Generating Recursive Formulas from Non-Recursive Formulas

Introduction

Consider a sequence of numbers with a distinct pattern, such as

1, 4, 7, 10, 13, 16, 19, …

For this thought exercise, the sequence needs to be chosen with some purpose, rather than at complete random. When this is achieved, the sequence can be described by two types of formulas. For this specific example, those formulas take the following two forms:

S_n = 3n + 1

AND