Vol. 10, No. 1, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11, 1 issue

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
Cover Page
Editorial Board
Editors’ Addresses
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Ethics Statement
Subscriptions
Editorial Login
Author Index
Coming Soon
Contacts
 
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
A generalization of Zeckendorf's theorem via circumscribed $m$-gons

Robert Dorward, Pari L. Ford, Eva Fourakis, Pamela E. Harris, Steven J. Miller, Eyvindur Palsson and Hannah Paugh

Vol. 10 (2017), No. 1, 125–150
Abstract

Zeckendorf’s theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy F1 = 1, F2 = 2, and Fn = Fn1 + Fn2 for n 3. The distribution of the number of summands in such a decomposition converges to a Gaussian, the gaps between summands converge to geometric decay, and the distribution of the longest gap is similar to that of the longest run of heads in a biased coin; these results also hold more generally, though for technical reasons previous work is needed to assume the coefficients in the recurrence relation are nonnegative and the first term is positive.

We extend these results by creating an infinite family of integer sequences called the m-gonal sequences arising from a geometric construction using circumscribed m-gons. They satisfy a recurrence where the first m+1 leading terms vanish, and thus cannot be handled by existing techniques. We provide a notion of a legal decomposition, and prove that the decompositions exist and are unique. We then examine the distribution of the number of summands used in the decompositions and prove that it displays Gaussian behavior. There is geometric decay in the distribution of gaps, both for gaps taken from all integers in an interval and almost surely in distribution for the individual gap measures associated to each integer in the interval. We end by proving that the distribution of the longest gap between summands is strongly concentrated about its mean, behaving similarly as in the longest run of heads in tosses of a coin.

Keywords
Zeckendorf decompositions, longest gap
Mathematical Subject Classification 2010
Primary: 11B39, 11B05
Secondary: 65Q30, 60B10
Milestones
Received: 10 September 2015
Revised: 5 December 2015
Accepted: 13 December 2015
Published: 11 October 2016

Communicated by Stephan Garcia
Authors
Robert Dorward
Department of Mathematics
Oberlin College
Oberlin, OH 44074
United States
Pari L. Ford
Department of Mathematics and Physics
Bethany College
Lindsborg, KS 67456
United States
Eva Fourakis
Department of Mathematics and Statistics
Williams College
Williamstown, MA 01267
United States
Pamela E. Harris
Department of Mathematical Sciences
United States Military Academy
West Point, NY 10996
United States
Department of Mathematics and Statistics
Williams College
Williamstown, MA 01267
United States
Steven J. Miller
Department of Mathematics and Statistics
Williams College
Williamstown, MA 01267
United States
Eyvindur Palsson
Department of Mathematics and Statistics
Williams College
Williamstown, MA 01267
United States
Department of Mathematics
Virginia Tech
Blacksburg, VA 24061
United States
Hannah Paugh
Department of Mathematical Sciences
United States Military Academy
West Point, NY 10996
United States