A generalization of the well-known Fibonacci sequence $\{F_n\}_{n\ge 0}$ given by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_{n}$ for all $n\ge 0$ is the $k$-generalized Fibonacci sequence $\{F_n^{(k)}\}_{n\geq -(k-2)}$ whose first $k$ terms are $0, \l
