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

TL;DR

This paper analyzes the differences in the number of Fibonacci summands in Zeckendorf partitions of consecutive integers, characterizing all cases of increases, decreases, peaks, and divots in these counts.

Contribution

It provides a complete characterization of when the number of summands in Zeckendorf partitions of consecutive integers increases, decreases, or remains the same, including peaks and divots.

Findings

Characterization of all integers where $L(n) > L(n+1)$, $L(n) < L(n+1)$, and $L(n) = L(n+1)$.

Identification of all peaks and divots in the sequence of summand counts.

Insights into the structure of Zeckendorf partitions for 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 $L$.