First toss heads given no two heads in a row

A fair coin is tossed $n$ times, and you are told the resulting sequence never has two heads next to each other. Find, as a function $p(n)$, the conditional probability that the first toss was heads, and then give $p(10)$.

$p(10)=55/144$. Valid sequences of length $n$ satisfy $A(n)=A(n-1)+A(n-2)$ (start with $T$, or with $HT$), so $A(n)=F_{n+2}$ (Fibonacci). Those starting with heads must start $HT$, giving $p(n)=\tfrac{F_n}{F_{n+2}}$ and $p(10)=\tfrac{55}{144}$.