Question 132
Let $n \geq 0$ be an integer. A sequence $a_{0}, a_{1}, a_{2}, \ldots$ of integers is defined as follows: $a_{0}=n$ and for $k \geq 1$, $a_{k}$ is the smallest integer greater than $a_{k-1}$ for which $a_{k}+a_{k-1}$ is the square of an integer. Prove that there are exactly $\lfloor\sqrt{2 n}\rfloor$ positive integers that cannot be written in the form $a_{k}-a_{\ell}$ with $k\gt \ell \geq 0$.