Induction · Superchallenge problem

Let $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(n) + 1$. Strong induction shows $f(n)$ is the number of $1$s in the binary expansion of $n$. What is $f(1) + f(2) + \cdots + f(255)$?