Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 3 (2019), 503-519.

Bin decompositions

Daniel Gotshall, Pamela E. Harris, Dawn Nelson, Maria D. Vega, and Cameron Voigt

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Abstract

It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy F n = F n 1 + F n 2 for n 3 , F 1 = 1 and F 2 = 2 . For any n , m we create a sequence called the ( n , m ) -bin sequence with which we can define a notion of a legal decomposition for every positive integer. These sequences are not always positive linear recurrences, which have been studied in the literature, yet we prove, that like positive linear recurrences, these decompositions exist and are unique. Moreover, our main result proves that the distribution of the number of summands used in the ( n , m ) -bin legal decompositions displays Gaussian behavior.

Article information

Source
Involve, Volume 12, Number 3 (2019), 503-519.

Dates
Received: 18 April 2018
Revised: 10 July 2018
Accepted: 22 July 2018
First available in Project Euclid: 5 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.involve/1549335636

Digital Object Identifier
doi:10.2140/involve.2019.12.503

Mathematical Reviews number (MathSciNet)
MR3905344

Zentralblatt MATH identifier
07033145

Subjects
Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 65Q30: Recurrence relations 60B10: Convergence of probability measures

Keywords
Zeckendorf decompositions bin decompositions Gaussian behavior integer decompositions

Citation

Gotshall, Daniel; Harris, Pamela E.; Nelson, Dawn; Vega, Maria D.; Voigt, Cameron. Bin decompositions. Involve 12 (2019), no. 3, 503--519. doi:10.2140/involve.2019.12.503. https://projecteuclid.org/euclid.involve/1549335636


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References

  • M. Catral, P. Ford, P. Harris, S. J. Miller, and D. Nelson, “Generalizing Zeckendorf's theorem: the Kentucky sequence”, Fibonacci Quart. 52:5 (2014), 68–90.
  • M. Catral, P. L. Ford, P. E. Harris, S. J. Miller, and D. Nelson, “Legal decomposition arising from non-positive linear recurrences”, Fibonacci Quart. 54:4 (2016), 348–365.
  • M. Catral, P. L. Ford, P. E. Harris, S. J. Miller, D. Nelson, Z. Pan, and H. Xu, “New behavior in legal decompositions arising from non-positive linear recurrences”, Fibonacci Quart. 55:3 (2017), 252–275.
  • P. Demontigny, T. Do, A. Kulkarni, S. J. Miller, D. Moon, and U. Varma, “Generalizing Zeckendorf's Theorem to $f$-decompositions”, J. Number Theory 141 (2014), 136–158.
  • P. Demontigny, T. Do, A. Kulkarni, S. J. Miller, and U. Varma, “A generalization of Fibonacci far-difference representations and Gaussian behavior”, Fibonacci Quart. 52:3 (2014), 247–273.
  • R. Dorward, P. L. Ford, E. Fourakis, P. E. Harris, S. J. Miller, E. Palsson, and H. Paugh, “A generalization of Zeckendorf's theorem via circumscribed $m$-gons”, Involve 10:1 (2017), 125–150.
  • R. Dorward, P. L. Ford, E. Fourakis, P. E. Harris, S. J. Miller, E. A. Palsson, and H. Paugh, “Individual gap measures from generalized Zeckendorf decompositions”, Unif. Distrib. Theory 12:1 (2017), 27–36.
  • M. Koloğlu, G. S. Kopp, S. J. Miller, and Y. Wang, “On the number of summands in Zeckendorf decompositions”, Fibonacci Quart. 49:2 (2011), 116–130.
  • C. G. Lekkerkerker, “Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci”, Simon Stevin 29 (1952), 190–195.
  • T. Lengyel, “A counting based proof of the generalized Zeckendorf's theorem”, Fibboniacci Quart. 44:4 (2006), 324–325.
  • S. J. Miller and Y. Wang, “Gaussian behavior in generalized Zeckendorf decompositions”, pp. 159–173 in Combinatorial and additive number theory: CANT 2011 and 2012, edited by M. B. Nathanson, Springer Proc. Math. Stat. 101, Springer, 2014.
  • 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.