Solving Skolem problem for negative indexed $k-$generalized Pell numbers

In this paper, we address the Skolem problem for the $k$-generalized Pell sequence $(P_n^{(k)})_{n\geq2-k}$ extended to negative indices. We focus on identifying and bounding the indices $n<0$ for which $P_n^{(k)}=0.$ In particular, we establish that the zero multiplicity of $P_n^{(k)}$ is $\chi_k = \lfloor k^2/4\rfloor$ for all $k \in [4, 500].$