Resolution of the Skolem Problem for $k$-Generalized Lucas Sequences

This paper provides a complete solution to Skolem’s problem for the $k$-generalized Lucas sequence $(L_n^{(k)}{)}_{n\in \mathbb{Z}}$ with a primary focus on its behavior at negative indices. We characterize the zero-distribution of this sequence by identifying and bounding all indices $n<0$ such that $L_n^{(k)}=0$. Our central result establishes that the zero-multiplicity ${\delta }_k$ of the sequence is $(k-1)(k-2)\mathrm{/}2$ for all $k\mathrm{.}$