Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct?

1sorry ↔2  ∃ f,3    Set.InjOn4      (fun x =>5        match x with6        | (a, b) => Polynomial.eval (↑a) f + Polynomial.eval (↑b) f)7      {(a, b) | a < b}