--description--

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

$$\begin{align} & f(1) = 1 \\ & f(3) = 3 \\ & f(2n) = f(n) \\ & f(4n + 1) = 2f(2n + 1) - f(n) \\ & f(4n + 3) = 3f(2n + 1) - 2f(n) \end{align}$$

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.

--hints--

weirdRecurrenceRelation() should return 808981553.

assert.strictEqual(weirdRecurrenceRelation(), 808981553);

--seed--

--seed-contents--

function weirdRecurrenceRelation() {

  return true;
}

weirdRecurrenceRelation();

--solutions--

// solution required