  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.
