Base phi representations and golden mean beta-expansions

Michel Dekking

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arXiv · 1906.08437

Base phi representations and golden mean beta-expansions

Michel Dekking

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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.

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Michel Dekking. 2019-06-20. Base phi representations and golden mean beta-expansions. https://arxiv.org/abs/1906.08437

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