Bin decompositions

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

doi:10.2140/involve.2019.12.503

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Bin decompositions

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

Vol. 12 (2019), No. 3, 503–519

DOI: 10.2140/involve.2019.12.503

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 $n \geq 3$ , $F_{1} = 1$ and $F_{2} = 2$ . For any $n, m \in ℕ$ 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.

Keywords

Zeckendorf decompositions, bin decompositions, Gaussian behavior, integer decompositions

Mathematical Subject Classification 2010

Primary: 11B39, 65Q30, 60B10

Milestones

Received: 18 April 2018

Revised: 10 July 2018

Accepted: 22 July 2018

Published: 14 December 2018

Communicated by Stephan Garcia

Authors

Daniel Gotshall
Department of Mathematics Saint Peter’s University Jersey City, NJ United States

Pamela E. Harris
Department of Mathematics and Statistics Williams College Williamstown, MA United States

Dawn Nelson
Department of Mathematics Saint Peter’s University Jersey City, NJ United States

Maria D. Vega
Department of Mathematical Sciences United States Military Academy West Point, NY United States

Cameron Voigt
Department of Mathematical Sciences United States Military Academy West Point, NY United States

Vol. 12, No. 3, 2019