# 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} }

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

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

- Mathematics
- 2019

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.…

### The sum of digits functions of the Zeckendorf and the base phi expansions

- MathematicsTheor. Comput. Sci.
- 2021

### On the representation of the natural numbers by powers of the golden mean

- 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…

### Points of increase of the sum of digits function of the base phi expansion

- Mathematics
- 2020

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…

### How to add two natural numbers in base phi

- Mathematics
- 2020

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…

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