Lucas generalized numbers in Narayana's cows sequence

Abstract

Let $\{N_n\}_{n≥0}$ be the Narayana’s cows sequence given by $N_0= 0,\;N_1=N_2=1$ and $N_{n+3}=N_{n+2}+N_n$, for integers $n≥0$ and let $\{U_n\}_{n≥0}$ be the generalized Lucas sequence with parameters integers $a≥1,\; b=±1$ given by $U_0=0,\; U_1= 1$ and $U_{n+2}=aU_{n+1}+bU_n$, for integers $n≥0$. In this paper we give effective bounds for the Diophantine equation $N_m=U_n$, in positive unknowns $m$ and $n$. We then solve explicitly that equation with Fibonacci, Pell and Balancing sequences cases.