On the fibbinary numbers and the Wythoffarray
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910461013786624 |
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| author | Macfarlane, A. J. |
| author_facet | Macfarlane, A. J. |
| contents | This paper defines the set fib of fibbinary numbers and displays its structure in the form of a table of a specialised type, and in array form. It uses the Zeckendorf representation $n \in \mathbf{N}$ to define a bijection $\mathcal{Z}$ between $\mathbf{N}$ and fib. It is proved that the fibbinary array is the image under $\mathcal{Z}$ of the famous Wythoff array. The fibbinary table proves useful pictorial insight into the fractal defined by the Wythoff array. The Wythoff table, obtained as the image under the inverse of $\mathcal{Z}$ of the fibbinary table, leads to a simpler view of the fractal, and may be compared with the (1938) Steinhaus tree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18128 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the fibbinary numbers and the Wythoffarray Macfarlane, A. J. Combinatorics Number Theory 11A67, 11B83 This paper defines the set fib of fibbinary numbers and displays its structure in the form of a table of a specialised type, and in array form. It uses the Zeckendorf representation $n \in \mathbf{N}$ to define a bijection $\mathcal{Z}$ between $\mathbf{N}$ and fib. It is proved that the fibbinary array is the image under $\mathcal{Z}$ of the famous Wythoff array. The fibbinary table proves useful pictorial insight into the fractal defined by the Wythoff array. The Wythoff table, obtained as the image under the inverse of $\mathcal{Z}$ of the fibbinary table, leads to a simpler view of the fractal, and may be compared with the (1938) Steinhaus tree. |
| title | On the fibbinary numbers and the Wythoffarray |
| topic | Combinatorics Number Theory 11A67, 11B83 |
| url | https://arxiv.org/abs/2405.18128 |