Sprague's Property of Root 2

Theorem
Let $S = \left\langle{s_n}\right\rangle$ be the sequence of fractions defined as follows:

Let the numerator of $s_n$ be:
 * $\left\lfloor{n \sqrt 2}\right\rfloor$

where $\left\lfloor{x}\right\rfloor$ denotes the floor of $x$.

Let the denominators of the terms of $S$ be the (strictly) positive integers missing from the numerators of $S$:


 * $S := \dfrac 1 3, \dfrac 2 6, \dfrac 4 {10}, \dfrac 5 {13}, \dfrac 7 {17}, \dfrac 8 {20}, \ldots$

Then the difference between the numerator and denominator of $s_n$ is equal to $2 n$.