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Integers with a maximal number of Fibonacci representations

Published online by Cambridge University Press:  15 April 2005

Petra Kocábová ,
Zuzana Masáková  and
Edita Pelantová
Petra Kocábová
Affiliation:
Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; petra.kocabova@centrum.cz, masakova@km1.fjfi.cvut.cz, pelantova@km1.fjfi.cvut.cz
Zuzana Masáková
Affiliation:
Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; petra.kocabova@centrum.cz, masakova@km1.fjfi.cvut.cz, pelantova@km1.fjfi.cvut.cz
Edita Pelantová
Affiliation:
Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; petra.kocabova@centrum.cz, masakova@km1.fjfi.cvut.cz, pelantova@km1.fjfi.cvut.cz
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Abstract

We study the properties of the function R(n) which determines the number of representations of an integer n as a sum of distinct Fibonacci numbers Fk. We determine the maximum and mean values of R(n) for Fk ≤ n < Fk+1.

Keywords

Fibonacci numbersZeckendorf representation.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 39 , Issue 2 , April 2005 , pp. 343 - 359
Copyright
© EDP Sciences, 2005

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References

Berstel, J., An exercise on Fibonacci representations. RAIRO-Inf. Theor. Appl. 35 (2001) 491–498. CrossRef
M. Bicknell-Johnson, The smallest positive integer having F k representations as sums of distinct Fibonacci numbers, in Applications of Fibonacci numbers. Vol. 8, Kluwer Acad. Publ., Dordrecht (1999) 47–52.
Bicknell-Johnson, M. and Fielder, D.C., The number of representations of N using distinct Fibonacci numbers, counted by recursive formulas. Fibonacci Quart. 37 (1999) 47–60.
M. Edson and L. Zamboni, On representations of positive integers in the Fibonacci base. Preprint University of North Texas (2003).