Could I know how to start with this kind of problem? Thanks.
Let $T_n$ be the number of ways of tiling a $2\times n$ rectangle using tiles of dimensions $1 \times 2$ or $2 \times 2$.
For example, the picture below shows one of the tiling counted by $T_{12}$
Find (and justify!) a recurrence relation and initial conditions that $T_n$ satisfies.
Suppose you already have a tiled rectangle. How can you extend it to the right to get a bigger rectangle? There are $3$ ways. You could add a single $1\times2$ tile vertically, or a single $2\times2$ tile, or two $1\times2$ tiles horizontally.
Now just turn this around to get $T_n$ in terms of $T_{n-1}$ and $T_{n-2}$. You need $2$ initial conditions, the values for $T_1$ and $T_2$. Do you see why?