The sum of digits function of the base phi expansion of the natural numbers

Journal article (2020)

Authors

F.M. Dekking Applied Probability - EEMCS
Research Group
Applied Probability (EEMCS) (TU Delft)
Copyright: © 2020 F.M. Dekking
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Published Date

2020

Language

English

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

Abstract

In the base phi expansion any natural number is written uniquely as a sum of 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 show that the sum of digits function modulo 2 of these expansions is a morphic sequence. In particular we prove that — like for the Thue-Morse sequence — the frequency of 0’s and 1’s in this sequence is equal to 1/2.