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

Adédji, Kouèssi Norbert, Bachabi, Mohamadou, and Togbé, Alain. "On $k$-Pell numbers which are sum of two Narayana's cows numbers." Mathematica Bohemica (2025): 25-47. <http://eudml.org/doc/299890>.

@article{Adédji2025,
abstract = {For any positive integer $k\geq 2$, let $(P_n^\{(k)\})_\{n\geq 2-k\}$ be the $k$-generalized Pell sequence which starts with $0,\cdots ,0,1$ ($k$ terms) with the linear recurrence $$ P\_\{n\}^\{(k)\} = 2P\_\{n-1\}^\{(k)\}+P\_\{n-2\}^\{(k)\}+\cdots +P\_\{n-k\}^\{(k)\}\quad \text \{for\}\ n\geq 2. $$ Let $(N_n)_\{n\geq 0\}$ 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},
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 $k\geq 2$, let $(P_n^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots ,0,1$ ($k$ terms) with the linear recurrence $$ P_{n}^{(k)} = 2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}\quad \text {for}\ n\geq 2. $$ Let $(N_n)_{n\geq 0}$ 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
UR - http://eudml.org/doc/299890
ER -