Problem 918

Recursive Sequence Summation

The sequence $a_{n}$ is defined by $a_{1} = 1$ , and then recursively for $n \geq 1$ :
$\begin{aligned} a_{2 n} & = 2 a_{n} \\ a_{2 n + 1} & = a_{n} - 3 a_{n + 1} \end{aligned}$
The first ten terms are $1, 2, - 5, 4, 17, - 10, - 17, 8, - 47, 34$ .

Define $S (N) = \sum_{n = 1}^{N} a_{n}$ . You are given $S (10) = - 13$ .

Find $S (10^{12})$ .

递归序列求和

序列 $a_{n}$ 的定义为： $a_{1} = 1$ ；对于 $n \geq 1$ ，递归定义：
$\begin{aligned} a_{2 n} & = 2 a_{n} \\ a_{2 n + 1} & = a_{n} - 3 a_{n + 1} \end{aligned}$
该序列的前十项是 $1, 2, - 5, 4, 17, - 10, - 17, 8, - 47, 34$ 。

定义 $S (N) = \sum_{n = 1}^{N} a_{n}$ 。已知 $S (10) = - 13$ 。

求 $S (10^{12})$ 。