Best Rational Approximations to Root 2 generate Pythagorean Triples

Theorem
Consider the Sequence of Best Rational Approximations to Square Root of 2:
 * $\left\langle{S}\right\rangle := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$

Every other term of $\left\langle{S}\right\rangle$ can be expressed as:
 * $\dfrac {2 a + 1} b$

such that:
 * $a^2 + \left({a + 1}\right)^2 = b^2$
 * $b$ is odd.

Proof
First the parity of the numerators and denominators of the terms of $\left\langle{S}\right\rangle$ is established.

Let $\dfrac {p_n} {q_n}$ be a general term of $\left\langle{S}\right\rangle$.

By Relation between Adjacent Best Rational Approximations to Root 2:
 * $p_{n + 1} = p_n + 2 q_n$

Thus if $p_n$ is odd then so is $p_{n + 1}$.

But $p_1 = 1$ is odd.

So $p_n$ is odd for all $n$, by Principle of Mathematical Induction.

By Relation between Adjacent Best Rational Approxmimations to Root 2:
 * $q_{n + 1} = p_n + q_n$

We have that $p_n$ is odd for all $n$.

Thus if $q_n$ is odd then $q_{n + 1}$ is even