Abstract
Zeckendor’s Theorem guarantees that every positive integer can be uniquely expressed as a sum of Fibonacci numbers, provided no two consecutive numbers are taken. The same holds for Lucas numbers, with the one additional condition that L2 and L0 not occur in the same representation (or else, 5 = L3 + L1 = L2 + L0). This note deals with sets of integers {F kn /F n }, {F kn /L n }, {L kn /L n }, and {L kn /F n } where n and k are positive integers obeying appropriate conditions to assure that all elements in our sequence are integral. Functions φ and λ are displayed where φ denotes the NUMBER OF addends in the FIBONACCI representation and λ the NUMBER OF LUCAS terms.
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. “Zeckendorf’s Theorem and Some Applications.” The Fibonacci Quarterly 2, No. 3 (1964): pp. 163–168.
Hoggatt, V. E. Jr. Fibonacci and Lucas Numbers. Houghton Mifflin Co., Boston (1969).
Filipponi, P. “Sulle Proprietà dei Rapporti fra Particolari Numeri di Fibonacci e di Lucas”. Note Recensioni Notizie 33, No. 3–4 (1984): pp. 91–96.
Filipponi, P. “The Representation of Certain Integers as a Sum of Distinct Fibonacci Numbers.” Fondazione Ugo Bordoni, Int. Report 280985, Roma (1985).
Freitag, Herta T. and Filipponi, P. “On the Representation of Integers in Terms of Sums of Lucas Numbers.” Note Recensioni Notizie 34, No. 3 (1985): pp. 145–150.
Freitag, Herta 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.” Applications of Fibonacci Numbers. Kluwer Academic Publishers, (1988).
Freitag, Herta T. and Filipponi, P. “On the Representation of Integral Sequences {F 2 n /d} and {L 2 n /d} where d is either a Fibonacci Number or a Lucas Number.” The Fibonacci Quarterly. (To appear).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Kluwer Academic Publishers
About this chapter
Cite this chapter
Freitag, H.T. (1990). On the Representation of {F kn /F n }, {F kn /L n }, {L kn /L n }, and {L kn /F n } as Zeckendorf Sums. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1910-5_11
Download citation
DOI: https://doi.org/10.1007/978-94-009-1910-5_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7352-3
Online ISBN: 978-94-009-1910-5
eBook Packages: Springer Book Archive