On $k$-Pell numbers which are sum of two Narayana’s cows numbers

@article{Adédji2025,
abstract = {For any positive integer $k\ge 2$, let $(P_n^\{(k)\})_\{n\ge 2-k\}$ be the $k$-generalized Pell sequence which starts with $0,\cdots ,0,1$ ($k$ terms) with the linear recurrence \[ P\_\{n\}^\{(k)\} = 2P\_\{n-1\}^\{(k)\}+P\_\{n-2\}^\{(k)\}+\cdots +P\_\{n-k\}^\{(k)\}\quad \text\{for\}\ n\ge 2. \] Let $(N_n)_\{n\ge 0\}$ 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 $m$.},
author = {Adédji, Kouèssi Norbert, Bachabi, Mohamadou, Togbé, Alain},
journal = {Mathematica Bohemica},
keywords = {Diophantine equation; Narayana’s cows sequence; $k$-Pell number; linear form in logarithms; reduction method},
language = {eng},
number = {1},
pages = {25-47},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On $k$-Pell numbers which are sum of two Narayana’s cows numbers},
url = {http://eudml.org/doc/299890},
year = {2025},
}

TY - JOUR
AU - Adédji, Kouèssi Norbert
AU - Bachabi, Mohamadou
AU - Togbé, Alain
TI - On $k$-Pell numbers which are sum of two Narayana’s cows numbers
JO - Mathematica Bohemica
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 25
EP - 47
AB - For any positive integer $k\ge 2$, let $(P_n^{(k)})_{n\ge 2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots ,0,1$ ($k$ terms) with the linear recurrence \[ P_{n}^{(k)} = 2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}\quad \text{for}\ n\ge 2. \] Let $(N_n)_{n\ge 0}$ 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 $m$.
LA - eng
KW - Diophantine equation; Narayana’s cows sequence; $k$-Pell number; linear form in logarithms; reduction method
UR - http://eudml.org/doc/299890
ER -