Head density in tosses with no two heads adjacent

A fair coin is tossed $n$ times, and you learn that the resulting sequence never has two heads in a row. Let $E(n)$ be the conditional expected number of heads. The limit $\lim_{n\to\infty} E(n)/n$ equals $\dfrac{a}{b+\sqrt{c}}$ for integers $a,b,c$ with $c$ as small as possible. Compute $abc$.

$50$. Conditioning makes all valid strings equally likely, and they are counted by Fibonacci numbers. Counting heads through the recursion (a string starts with T or with HT) shows the head fraction tends to $\tfrac{1}{\varphi+2}=\tfrac{2}{5+\sqrt5}$, where $\varphi$ is the golden ratio. So $a=2$, $b=5$, $c=5$ and $abc=50$.