My brain had twisted because of this nasty problem.
let
$$r_{n}=\sqrt{n^2+n+\frac{3}{4}}$$
$$x_{n}=\left \lfloor \frac{r_{n}}{\sqrt{2}}-\frac{1}{2} \right \rfloor$$
$$a_{n}=\sum_{k=1}^{\left \lfloor r_{n}-x_{n} \right \rfloor} \left \lfloor \sqrt{n^2+n-k^2-k+\frac{1}{2}-(x_{n})^2-(2k+1)x_{n}}-\frac{1}{2} \right \rfloor $$
$$A_{n}=4 ( (x_{n})^2+2a_n+n )+1$$
Question. How can I find the limit of below?
$$\lim_{n\to\infty}\frac{A_{n}}{n^2}$$