(noch nicht übersetzt)

The sequence $a_n$ is defined by $a_1=1$, and then recursively for $n\geq1$: $$\begin{align*} a_{2n} &=2a_n\\ a_{2n+1} &=a_n-3a_{n+1} \end{align*}$$ The first ten terms are $1, 2, -5, 4, 17, -10, -17, 8, -47, 34$.
Define $\displaystyle S(N) = \sum_{n=1}^N a_n$. You are given $S(10) = -13$
Find $S(10^{12})$.