  In a base phi representation a natural number is written as a sum of powers
of the golden mean $\varphi$. There are many ways to do this. How many? Even if
the number of powers of $\varphi$ is finite, then any number has infinitely
many base phi representations. By not allowing an expansion to end with the
digits 0,1,1, the number of expansions becomes finite, a solution proposed by
Ron Knott. Our first result is a recursion to compute this number of
expansions. This recursion is closely related to the recursion given by Neville
Robbins to compute the number of Fibonacci representations of a number, also
known as Fibonacci partitions. We propose another way to obtain finitely many
expansions, which we call the natural base phi expansions. We prove that these
are closely connected to the Fibonacci partitions.
