The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1's. The Lucas cube $\Lambda_n$ is obtained from $\Gamma_n$ by removing vertices that start and end with 1. The eccentricity of a vertex $u$,denoted $e_G(u)$ is the greatest distance between $u$ and any other vertex $v$ in the graph $G$. We characterize the vertices that satisfy the eccentricity of a given vertex of $\Gamma_n$. We then obtain the generating functions of the eccentricity sequences of $\Gamma_n$ and $\Lambda_n$. As a corollary we deduce the number of vertices of a given eccentricity.
