On the Sum of the Sixth Powers of Fibonacci Numbers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916096977666048 |
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| author | Adegoke, Kunle Layeni, Olawanle |
| author_facet | Adegoke, Kunle Layeni, Olawanle |
| contents | Let $(G_k)_{k\in\mathbb Z}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We derive formulas for $\sum_{j=1}^n{G_{j + t}^6}$ and $\sum_{j=1}^n{(-1)^{j - 1}G_{j + t}^5(G_{j + t - 1} + G_{j + t + 1})}$, thereby extending the results of Ohtsuka and Nakamura who found simple formulas for $\sum_{j=1}^n{F_j^6}$ and $\sum_{j=1}^n{L_j^6}$, where $F_k$ and $L_k$ are the $k$th Fibonacci and Lucas numbers. We also evaluate $\sum_{j = 1}^n {G_{j + t}^3 G_{j + t + 1}^3 } $ and $\sum_{j = 1}^n {G_{j + t - 1}^2 G_{j + t} G_{j + t + 1} G_{j + t + 2}^2 } $, of which the results of Treeby are particular cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Sum of the Sixth Powers of Fibonacci Numbers Adegoke, Kunle Layeni, Olawanle General Mathematics 11B39 (Primary), 11B37 (Secondary) Let $(G_k)_{k\in\mathbb Z}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We derive formulas for $\sum_{j=1}^n{G_{j + t}^6}$ and $\sum_{j=1}^n{(-1)^{j - 1}G_{j + t}^5(G_{j + t - 1} + G_{j + t + 1})}$, thereby extending the results of Ohtsuka and Nakamura who found simple formulas for $\sum_{j=1}^n{F_j^6}$ and $\sum_{j=1}^n{L_j^6}$, where $F_k$ and $L_k$ are the $k$th Fibonacci and Lucas numbers. We also evaluate $\sum_{j = 1}^n {G_{j + t}^3 G_{j + t + 1}^3 } $ and $\sum_{j = 1}^n {G_{j + t - 1}^2 G_{j + t} G_{j + t + 1} G_{j + t + 2}^2 } $, of which the results of Treeby are particular cases. |
| title | On the Sum of the Sixth Powers of Fibonacci Numbers |
| topic | General Mathematics 11B39 (Primary), 11B37 (Secondary) |
| url | https://arxiv.org/abs/2401.10221 |