Domino tilings of a three by eight board

In how many ways can a board of $3$ rows and $8$ columns be completely covered by non-overlapping dominoes, each covering two adjacent cells (horizontal or vertical)?

$153$. Let $t_n$ count tilings of a $3\times n$ board and $s_n$ tilings with one corner cell removed. Looking at the leftmost column gives $t_n=t_{n-2}+2s_{n-1}$ and $s_n=t_{n-1}+s_{n-2}$, which combine to $t_n=4t_{n-2}-t_{n-4}$. Starting from $t_0=1,t_2=3$ this gives $11, 41, 153$.