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 http://math.stackexchange.com/posts/938147/edit.)

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?