Difference in the Number of Summands in the Zeckendorf Partitions of Consecutive Integers

Abstract

Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. We study the difference between the number of summands in the partition of two consecutive integers. In particular, let $L(n)$ be the number of summands in the partition of $n$. We characterize all positive integers such that $L(n)>L(n+1)$, $L(n)<L(n+1)$, and $L(n)=L(n+1)$. Furthermore, we call $n+1$ a peak of $L$ if $L(n)<L(n+1)>L(n+2)$ and a divot of $L$ if $L(n)>L(n+1)<L(n+2)$. We characterize all such peaks and divots of $L$.

6 pages

References in corpus (2)

• On the number of summands in Zeckendorf decompositions
• An Introduction to Completeness of Positive Linear Recurrence Sequences