2018-09-30 22:01:58 +00:00
---
id: 5900f4141000cf542c50ff26
title: 'Problem 167: Investigating Ulam sequences'
2020-11-27 18:02:05 +00:00
challengeType: 5
2019-08-05 16:17:33 +00:00
forumTopicId: 301801
2021-01-13 02:31:00 +00:00
dashedName: problem-167-investigating-ulam-sequences
2018-09-30 22:01:58 +00:00
---
2020-11-27 18:02:05 +00:00
# --description--
2021-07-12 14:19:03 +00:00
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 $U(a,b)$.
2020-11-27 18:02:05 +00:00
2021-07-12 14:19:03 +00:00
For example, the sequence $U(1,2)$ begins with
2020-11-27 18:02:05 +00:00
2021-07-12 14:19:03 +00:00
$$1, 2, 3 = 1 + 2, 4 = 1 + 3, 6 = 2 + 4, 8 = 2 + 6, 11 = 3 + 8$$
2020-11-27 18:02:05 +00:00
2021-07-12 14:19:03 +00:00
5 does not belong to it because $5 = 1 + 4 = 2 + 3$ has two representations as the sum of two previous members, likewise $7 = 1 + 6 = 3 + 4$.
2018-09-30 22:01:58 +00:00
2021-07-12 14:19:03 +00:00
Find $\sum {U(2, 2n + 1)_k}$ for $2 ≤ n ≤ 10$, where $k = {10}^{11}$.
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
# --hints--
2018-09-30 22:01:58 +00:00
2021-07-12 14:19:03 +00:00
`ulamSequences()` should return `3916160068885` .
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
```js
2021-07-12 14:19:03 +00:00
assert.strictEqual(ulamSequences(), 3916160068885);
2018-09-30 22:01:58 +00:00
```
2020-11-27 18:02:05 +00:00
# --seed--
2018-09-30 22:01:58 +00:00
2020-11-27 18:02:05 +00:00
## --seed-contents--
2018-09-30 22:01:58 +00:00
```js
2021-07-12 14:19:03 +00:00
function ulamSequences() {
2020-09-15 16:57:40 +00:00
2018-09-30 22:01:58 +00:00
return true;
}
2021-07-12 14:19:03 +00:00
ulamSequences();
2018-09-30 22:01:58 +00:00
```
2020-11-27 18:02:05 +00:00
# --solutions--
2018-09-30 22:01:58 +00:00
```js
// solution required
```