Once floored, always floored

Calculus Level 5

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

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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