Let $U \subset \mathbb{N}$ and let $T$ be a linear tiling of a $1 \times n$ board using $1 \times u$ polyomino tiles where $u \in U$. We define $\mathcal{T}_U(n)$ to be the set of linear tilings of length $n$ using tile sizes in $U$. Classically, when $U = \{ 1, 2\}$ these tilings are known as the Lucanomial tilings and they follow the Lucas sequence. Now, let $U$ and $V$ be distinct subsets of the natural numbers. In this talk, we will provide some general results and supportive examples of a large family of linear recurrence relations between the sequences $\#\mathcal{T}_U(n)$ and $\#\mathcal{T}_V(n)$ for fixed sets $U$ and $V$.