data/problems/167.yml
---
:id: 167
:name: Investigating Ulam sequences
:url: https://projecteuler.net/problem=167
:content: "For two positive integers a and b, the Ulam sequence U(a,b) is defined
by U(a,b)<sub>1</sub> = a, U(a,b)<sub>2</sub> = b and for k \\> 2, U(a,b)<sub>k</sub>
is the smallest integer greater than U(a,b)<sub>(k-1)</sub> which can be written
in exactly one way as the sum of two distinct previous members of U(a,b).\n\nFor
example, the sequence U(1,2) begins with \n1, 2, 3 = 1 + 2, 4 = 1 + 3, 6 = 2 +
4, 8 = 2 + 6, 11 = 3 + 8; \n5 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.\n\nFind ∑U(2,2<var>n</var>+1)<sub><var>k</var></sub> for 2 ≤ <var>n</var> ≤10,
where <var>k</var> = 10<sup>11</sup>.\n\n"