Involve: A Journal of Mathematics

  • Involve
  • Volume 4, Number 1 (2011), 75-89.

Combinatorial proofs of Zeckendorf representations of Fibonacci and Lucas products

Duncan McGregor and Michael Rowell

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Abstract

In 1998, Filipponi and Hart introduced many Zeckendorf representations of Fibonacci, Lucas and mixed products involving two variables. In 2008, Artz and Rowell proved the simplest of these identities, the Fibonacci product, using tilings. This paper extends the work done by Artz and Rowell to many of the remaining identities from Filipponi and Hart’s work. We also answer an open problem raised by Artz and Rowell and present many Zeckendorf representations of mixed products involving three variables.

Article information

Source
Involve, Volume 4, Number 1 (2011), 75-89.

Dates
Received: 10 August 2010
Accepted: 24 October 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513733364

Digital Object Identifier
doi:10.2140/involve.2011.4.75

Mathematical Reviews number (MathSciNet)
MR2838263

Zentralblatt MATH identifier
1233.05039

Subjects
Primary: 05A19: Combinatorial identities, bijective combinatorics 11B39: Fibonacci and Lucas numbers and polynomials and generalizations

Keywords
number theory Fibonacci numbers Zeckendorf representations combinatorics

Citation

McGregor, Duncan; Rowell, Michael. Combinatorial proofs of Zeckendorf representations of Fibonacci and Lucas products. Involve 4 (2011), no. 1, 75--89. doi:10.2140/involve.2011.4.75. https://projecteuclid.org/euclid.involve/1513733364


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References

  • J. Artz and M. Rowell, “A tiling approach to Fibonacci product identities”, Involve 2:5 (2009), 581–587.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, The Dolciani Mathematical Expositions 27, Mathematical Association of America, Washington, DC, 2003.
  • P. Filipponi and E. L. Hart, “The Zeckendorf decomposition of certain Fibonacci–Lucas products”, Fibonacci Quart. 36:3 (1998), 240–247. http://www.emis.de/cgi-bin/MATH-item?0942.11012Zbl 0942.11012
  • D. Gerdemann, “Combinatorial proofs of Zeckendorf family identities”, Fibonacci Quart. 46/47:3 (2009), 249–261.
  • P. M. Wood, “Bijective proofs for Fibonacci identities related to Zeckendorf's theorem”, Fibonacci Quart. 45:2 (2007), 138–145.
  • E. Zeckendorf, “Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas”, Bull. Soc. Roy. Sci. Liège 41 (1972), 179–182.