$$A=\sum_{i=1}^{\infty} \frac{1}{3^i 10^{3^i}}$$

Define $A(n)$ to be the $10$ decimal digits from the $n$th digit onward. For example, $A(100) = 4938271604$ and $A(10^8)=2584642393$.

Find $A(10^{16})$.