Olry Terquem’s Forgotten Problem and an Enumerative-Combinatorial Perspective on the Euclidean Algorithm

“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 chapter, we highlight this distinction and also make connections between Terquem’s “forgotten” problem, the Fibonacci sequence, a one-player game of numbers, continuant polynomials, (an extended version of) the Euclidean Algorithm, and Bézout’s Lemma.