Head density with no two in a row

Flip a fair coin $n$ times, conditioned on the event that no two consecutive heads ever appear. Let $E(n)$ be the expected number of heads under this conditioning. Compute $\displaystyle\lim_{n\to\infty}\frac{E(n)}{n}$. It equals $\dfrac{a}{b + \sqrt{c}}$ for integers $a, b, c$ with $c$ minimal; find $a\,b\,c$.