Dyck Numbers, II. Triplets and Rooted Trees in OEIS A036991

TL;DR

This paper explores the structure of Dyck numbers within OEIS sequence A036991, revealing triplet patterns, their distribution, and the infinite trees rooted at lone terms, and tests the twin prime conjecture within this framework.

Contribution

It introduces a detailed analysis of triplets and lone terms in Dyck numbers, connecting them to infinite ternary trees and testing prime gap conjectures.

Findings

Triplets cover 80% of A036991 and include all Mersenne numbers.

Lone terms are roots of infinite ternary trees of triplets.

The sequence forms a forest of directed trees, enabling prime gap analysis.

Abstract

Dyck paths are among the most heavily studied Catalan families. This paper is a continuation of [2]. In the paper we are dealing with the numbering of Dyck paths, the terms of the OEIS sequence A036991 or Dyck numbers. We consider triplets of terms of the form (t-4, t-2, t) (t is the senior term); triplets cover 80% of A036991. Triplets include all Mersenne numbers, with each Mersenne number being a senior term in some triplet. Triples are distributed over A036991 ranges; both the length of the ranges and the number of triplets in the range are counted by the terms of the OEIS sequence A001405. In addition to triplets, there are many lone terms in the ranges, which are counted by the terms of the OEIS sequence A116385. Each lone term (there are an infinite number of them) is the root of an infinite ternary tree of triplets. As a result, the sequence A036991 is a forest of such directed trees. We test the twin prime conjecture on infinite ternary trees of A036991. We consider the infinity theorem for pairs of prime numbers that differ by no more than 4.