Infinite String

Let $s_0$ be the string "01" and $s_i=s_{i-1}+s_{i-1}[0:i]$ for $i\ge 1$, where $s[0:i]$ means the first $i$ characters of $s$.

For example, $s_1=$ "010", $s_2=$ "01001" and $s_3=$ "01001010".

Let $C(i)$ be the number of "1" in the $s_i[0:i]$.

You are given that $C(10)=4,C({10}^4)=3576$ and $C({10}^8)=35807401$.

Find $C({10}^{18})$.

[My Timing: instant]