Domino and Tromino Combined Tiling
There is a mutual recursion in two variables $f(n)$  for the $n\times 2$ grid and $g(n)$ for the $n\times 2$ grid with a single square sticking out. It does not matter where the square sticks out (first or second row) because their number of tilings is the same. 



Note the trominos indeed come in pairs, but they do not need to form a $3\times 2$ block. You can place horizontal dominoes in between. Now that I look at it, it seems there is a recursion with one $f(n)$ if you look at a tromino and its matching partner. This picture suggests $f(7) = f(6)+f(5)+2f(4)+2f(3)+ \dots +2f(1)+2f(0)$. Sounds like a promising recursion to me. 



If my initial computations are right, this suggests 1,1,2,5,11,24,53,117, ... which is OEISA052980. This lists another recurrence however. Would be fun if they match.

PS. Yes! Recursions are the same.
$f(n) = f(n-1) + f(n-2) + 2\sum_{k=0}^{n-3} f(k)$. Then subtract $f(n)-f(n-1) = f(n-1)+f(n-3)$. Thus $f(n) = 2f(n-1)+f(n-3)$, which is in the OEIS.