## Investigating Ulam sequences

### Problem 167

Published on Friday, 9th November 2007, 01:00 pm; Solved by 1009; Difficulty rating: 75%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 ∑U(2,2`n`+1)_{k} for 2 ≤ `n` ≤10, where `k` = 10^{11}.