Finding Structure in Sequences of Real Numbers via Graph Theory: a Problem List

TL;DR

This paper explores a graph-based method to uncover hidden structures in sequences of real numbers, revealing intricate patterns in well-known mathematical sequences through empirical analysis.

Contribution

It introduces a novel graph-theoretic approach to analyze real number sequences and presents a list of sequences with discovered hidden structures, framing each as an open problem.

Findings

Sequences exhibit intricate hidden structures.

Graph-based analysis reveals patterns in famous sequences.

Each sequence remains an open problem for further research.

Abstract

We investigate a method of generating a graph $G=(V,E)$ out of an ordered list of $n$ distinct real numbers $a_1, \dots, a_n$. These graphs can be used to test for the presence of interesting structure in the sequence. We describe sequences exhibiting intricate hidden structure that was discovered this way. Our list includes sequences of Deutsch, Erd\H{o}s, Freud & Hegyvari, Recaman, Quet, Zabolotskiy and Zizka. Since our observations are mostly empirical, each sequence in the list is an open problem.