On $k$-Pell numbers which are sum of two Narayana's cows numbers
For any positive integer $k\geq2$, let $(P_n^{(k)})_{n\geq2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots,0,1$ ($k$ terms) with the linear recurrence P_n^{(k)} = 2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots+P_{n-k}^{(k)}\quad\text{for} n\geq2. Let $(N_n)_{n\geq0}$ be Narayana...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics of the Czech Academy of Science
2025-04-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | https://mb.math.cas.cz/full/150/1/mb150_1_2.pdf |
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| Summary: | For any positive integer $k\geq2$, let $(P_n^{(k)})_{n\geq2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots,0,1$ ($k$ terms) with the linear recurrence
P_n^{(k)} = 2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots+P_{n-k}^{(k)}\quad\text{for} n\geq2.
Let $(N_n)_{n\geq0}$ be Narayana's sequence given by
N_0=N_1=N_2=1\quad\text{and}\quad N_{n+3}=N_{n+2}+N_n.
The purpose of this paper is to determine all $k$-Pell numbers which are sums of two Narayana's numbers. More precisely, we study the Diophantine equation
P_p^{(k)}=N_n+N_m
in nonnegative integers $k$, $p$, $n$ and $m$. |
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| ISSN: | 0862-7959 2464-7136 |