Ten tosses where tails only come in pairs

A fair coin is tossed $10$ times. What is the probability that the tails can be split into adjacent pairs, i.e. every maximal block of consecutive tails has even length? (With $4$ tosses, $TTHH$, $TTTT$ and $HHHH$ qualify, while $HTHH$ and $TTTH$ do not.)

$89/1024$. A valid string ends in either $H$ or $TT$, so the count $t_n$ satisfies $t_n=t_{n-1}+t_{n-2}$ with $t_1=1$, $t_2=2$: Fibonacci numbers. Thus $t_{10}=89$ out of $2^{10}=1024$.