---
id: 5900f5241000cf542c510037
title: 'Problem 440: GCD and Tiling'
challengeType: 5
forumTopicId: 302112
dashedName: problem-440-gcd-and-tiling
---

# --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 ∑a,b,c gcd(T(ca), T(cb)) for 1 ≤ a, b, c ≤ L. For example: S(2) = 10444 S(3) = 1292115238446807016106539989 S(4) mod 987 898 789 = 670616280.

Find S(2000) mod 987 898 789.

# --hints--

`euler440()` should return 970746056.

```js
assert.strictEqual(euler440(), 970746056);
```

# --seed--

## --seed-contents--

```js
function euler440() {

  return true;
}

euler440();
```

# --solutions--

```js
// solution required
```