Problem 312

Cyclic paths on Sierpiński graphs

A Sierpiński graph of order-1 (S₁) is an equilateral triangle.
S_n+1 is obtained from S_n by positioning three copies of S_n so that every pair of copies has one common corner.

Let C(n) be the number of cycles that pass exactly once through all the vertices of S_n.
For example, C(3) = 8 because eight such cycles can be drawn on S₃, as shown below:

It can also be verified that :
C(1) = C(2) = 1
C(5) = 71328803586048
C(10 000) mod 10⁸ = 37652224
C(10 000) mod 13⁸ = 617720485

Find C(C(C(10 000))) mod 13⁸.

谢尔宾斯基图的回路

1阶谢尔宾斯基图（S₁）是一个等边三角形。
将三个S_n摆在一起，使得任意两个S_n都共用一个角，就得到了S_n+1。

记C(n)是经过S_n中所有顶点恰好一次的回路的数目。
例如，C(3) = 8，因为在S₃上恰好可以画出8条这样的回路，如下图所示：

同样可以验证：
C(1) = C(2) = 1
C(5) = 71328803586048
C(10 000) mod 10⁸ = 37652224
C(10 000) mod 13⁸ = 617720485

求C(C(C(10 000))) mod 13⁸。