Counting domino coverings of a two-row strip

In how many ways can a board with $2$ rows and $n$ columns be covered exactly by $1\times2$ dominoes?

$F_{n+1}$, the $(n+1)$-th Fibonacci number ($1,2,3,5,8,\dots$ for $n=1,2,3,\dots$). Look at the left end: either one vertical domino fills column $1$, leaving $T_{n-1}$ ways, or two stacked horizontal dominoes fill columns $1$-$2$, leaving $T_{n-2}$. So $T_n=T_{n-1}+T_{n-2}$ with $T_1=1$, $T_2=2$.