Lagarias' bibliographies give some references from where you might search forward. For instance see this
This is from the year-2004 version; Lagarias updated this up to version 2011; the Lagarias bibliogrphies are downloadable from the net.
Update (7'22): Just found an old downloaded article
Maurice Margenstern and Yuri Matiyasevich
Abstract:
We show how the 3 x + 1 conjecture can be expressed
in the language of arithmetical formulas with
binomial coefficients.
e-mail: [email protected]
e-mail: [email protected]
URL: http://logic.pdmi.ras.ru/~yumat
This is of 1998; don't know whether the URL or the email is still working; maybe it is only tangential to your question, but thought it might be of interest from a wider perspective in the combinatorical context.