Vol. 12, No. 3, 2019

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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
Abstract

It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy Fn = Fn1 + Fn2 for n 3, F1 = 1 and F2 = 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.

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