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(2,2n+1)_k for 2 ≤ n ≤10, where k = 10¹¹.