To find positive integers $n$ such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square

Question

How many positive integers $n$ are there such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square ? I know $n=1 , 2$ works ; are there any more ? Are there only finitely many such $n$ ?

Answer 1

Your equation is the same as finding integer solutions to the equation $$ \binom{n+2}{3}=m^2. $$

The top answer to this question tells us that there are only solutions for $n\in\{1,2,48\}$.