This problem is about numbers of tilings of a grid by polyominoes.
$a(n)$ is the number of tilings of a $3\times n$ grid by $n$ monominoes and $n$ dominoes.
$b(n)$ is the number of tilings of a $5\times n$ grid by $n$ dominoes and $n$ straight trominoes.
Obviously, $a(1)=b(1)=2$. $a(2)=11$ and they are in pictured in following figure.
Find $a(n)$ and $b(n)$.