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A weird recurrence relation

Problem 463

Published on Saturday, 15th March 2014, 10:00 pm; Solved by 487; Difficulty rating: 30%

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.