On Zeckendorf Related Partitions Using the Lucas Sequence

TL;DR

This paper explores partitions of natural numbers into non-consecutive Lucas sequence terms, establishing limits on the number of such partitions, characterizing fixed-term partitions, and analyzing their frequency.

Contribution

It proves that each number has at most two such partitions, characterizes numbers with fixed terms, and determines the asymptotic proportion of non-unique partitions.

Findings

Each number has at most two non-consecutive Lucas partitions.

Characterization of numbers with a fixed term in their partition.

Calculated the limiting proportion of numbers with non-unique partitions.

Abstract

Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive terms of the Lucas sequence, although such partitions need not be unique. In this paper, we prove that a natural number can have at most two distinct non-consecutive partitions in the Lucas sequence, find all positive integers with a fixed term in their partition, and calculate the limiting value of the proportion of natural numbers that are not uniquely partitioned into the sum of non-consecutive terms in the Lucas sequence.