Pairwise sums are equal

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with
$n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the
same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). Prove that $n$ is
a power of $2$.
I've seen a solution using generating function and some calculus, but is
there a more intuitive proof why $n$ has to be a power of $2$?