--description--

• A Sierpiński graph of order-1 ($S_1$) 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 $S_3$, as shown below:

It can also be verified that:

$$\begin{align} & C(1) = C(2) = 1 \\ & C(5) = 71\,328\,803\,586\,048 \\ & C(10 000)\bmod {10}^8 = 37\,652\,224 \\ & C(10 000)\bmod {13}^8 = 617\,720\,485 \\ \end{align}$$

Find $C(C(C(10\,000)))\bmod {13}^8$.

--hints--

pathsOnSierpinskiGraphs() should return 324681947.

assert.strictEqual(pathsOnSierpinskiGraphs(), 324681947);

--seed--

--seed-contents--

function pathsOnSierpinskiGraphs() {

  return true;
}

pathsOnSierpinskiGraphs();

--solutions--

// solution required