• Corpus ID: 195218850

# Base phi representations and golden mean beta-expansions

@article{Dekking2019BasePR,
title={Base phi representations and golden mean beta-expansions},
author={Michel Dekking},
journal={arXiv: Number Theory},
year={2019}
}
• M. Dekking
• Published 20 June 2019
• Mathematics
• arXiv: Number Theory
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.
5 Citations
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
• Mathematics
• 2021
In a base phi representation a natural number is written as a sum of powers of the golden mean φ. There are many ways to do this. Well known is the standard representation, introduced by George
We prove that the sequence of first differences of the points of increase of the sum of digits function of the phi expansions of the natural numbers is a morphic sequence. We also show that it is the
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.

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A generalized Beatty sequence is a sequence $V$ defined by $V(n)=p\lfloor{n\alpha}\rfloor+qn +r$, for $n=1,2,\dots$, where $\alpha$ is a real number, and $p,q,r$ are integers. These occur in several
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The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
Editors' Note: This article was written by the author when he was a 12-year old student at Junior High School 246 in Brooklyn, New York. Here he explores using the golden mean, which he calls τ, more

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