The extended Frobenius problem for Lucas series incremented by a Lucas number

We study the extended Frobenius problem for sequences of the form $\{l_a\}\cup \{l_a+l_n{\}}_{n\in \mathbb{N}}$ and $\{l_a+l_n{\}}_{n\in \mathbb{N}}$, where $\{l_n{\}}_{n\in \mathbb{N}}$ is the Lucas series and $l_a$ is a Lucas number. As a consequence, we show that the families of numerical semigroups associated to both sequences satisfy the Wilf’s conjecture.