Erdős Problem #29 — Explicit sequence with $f(n)>0$ but $f(n)=o(n^\varepsilon)$

Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence of integers, and denote by $f(n)$ the number of solutions of $n = a_i + a_j$. Also I offer 100 dollars for an explicit construction of a sequence $A$ for which $f(n) > 0$ for all $n$ but $f(n)/n^{\varepsilon} \to 0$ for every $\varepsilon > 0$, i.e., for a constructive solution to Sidon's original question.