Sprague's Property of Root 2

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

Let the numerator of $s_n$ be:
 * $\floor {n \sqrt 2}$

where $\floor x$ 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 $2 n$.

Proof
Denote the numerators of the terms of $S$ as $\sequence {N_n}$.

Denote the denominators of the terms of $S$ as $\sequence {D_n}$.

From the definition:


 * $\sequence {N_n}$ is a Beatty sequence, where $\sequence {N_n} = \BB_{\sqrt 2} = \sequence{\floor{n \sqrt 2} }_{n \mathop \in \Z_{> 0} }$


 * $\sequence {N_n}$ and $\sequence {D_n}$ are complementary Beatty sequences.

Then by Beatty's Theorem, $\sequence {D_n}$ is a Beatty sequence.

Define $\sequence {D_n} = \BB_y = \sequence{\floor{n y} }_{n \mathop \in \Z_{> 0} }$.

Then we have:

So we have $s_n = \dfrac {N_n} {D_n} = \dfrac {\floor {n \sqrt 2} } {\floor {n \paren {2 + \sqrt 2} } }$.

The difference between the numerator and denominator of $s_n$ is $\floor {n \paren {2 + \sqrt 2} } - \floor {n \sqrt 2}$.

From Integer equals Floor iff Number between Integer and One More:

The only integer strictly between $2 n - 1$ and $2 n + 1$ is $2 n$, and $\floor {n \paren {2 + \sqrt 2} } - \floor {n \sqrt 2}$ is an integer.

Therefore $\floor {n \paren {2 + \sqrt 2} } - \floor {n \sqrt 2} = 2 n$.