Stacking 1-high and 2-high blocks to height 12

You build a tower exactly $12$ units tall by stacking blocks of height $1$ and height $2$, with plenty of each available. Towers differ if their sequences of block heights from bottom to top differ. How many towers are possible?

$233$. Looking at the top block, $T_n=T_{n-1}+T_{n-2}$ with $T_1=1$, $T_2=2$. These are Fibonacci numbers shifted by one, so $T_{12}=F_{13}=233$.