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!



Palindromic continued fraction
It's straightforward to prove by induction on $k$ that $[a\_k; a\_{k-1},\ldots,a\_1] = q\_k/q\_{k-1}$. (Let the left-hand side be $r\_k$, observe that $r\_{k+1} = a\_{k+1} + 1/r\_k$, and then note that the sequence $q\_k/q\_{k-1}$ satisfies the same recurrence.) Then $[0; a\_k,\ldots,a\_1] = q\_{k-1}/q\_k$; deduce for your palindromic $\alpha$ that $\alpha = q\_{k-1}/q\_k$. But of course $\alpha = p\_k/q\_k$ as well, and so we conclude.


