Sequence of Best Rational Approximations to Square Root of 2

Theorem
A sequence of best rational approximations to the square root of $2$ starts:
 * $\dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \ldots$

The denominators are the Lucas numbers, while the numerators are half of the corresponding Pell-Lucas numbers.