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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Zeckendorf decompositions bin decompositions Gaussian behavior integer decompositions


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.

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