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Problem 843

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Periodic Circles

This problem involves an iterative procedure that begins with a circle of $n\ge 3$ integers. At each step every number is simultaneously replaced with the absolute difference of its two neighbours.

For any initial values, the procedure eventually becomes periodic.

Let $S(N)$ be the sum of all possible periods for $3\le n \le N$. For example, $S(6) = 6$, because the possible periods for $3\le n \le 6$ are $1, 2, 3$. Specifically, $n=3$ and $n=4$ can each have period $1$ only, while $n=5$ can have period $1$ or $3$, and $n=6$ can have period $1$ or $2$.

You are also given $S(30) = 20381$.

Find $S(100)$.

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周期性的环形数列

从一个包含$n\ge3$个整数的环形数列开始，不断重复以下过程：在每一步中，同时将每一个数替换为与其相邻的两个数之差的绝对值。

无论初始数列的取值，这一过程最终都会进入周期性的循环。

记$S(N)$为所有$3\le n \le N$的环形数列可能进入的循环周期之和。例如，$S(6) = 6$，因为对于所有$3 \le n \le 6$的环形数列，其可能进入的循环周期只有$1$、$2$、$3$。具体来说，$n=3$和$n=4$时只会进入周期为$1$的循环，$n=5$时可能进入周期为$1$或$3$的循环，而$n=6$时可能进入周期为$1$或$2$的循环。

已知$S(30) = 20381$。

求$S(100)$。

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