---
id: 5900f4141000cf542c50ff26
title: 'Problem 167: Investigating Ulam sequences'
challengeType: 5
forumTopicId: 301801
dashedName: problem-167-investigating-ulam-sequences
---

# --description--

For two positive integers $a$ and $b$, the Ulam sequence $U(a,b)$ is defined by ${U{(a,b)}\_1} = a$, ${U{(a,b)}\_2} = b$ and for $k > 2$, ${U{(a,b)}\_k}$ is the smallest integer greater than ${U{(a,b)}\_{(k-1)}}$ which can be written in exactly one way as the sum of two distinct previous members of $U(a,b)$.

For example, the sequence $U(1,2)$ begins with

$$1, 2, 3 = 1 + 2, 4 = 1 + 3, 6 = 2 + 4, 8 = 2 + 6, 11 = 3 + 8$$

5 does not belong to it because $5 = 1 + 4 = 2 + 3$ has two representations as the sum of two previous members, likewise $7 = 1 + 6 = 3 + 4$.

Find $\sum {U(2, 2n + 1)_k}$ for $2 ≤ n ≤ 10$, where $k = {10}^{11}$.

# --hints--

`ulamSequences()` should return `3916160068885`.

```js
assert.strictEqual(ulamSequences(), 3916160068885);
```

# --seed--

## --seed-contents--

```js
function ulamSequences() {

  return true;
}

ulamSequences();
```

# --solutions--

```js
// solution required
```