I am getting ready to publish the manuscript
http://arxiv.org/pdf/1408.4631v2.pdf
and I am trying to do due diligence on a quantity I study before it gets published. (This is cross-posted from Name/properties of a difference of continuants.)
Suppose that $q_1$, $\ldots$, $q_s$ is a sequence of positive integers. Denote by $[q_1, \ldots, q_s]$ the numerator (in lowest terms) of the rational number represented by the continued fraction
$$ q_1 + \cfrac{1}{q_2 + \cfrac{1}{q_3 + \cfrac{1}{\ddots \, + \cfrac{1}{q_s}}}} $$ The expression $[q_1, \ldots, q_s]$ is a polynomial in $q_1$, $\ldots$, $q_s$, called the continuant of $q_1$, $\ldots$, $q_s$.
It is not hard to show using standard properties of continuants that for a given positive integer $n$, there are only finitely many sequences of positive integers $q_1$, $\ldots$, $q_s$ with $$ [q_1, \ldots, q_s] - [q_2, \ldots, q_{s-1}] = n $$ (Let's make the conventions that the left side is just $q_1$ and $q_1 q_2$ in the cases $s=1$ and $s=2$ respectively). The proof I give in the paper instead puts them in one-to-one correspondence with a finite collection of binary quadratic forms.
Because the above fact is not hard to prove, I would surmise it has appeared before. My questions:
1) Does the expression $[q_1, \ldots, q_s] - [q_2, \ldots, q_{s-1}]$ have a standard name?
2) What use has been made of it?
