# 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

### 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…

### 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…

### 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.…

## References

SHOWING 1-10 OF 11 REFERENCES

### ON USING PATTERNS IN BETA-EXPANSIONS TO STUDY FIBONACCI-LUCAS PRODUCTS

- Mathematics
- 2010

The Zeckendorf decomposition of a natural number n is the unique expression of n as a sum of Fibonacci numbers with nonconsecutive indices and with each index greater than 1, where F0 = 0, Fx = 1,…

### Generalized Beatty sequences and complementary triples

- MathematicsMoscow Journal of Combinatorics and Number Theory
- 2019

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…

### The On-Line Encyclopedia of Integer Sequences

- Computer ScienceElectron. J. Comb.
- 1994

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.

### A Number System with an Irrational Base

- Mathematics
- 1957

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…

### On the frequency of occurrence of α in the α-expansions of the positive integers

- The Fibonacci Quarterly 39
- 2001

### On the occurrence of Fn in the Zeckendorf decomposition of nFn

- The Fibonacci Quarterly 37
- 1999

### Canonical Diophantine representations of natural numbers with respect to quadratic “bases”☆

- Mathematics
- 2013

### On the frequency of occurrence of α i in the α-expansions of the positive integers

- The Fibonacci Quarterly
- 2001

### On the occurrence of F n in the Zeckendorf decomposition of nF n

- The Fibonacci Quarterly