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On the Representation of Integral Sequences {Fn/d} and {Ln/d} as Sums of Fibonacci Numbers and as Sums of Lucas Numbers

In: Applications of Fibonacci Numbers

Author

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  • Herta T. Freitag
  • Piero Filipponi

Abstract

Based on Zeckendorf’s theorem concerning the unique sum-representation of any positive integer in terms of Fibonacci numbers as well as Lucas numbers /1/, the purpose of this study is the development of relationships which enable prediction of the NUMBER of addends in these representations. Integral sequences {Fn/d} and {Ln/d} are considered such that d, with 2 is a predetermined integer and n is subject to appropriate conditions to assure integral elements in these sequences. Restrictions on n such that Fn = 0 (mod d) can always be determined. However, for n ε{5, 8, 10, 12, 13, 15, 16, 17, 20} there does not exist an n-value such that Ln = 0 (mod d).

Suggested Citation

  • Herta T. Freitag & Piero Filipponi, 1988. "On the Representation of Integral Sequences {Fn/d} and {Ln/d} as Sums of Fibonacci Numbers and as Sums of Lucas Numbers," Springer Books, in: A. N. Philippou & A. F. Horadam & G. E. Bergum (ed.), Applications of Fibonacci Numbers, pages 97-112, Springer.
    • Handle: RePEc:spr:sprchp:978-94-015-7801-1_11
    DOI: 10.1007/978-94-015-7801-1_11
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