Quadratic Irrationals

Definition 1.1   An element $\alpha\in\mathbb{R}$ is a quadratic irrational if it is irrational and satisfies a quadratic polynomial.

Thus, e.g., $(1+\sqrt{5})/2$ is a quadratic irrational. Recall that

$$\frac{1+\sqrt{5}}{2} = [1,1,1,\ldots].$$

The continued fraction of $\sqrt{2}$ is $[1,2,2,2,2,2,\ldots]$, and the continued fraction of $\sqrt{389}$ is

$$[19,1,2,1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38,\ldots].$$

Does the $[1,2,1, 1, 1, 1, 2, 1, 38]$ pattern repeat over and over again??