On Collatz' Words, Sequences and Trees

TL;DR

This paper explores generalized Collatz words and sequences, deriving recurrence relations and solutions, and discusses the structure of the Collatz tree to deepen understanding of the problem.

Contribution

It introduces a general framework for Collatz words and sequences, deriving explicit recurrences and solutions, and analyzes the Collatz tree structure.

Findings

Derived recurrence relations for Collatz sequences

Solved the recurrences explicitly

Discussed the structure of the Collatz tree

Abstract

Motivated by a recent work of Tr\"umper we consider the general Collatz word (up-down pattern) and the sequences following this pattern. The recurrences for the first and last sequence entries are given, obtained from repeated application of the general solution of a binary linear inhomogeneous Diophantine equation. These recurrences are then solved. The Collatz tree is also discussed.