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Applications of Fibonacci Numbers pp 321–337Cite as

The Wythoff and the Zeckendorf Representations of Numbers are Equivalent

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Abstract

The quintessence of many application of Fibonacci numbers is the binary substitution sequence 1→10, 0→1. The infinite sequence generated this way is self-similar and quasiperiodic. See refs. [7, 16] for details on this rabbit or golden sequences. It is intimately related to Wythoffs complementary sequences which cover the natural numbers ([·] is the greatest integer function)

$$ A(m): = \left\lfloor {m\varphi } \right\rfloor ,\,B(m): = \left\lfloor {m{\varphi ^2}}\right\rfloor ,\,m \in \,N,{\varphi ^2} = \varphi + 1,\,\varphi > 0. $$
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References

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Authors
  1. Wolfdieter Lang
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Editors and Affiliations

  1. Computer Science Department, South Dakota State University, Box 2201, 57007-1596, Brookings, South Dakota, USA

    Gerald E. Bergum

  2. 26 Atlantis Street, Aglangia, Nicosia, Cyprus

    Andreas N. Philippou

  3. Department of Mathematics, Statistics and Computer Science, The University of New England, Armidale, New South Wales, 2351, Australia

    Alwyn F. Horadam

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© 1996 Kluwer Academic Publishers

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Lang, W. (1996). The Wythoff and the Zeckendorf Representations of Numbers are Equivalent. 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_27

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  • DOI: https://doi.org/10.1007/978-94-009-0223-7_27

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6583-2

  • Online ISBN: 978-94-009-0223-7

  • eBook Packages: Springer Book Archive

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