In a base phi representation, a natural number is written as a sum of powers of the golden mean φ. There are many ways to do this. How many? Even if the number of powers of φ 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.

In a base phi representation, a natural number is written as a sum of powers of the golden mean φ. There are many ways to do this. Well known is the standard representation, introduced by George Bergman in 1957, where a unique representation is obtained by requiring that no consecutive powers, φⁿ and φⁿ⁺¹, occur in the representation. In this paper, we introduce a new representation by allowing that the powers φ⁰ and φ¹ may occur at the same time, but no other consecutive powers. We then argue that this representation is much closer to the classical representation of the natural numbers by powers of an integer than Bergman’s standard representation.