Here's what I hope is a final question outside of my area that I need to understand a problem about stable vector bundles on $\mathbb{P}^2$. Thank you everybody for your help so far!

Suppose I have a rational number $0<\alpha<1$ with a palindromic continued fraction expansion, i.e.

$$\alpha = [0; a_1,\ldots,a_k],$$ where $a_i = a_{k+1-i}$, so that the sequence $a_1,\ldots,a_k$ is a palindrome. I believe from working with several examples that the last two convergents $p_k/q_k$, $p_{k-1}/q_{k-1}$ of this continued fraction satisfy $p_k = q_{k-1}$. That is, the denominator of the penultimate convergent is the numerator of $\alpha$.

I assume this is well known, and if so a reference would be great! Thanks!