Let $A$ be the adjacency matrix of the path graph. The characteristic polynomial of $A^k$ evaluated at $1$ gives the following for $0\leq k\leq 182$ (we take everything over the ring $\mathbb{Z}/6\mathbb{Z}$): $$[0,1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 3, 4, 1, 1, 5, 1, 1, 1, 2, 1, 5, 1, 5, 3, 1, 4, 1, 1, 1, 1, 5, 1, 4, 1, 5, 1, 3, 1, 1, 4, 5, 1, 1, 1, 5, 1, 2, 1, 5, 3, 1, 1, 1, 4, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 1, 1, 5, 4, 1, 1, 5, 1, 5, 1, 2, 3, 1, 1, 1, 1, 1, 4, 5, 1, 1, 1, 5, 1, 0, 1, 1, 1, 5, 1, 1, 4, 5, 1, 5, 1, 5, 3, 4, 1, 1, 1, 1, 1, 5, 4, 1, 1, 5, 1, 3, 1, 4, 1, 5, 1, 1, 1, 5, 4, 5, 1, 5, 3, 1, 1, 4, 1, 1, 1, 5, 1, 1, 4, 5, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 5, 4, 5, 3, 1, 1, 1, 1, 4, 1, 5, 1, 1, 1, 5, 4, 3, 1, 1, 1, 5, 1, 4, 1, 5, 1, 5, 1, 5, 0]$$The smallest $k$ where this happens to be $0$ is the 182nd entry in the list.
$$A=\left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right)$$