Once floored, always floored

Calculus Level 4

$S_n(x) = \sum_{r=1}^{n} \lfloor rx \rfloor$

For $x$ $\in \mathbb{R}$ , define $S_n(x)$ as above.

Evaluate

$\displaystyle \left\lfloor \left. \lim_{n \to \infty } \frac{S_n(x)}{n^2} \right|_{x = 2015.20162017} \right\rfloor.$

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