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
From Parity of Best Rational Approximations to Root 2‎:


 * The numerators of the terms of $\left\langle{S}\right\rangle$ are all odd.


 * For all $n$, the parity of the denominator of term $S_n$ is the same as the parity of $n$.

Thus it follows that every other term of $\left\langle{S}\right\rangle$ has a numerator and a denominator which are both odd.