Start counting

Let $n + 1 = k^{2}$ and $\frac{n}{2} + 1 = m^{2}$ . Then $k^{2} - 2m^{2} = -1$ .

Now this is a negative Pell's equation, i.e., it is of the form $k^{2} - dm^{2} = -1$ where $d$ is a non-square integer and integer solutions are sought for $k$ and $m$ . We know that this equation has an infinite number of solutions when $\sqrt{d}$ has a continued fraction expansion with an odd period. In this case we have $d = 2$ , and since $\sqrt{2}$ has a continued fraction expansion of period $1$ we can conclude that there are an infinite number of solutions $(k,m)$ , This in turn implies that there are an infinite number of positive even integers $n$ such that both $(n + 1)$ and $(\frac{n}{2} + 1)$ are perfect squares.