstel, Jean. An exercise on Fibonacci representations. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 6, pp. 491-498. http://www.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 1205889 | Zbl 0804.11021

[2] L. Carlitz, Fibonacci representations. Fibonacci Quarterly 6 (1968) 193-220. | MR 236094 | Zbl 0167.03901

[3] S. Eilenberg, Automata, Languages, and Machines, Vol. A. Academic Press (1974). | MR 530382 | Zbl 0317.94045

[4] A.S. Fraenkel, Systems of numeration. Amer. Math. Monthly 92 (1985) 105-114. | MR 777556 | Zbl 0568.10005

[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 1723473 | Zbl 1040.68061

[6] A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximation I. Abh. Math. Sem. Hamburg 1 (1922) 77-98. | JFM 48.0185.01 | MR 3069389

[7] J. Sakarovitch, Éléments de théorie des automates. Vuibert (to appear). | Zbl 1178.68002

[8] D. Simplot and A. Terlutte, Closure under union and composition of iterated rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 183-212. | EuDML 221990 | Numdam | MR 1796268 | Zbl 0970.68085

[9] D. Simplot and A. Terlutte, Iteration of rational transductions. RAIRO: Theoret. Informatics Appl. 34 (2000) 99-129. | EuDML 222099 | Numdam | MR 1774304 | Zbl 0962.68090

[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 308032 | Zbl 0252.10011