Sum of Consecutive Terms of Pell and Related Sequences
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arXiv
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| Autores principales: | , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Anand, Navvye Basistha, Amit Kumar Davenport, Kenny B. Gong, Alexander Luca, Florian Miller, Steven J. Zhu, Alexander |
| author_facet | Anand, Navvye Basistha, Amit Kumar Davenport, Kenny B. Gong, Alexander Luca, Florian Miller, Steven J. Zhu, Alexander |
| contents | We study new identities related to the sums of adjacent terms in the Pell sequence, defined by $P_{n} := 2P_{n-1}+P_{n-2}$ for $ n\geq 2$ and $P_{0}=0, P_{1}=1$, and generalize these identities for many similar sequences. We prove that the sum of $N>1$ consecutive Pell numbers is a fixed integer multiple of another Pell number if and only if $4\mid N$. We consider the generalized Pell $(k,i)$-numbers defined by $p(n) :=\ 2p(n-1)+p(n-k-1) $ for $n\geq k+1$, with $p(0)=p(1)=\cdots =p(i)=0$ and $p(i+1)=\cdots = p(k)=1$ for $0\leq i\leq k-1$, and prove that the sum of $N=2k+2$ consecutive terms is a fixed integer multiple of another term in the sequence. We also prove that for the generalized Pell $(k,k-1)$-numbers such a relation does not exist when $N$ and $k$ are odd. We give analogous results for the Fibonacci and other related second-order recursive sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12868 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sum of Consecutive Terms of Pell and Related Sequences Anand, Navvye Basistha, Amit Kumar Davenport, Kenny B. Gong, Alexander Luca, Florian Miller, Steven J. Zhu, Alexander Number Theory Combinatorics 11Bxx, 11B37, 11B39, 11B50 We study new identities related to the sums of adjacent terms in the Pell sequence, defined by $P_{n} := 2P_{n-1}+P_{n-2}$ for $ n\geq 2$ and $P_{0}=0, P_{1}=1$, and generalize these identities for many similar sequences. We prove that the sum of $N>1$ consecutive Pell numbers is a fixed integer multiple of another Pell number if and only if $4\mid N$. We consider the generalized Pell $(k,i)$-numbers defined by $p(n) :=\ 2p(n-1)+p(n-k-1) $ for $n\geq k+1$, with $p(0)=p(1)=\cdots =p(i)=0$ and $p(i+1)=\cdots = p(k)=1$ for $0\leq i\leq k-1$, and prove that the sum of $N=2k+2$ consecutive terms is a fixed integer multiple of another term in the sequence. We also prove that for the generalized Pell $(k,k-1)$-numbers such a relation does not exist when $N$ and $k$ are odd. We give analogous results for the Fibonacci and other related second-order recursive sequences. |
| title | Sum of Consecutive Terms of Pell and Related Sequences |
| topic | Number Theory Combinatorics 11Bxx, 11B37, 11B39, 11B50 |
| url | https://arxiv.org/abs/2407.12868 |