Problem 463: A weird recurrence relation

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The function $f$ is defined for all positive integers as follows:

$f(1)=1$

$f(3)=3$

$f(2n)=f(n)$

$f(4n + 1)=2f(2n + 1) - f(n)$

$f(4n + 3)=3f(2n + 1) - 2f(n)$

The function $S(n)$ is defined as $\sum_{i=1}^{n}f(i)$. $S(8)=22$ and $S(100)=3604$. Find $S(3^{37})$. Give the last 9 digits of your answer.