Abstract
That any positive integer N can be represented as a sum of distinct nonconsecutive Fibonacci numbers F n is a well-known fact. Apart from the equivalent use of F 2 instead of F 1, such a representation is unique [1] and is commonly referred to as the Zeckendorf Decomposition (or Representation) of N (ZD of N, in brief). Since the ZD of a class of integers is in general unpredictable, its discovery is always a pleasant surprise to the researcher. This fact led us to undertake this kind of investigations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Brown, J.L. Jr. “Zeckendorfs Theorem and Some Applications”. The Fibonacci Quarterly, Vol. 2.3 (1964): pp. 163–8.
Bucci, M., DiPorto, A. and Filipponi, P. “A Note on the Sequence {F 5n /(5F n )}”. Abstracts AMS, Vol. 10.4 (1991): p. 341.
Filipponi, P. “The Representation of Certain Integers as a Sum of Distinct Fibonacci Numbers”. Fondazione Ugo Bordoni, Technical Report 2B0985, Roma (1985).
Filipponi, P. and Freitag, H.T. “Some Properties of the Sequences {F kn /F n }”. Fondazione Ugo Bordoni, Technical Report 3T00292, Roma (1992).
Filipponi, P. and Freitag, H.T. “The Zeckendorf Representation of {F kn/F n}”. Application of Fibonacci Numbers. Volume 5. Edited by G.E. Bergum, A.N. Philippou and A.F. Horadam, Kluwer Acad. Publ., Dordrecht, The Netherlands, 1993, pp. 217–219.
Freitag, H.T. and Filipponi, P. “On the Representation of Integral Sequences {F n /d} and {L n /d} as Sums of Fibonacci Numbers and as Sums of Lucas Numbers”. Application of Fibonacci Numbers, Volume 2. Edited by G.E. Bergum, A.N. Philippou and A.F. Horadam, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988, pp. 97–112.
Freitag, H.T. and Filipponi, P. “On the F-Representation of Integral Sequences {F 2 n /d} and {F 2 n /d} Where d is either a Fibonacci or a Lucas Number”. The Fibonacci Quarterly, Vol. 27.3 (1989): pp. 276–282, 286.
Freitag, H.T. “On the Representation of {F kn /F n }, {F kn /L n }, {L kn /F n } and {L kn /L n } as Zeckendorf Sums”. Application of Fibonacci Numbers. Volume 3. Edited by G.E. Bergum, A.N. Philippou and A.F. Horadam. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990: pp. 107–114.
Hoggatt, V.E. Jr. Fibonacci and Lucas Numbers. Boston: Houghton Mifflin, 1969.
Jarden, D. Recurring Sequences. 3rd ed. Jerusalem (Israel): Riveon Lematematika, 1973.
Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. Chichester (UK): Ellis Horwood Ltd., 1989.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers
About this chapter
Cite this chapter
Filipponi, P., Freitag, H.T. (1996). The Zeckendorf Decomposition of Certain Classes of Integers. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0223-7_11
Download citation
DOI: https://doi.org/10.1007/978-94-009-0223-7_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6583-2
Online ISBN: 978-94-009-0223-7
eBook Packages: Springer Book Archive