  In this paper, we perform an arithmetization of well-formed parenthesis
strings with zeros (Motzkin words) and of corresponding Motzkin paths. The
transformations used are reminiscent of G\"odel numbering for mathematical
objects of some formal language. We construct a Motzkin series that is as close
as possible to natural numbers by many formal features. Parenthesis strings are
encoded by ternary codes and corresponding natural numbers, Motzkin numbers of
the 2nd kind, which made it possible to formalize and simplify the analysis of
the successor function, to specify and clarify the procedure for selecting a
successor. In the process of arithmetization of well-wormed parenthesis
strings, various special numbers appeared. In this regard, in conclusion we
will talk a little about numbers of the form $3^n+2$ and mirror numbers
$2*3^n+1$.
