Question 135
Determine all positive integers $n$ such that number $N$ is also integer, where
$$
N = \frac{2 \cdot 1}{\sqrt{1^2 + 1 + 4} + \sqrt{1^2 - 1 + 4}} + \frac{2 \cdot 2}{\sqrt{2^2 + 2 + 4} + \sqrt{2^2 - 2 + 4}} + \dots + \frac{2n}{\sqrt{n^2 + n + 4} + \sqrt{n^2 - n + 4}}
$$
(Anikushyn A., Rublyov B.)
$$
N = \frac{2 \cdot 1}{\sqrt{1^2 + 1 + 4} + \sqrt{1^2 - 1 + 4}} + \frac{2 \cdot 2}{\sqrt{2^2 + 2 + 4} + \sqrt{2^2 - 2 + 4}} + \dots + \frac{2n}{\sqrt{n^2 + n + 4} + \sqrt{n^2 - n + 4}}
$$
(Anikushyn A., Rublyov B.)