Olry Terquem's forgotten problem

`Terquem's problem' is a name given in the twentieth century to the problem of enumerating certain integer sequences whose entries alternate in parity. In particular, this problem asks for the count of strictly increasing length $m$ sequences of positive integers bounded above by some integer $n$ whose odd-indexed entries are odd and whose even-indexed entries are even. This problem and its generalizations have been well-studied. However, the putative original source for this problem, an 1839 paper by Olry Terquem, is subtly different from the problem that is now attributed to Terquem. In this paper, we highlight this distinction and also make connections between Terquem's `forgotten' problem and the Fibonacci sequence, continuants, Bézout's Lemma, and the extended version of the Euclidean Algorithm.