Hopping forever

After the previous exercise, Brilli the bug is ready for another session of hopping. It starts on the number line at the point \(0\). For all \(i = 1, 2, 3, \ldots\), on the \(i\)-th step, it hops a distance of \(2^{i-1}\) units, either to the left or to the right.

Let \(f\) be a function from the integers to the nonnegative integers, such that \(f(n)\) is \(0\) if Brilli can never reach \(n\) in a finite number of steps, and the minimum number of moves required to reach \(n\) otherwise. Compute \[\displaystyle\sum_{n=1}^{2015} f(n)\]

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