Extension: Fibonacci Sequence

Let us define the Fibonacci sequence as: $$F_1 = 1, F_2 = 2, F_{n + 1} = F_n + F_{n - 1}\text{ for integers }n \geq 2$$ Then every positive integer has a unique decomposition as a sum of non-consecutive elements of the sequence (known as a Zeckendorf decomposition). For example, the Zeckendorf decomposition of $15$ is $13+2$.