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.
@article{arxiv.2004.08316,
title = {On Zeckendorf Related Partitions Using the Lucas Sequence},
author = {Hung V. Chu and David C. Luo and Steven J. Miller},
journal= {arXiv preprint arXiv:2004.08316},
year = {2021}
}