The structure of base phi expansions

Abstract

In the base phi expansion, a natural number is written uniquely as a sum of powers of the golden mean with coefficients 0 and 1, where it is required that the product of two consecutive digits is always 0. We tackle the problem of describing these expansions in detail. We classify the positive parts of the base phi expansions according to their suffixes, and the negative parts according to their prefixes, specifying the sequences of occurrences of these digit blocks. We prove that the positive parts of the base phi expansions are a subsequence of the sequence of Zeckendorf expansions, giving an explicit formula in terms of a generalized Beatty sequence. The negative parts of the base phi expansions no longer appear lexicographically. We prove that all allowed digit blocks appear, and determine the order in which they do appear.