Base phi representations and golden mean beta-expansions

Abstract

In the base phi representation, any 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. In this paper, we give precise expressions for those natural numbers for which the kth digit is 1, proving two conjectures for k = 0,1. The expressions are all in terms of generalized Beatty sequences.