Narayana numbers that are products of two Fibonacci numbers

Abstract: Let ${N_m}<em>{m\ge0}$ be the Narayana's cows sequence given by $N_0=0$, $N_1=1=N_2=1$ and [ N{m+3}=N_{m+2}+N_m,\quad \text{ for }\; m\geq 0 ] and let ${F_n}_{n\ge0}$ be the Fibonacci sequence. In this paper we solve explicitely the Diophantine equation [ N_m=F_nF_k, ] in positive unknowns $m,\,n$ and $k$. That is, we find the non-zero narayana numbers that are products of two Fibonacci numbers.