A note on the power sums of the number of Fibonacci partitions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908347646607360 |
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| author | Sanna, Carlo |
| author_facet | Sanna, Carlo |
| contents | For every nonnegative integer $n$, let $r_F(n)$ be the number of ways to write $n$ as a sum of Fibonacci numbers, where the order of the summands does not matter. Moreover, for all positive integers $p$ and $N$, let \begin{equation*} S_{F}^{(p)}(N) := \sum_{n = 0}^{N - 1} \big(r_F(n)\big)^p . \end{equation*} Chow, Jones, and Slattery determined the order of growth of $S_{F}^{(p)}(N)$ for $p \in \{1,2\}$. We prove that, for all positive integers $p$, there exists a real number $λ_p > 1$ such that \begin{equation*} S^{(p)}_F(N) \asymp_p N^{(\log λ_p) /\!\log φ} \end{equation*} as $N \to +\infty$, where $φ:= (1 + \sqrt{5})/2$ is the golden ratio. Furthermore, we show that \begin{equation*} \lim_{p \to +\infty} λ_p^{1/p} = φ^{1/2} . \end{equation*} Our proofs employ automata theory and a result on the generalized spectral radius due to Blondel and Nesterov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12724 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on the power sums of the number of Fibonacci partitions Sanna, Carlo Number Theory Formal Languages and Automata Theory 11B39 (Primary) 05A16, 05A17, 11P99, 68Q45 (Secondary) For every nonnegative integer $n$, let $r_F(n)$ be the number of ways to write $n$ as a sum of Fibonacci numbers, where the order of the summands does not matter. Moreover, for all positive integers $p$ and $N$, let \begin{equation*} S_{F}^{(p)}(N) := \sum_{n = 0}^{N - 1} \big(r_F(n)\big)^p . \end{equation*} Chow, Jones, and Slattery determined the order of growth of $S_{F}^{(p)}(N)$ for $p \in \{1,2\}$. We prove that, for all positive integers $p$, there exists a real number $λ_p > 1$ such that \begin{equation*} S^{(p)}_F(N) \asymp_p N^{(\log λ_p) /\!\log φ} \end{equation*} as $N \to +\infty$, where $φ:= (1 + \sqrt{5})/2$ is the golden ratio. Furthermore, we show that \begin{equation*} \lim_{p \to +\infty} λ_p^{1/p} = φ^{1/2} . \end{equation*} Our proofs employ automata theory and a result on the generalized spectral radius due to Blondel and Nesterov. |
| title | A note on the power sums of the number of Fibonacci partitions |
| topic | Number Theory Formal Languages and Automata Theory 11B39 (Primary) 05A16, 05A17, 11P99, 68Q45 (Secondary) |
| url | https://arxiv.org/abs/2309.12724 |