Sequence of Best Rational Approximations to Square Root of 2

Theorem
A sequence of best rational approximations to the square root of $2$ starts:
 * $\dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$

where:
 * the denominators are the Pell numbers $P_n$
 * the numerators are half of the corresponding Pell-Lucas numbers $Q_n$

starting from $\dfrac {Q_1} {P_1}$.

Proof
By Continued Fraction Expansion of Root 2:
 * $\sqrt 2 = \left[{1, \left \langle{2}\right \rangle}\right] = \left[{1, 2, 2, 2, \ldots}\right]$

From Convergents are Best Approximations, the convergents of $\left[{1, \left \langle{2}\right \rangle}\right]$ are the best rational approximations of $\sqrt 2$.

It remains to be shown that:
 * the denominators of the convergents are the Pell numbers
 * twice the numerators of the convergents are the corresponding Pell-Lucas numbers.

Let $\dfrac {p_n} {q_n}$ be the $n$th convergent of $\left[{1, \left \langle{2}\right \rangle}\right]$.

From Value of Simple Continued Fraction:

The result follows by definition of Pell numbers and Pell-Lucas numbers.