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Problem 908

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Clock Sequence II

A clock sequence is a periodic sequence of positive integers that can be broken into contiguous segments such that the sum of the $n$-th segment is equal to $n$.

For example, the sequence
$$1234321234321234321\cdots$$
is a clock sequence with period $6$, as it can be broken into
$$1|2|3|4|32|123|43|2123|432|1234|32123|\cdots$$
Let $C(N)$ be the number of different clock sequences with period at most $N$. For example, $C(3)=3$, $C(4)=7$ and $C(10)=561$.

Find $C({10}^4)mod1111211113$.

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钟摆序列（二）

钟摆序列是一种具有周期性的正整数序列。这种序列可以被分成连续的片段，使得第$n$个片段之和等于$n$。

例如，序列
$$1234321234321234321\cdots$$
是一个周期为$6$的钟摆序列，因为它可以被分成
$$1|2|3|4|32|123|43|2123|432|1234|32123|\cdots$$

令$C(N)$为周期至多为$N$的不同钟摆序列的数量。例如，$C(3)=3$，$C(4)=7$，$C(10)=561$。

求$C({10}^4)mod1111211113$。

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