2016 is absolutely awesome 19

In terms of $a$ and $b$ , $f(f(x)) - x = x^4 - 2ax^3 + (a^2 - a + 2b)x^2 + (2a^2 - 2ab - 1)x + (b^2 - ab + b).$ At first, this seems like a mess to factor, but suppose we had a root $r$ of $f(x) - x = 0$ . Then we would also have a root of $f(f(x)) - x = 0$ since $f(r) = r \implies f(f(r)) - r = f(r) - r = 0.$ In other words, $f(x) - x$ is a factor of $f(f(x)) - x$ , so for some $u$ and $v$ , $f(f(x)) - x = \big( f(x) - x \big) \big(x^2 + ux + v \big) = \big( x^2 - (a + 1)x + b \big) \big( x^2 + ux + v \big).$ Consider the constant term and the quadratic coefficient in the expansion of $f(f(x)) - x$ above. Comparing those values to the factored form of $f(f(x)) - x$ , we see that we must have $bv = b^2 - ab + b, \quad b + v - (a + 1)u = a^2 - a + 2b,$ which we can solve to find $v = b - a + 1, \quad u = 1 - a.$ Now we are halfway to factoring $f(f(x)) - x$ : $f(f(x)) = \big( x^2 - (a + 1)x + b \big) \big( x^2 + (1 - a)x + (b - a + 1) \big).$ The roots of the two quadratic factors are $a + 1 \pm \sqrt{a^2 + 2a - 4b + 1}, \quad a - 1 \pm \sqrt{a^2 + 2a - 4b - 3}.$ The roots of $f(f(x)) - x = 0$ are integers if and only if the radicands are perfect squares. But the radicands differ by exactly 4, so this is only possible if $a^2 + 2a - 4b + 1 = 4, \quad a^2 + 2a - 4b - 3 = 0.$ Solving either equation for positive $a$ gives the relation we want. The roots of $f(f(x)) - x$ are integers exactly when $a = -1 + 2\sqrt{b + 1}, \qquad b = n^2 - 1 \geq 3,$ where $n$ is an integer. Based on this relation, there are $\boxed{43}$ pairs $(a,b)$ in the desired range.