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| Format: | Preprint |
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2011
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| Online Access: | https://arxiv.org/abs/1103.4507 |
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| _version_ | 1866917402130776064 |
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| author | Grinberg, Darij |
| author_facet | Grinberg, Darij |
| contents | Philip Matchett Wood and Doron Zeilberger have constructed identities for the Fibonacci numbers $f_n$ of the form $1f_n = f_n$ for all $n \geq 1$; $2f_n = f_{n-2} + f_{n+1}$ for all $n \geq 3$; $3f_n = f_{n-2} + f_{n+2}$ for all $n \geq 3$; $4f_n = f_{n-2} + f_{n} + f_{n+2}$ for all $n \geq 3$; ...; the general identity in this family has the form $kf_n = \sum_{s \in S_k} f_{n+s}$ (for all sufficiently high $n$), where $S_k$ is a finite set of integers that depends only on $k$ and contains no two consecutive integers. These identities are generalized, replacing the left-hand side $kf_n$ by arbitrary sums of the form $f_{n+a_1} + f_{n+a_2} + \cdots + f_{n+a_p}$ for arbitrary integers $a_1, a_2, \ldots, a_p$. The resulting theorem is proved using the connection between the Fibonacci numbers and the golden ratio. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1103_4507 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Zeckendorf family identities generalized Grinberg, Darij Combinatorics 05A19 Philip Matchett Wood and Doron Zeilberger have constructed identities for the Fibonacci numbers $f_n$ of the form $1f_n = f_n$ for all $n \geq 1$; $2f_n = f_{n-2} + f_{n+1}$ for all $n \geq 3$; $3f_n = f_{n-2} + f_{n+2}$ for all $n \geq 3$; $4f_n = f_{n-2} + f_{n} + f_{n+2}$ for all $n \geq 3$; ...; the general identity in this family has the form $kf_n = \sum_{s \in S_k} f_{n+s}$ (for all sufficiently high $n$), where $S_k$ is a finite set of integers that depends only on $k$ and contains no two consecutive integers. These identities are generalized, replacing the left-hand side $kf_n$ by arbitrary sums of the form $f_{n+a_1} + f_{n+a_2} + \cdots + f_{n+a_p}$ for arbitrary integers $a_1, a_2, \ldots, a_p$. The resulting theorem is proved using the connection between the Fibonacci numbers and the golden ratio. |
| title | Zeckendorf family identities generalized |
| topic | Combinatorics 05A19 |
| url | https://arxiv.org/abs/1103.4507 |