NMC 2013 Problem 1

Let $\left(a_{n}\right)_{n \geq 1}$ be a sequence with $a_{1}=1$ and

$$a_{n+1}=\left\lfloor a_{n}+\sqrt{a_{n}}+\frac{1}{2}\right\rfloor$$

for all $n \geq 1$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. Find all $n \leq 2013$ such that $a_{n}$ is a perfect square.