Best Rational Approximations to Root 2 generate Pythagorean Triples

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

Every odd term of $\sequence S$ can be expressed as:
 * $\dfrac {2 a + 1} b$

such that:
 * $a^2 + \paren {a + 1}^2 = b^2$