Reference request: an interpretation of Euler totient function.

Question

Let $\phi$ be the Euler totient function. Let $A_r$ be the number of coprime pairs of positive integers $a,b$ such that $2a+b=r$, $r \ge 3$. By direct computation, I verified a few examples that $\phi(r) = 2A_r$, $r \ge 3$. Is there a proof of this fact in the literature? Thank you very much.