Dyck Numbers, IV. Nested patterns in OEIS A036991

Abstract: Dyck paths (also balanced brackets and Dyck words) are among the most heavily studied Catalan families. This paper is a continuation of [2, 3, 4]. In the paper we are dealing with the numbering of Dyck paths, with the resulting numbers, the terms of the OEIS sequence A036991, which encode Dyck paths and which we have called the Dyck numbers. In this sequence, it turned out to be promising to investigate nested term patterns. The sequence is constructed entirely from nested patterns, with each pattern having an infinite number of copies. We have already considered repeated triplets of adjacent odd numbers; copies of such triplets in the process of expansion form quite complex structures, the analysis of which allows us to understand the structure of A036991. Additionally, we note that the composition of the terms of each pattern is repeated many times by a simple shift (offset) of Dyck numbers by some fixed value. As a result, pattern copies give us important parts of most levels and even give us new levels.