Problem Archive

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 $k \gt 2$, $U(a,b)_k$ 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 $\sum\limits_{n = 2}^{10} U(2,2n+1)_k$, where $k = 10^{11}$.