--description--

We want to tile a board of length $n$ and height 1 completely, with either 1 × 2 blocks or 1 × 1 blocks with a single decimal digit on top:

For example, here are some of the ways to tile a board of length $n = 8$:

Let $T(n)$ be the number of ways to tile a board of length $n$ as described above.

For example, $T(1) = 10$ and $T(2) = 101$.

Let $S(L)$ be the triple sum $\sum_{a, b, c} gcd(T(c^a), T(c^b))$ for $1 ≤ a, b, c ≤ L$.

For example:

$$\begin{align} & S(2) = 10\,444 \\ & S(3) = 1\,292\,115\,238\,446\,807\,016\,106\,539\,989 \\ & S(4)\bmod 987\,898\,789 = 670\,616\,280. \end{align}$$

Find $S(2000)\bmod 987\,898\,789$.

--hints--

gcdAndTiling() should return 970746056.

assert.strictEqual(gcdAndTiling(), 970746056);

--seed--

--seed-contents--

function gcdAndTiling() {

  return true;
}

gcdAndTiling();

--solutions--

// solution required