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?
$\lim_{n\to\infty}\frac{A_{n}}{n^2}$