A type of integer numbers set

Question

Is there any $A\subseteq \mathbb{Z}$ such that $A-A$ is a non-trivial co-finite subset?

Note that $A-A=\{a_1-a_2: a_1,a_2\in A\}$.

Answer 1

Let $$ A := \left\{\sum_{k=2}^n k : n \ge 1 \right\} $$ Then $A - A = \mathbf Z \setminus \{\pm 1\}$.