Base phi representations and golden mean beta-expansions

TL;DR

This paper investigates unique representations of natural numbers using the golden mean with specific digit constraints, providing explicit formulas and proving conjectures related to digit positions in these expansions.

Contribution

It offers precise formulas for digit positions in base phi representations and proves two conjectures for specific cases, advancing understanding of golden mean expansions.

Findings

Explicit expressions for when the kth digit is 1 in base phi representations.

Proof of two conjectures for k=0,1 regarding digit positions.

Representation formulas involve generalized Beatty sequences.

Abstract

In the base phi representation any natural number is written uniquely as a sum powers of the golden mean with digits 0 and 1, where one requires that the product of two consecutive digits is always 0. In this paper we give precise expressions for the those natural numbers for which the $k$th digit is 1, proving two conjectures for $k=0,1$. The expressions are all in terms of generalized Beatty sequences.