The Pell Tower and Ostronometry

TL;DR

This paper explores patterns in generalized Fibonacci sequences defined by a recurrence relation, revealing new phenomena like a Red Wall and exotic numeration systems.

Contribution

It extends previous work on Fibonacci patterns to a broader class of recurrence relations, uncovering novel mathematical structures.

Findings

Discovery of new patterns in generalized Fibonacci sequences

Identification of a Red Wall phenomenon

Introduction of exotic numeration systems

Abstract

Conway and Ryba considered a table of bi-infinite Fibonacci sequences and discovered new interesting patterns. We extend their considerations to tables that are defined by the recurrence $X_{n+1}=dX_n+X_{n-1}$ for natural numbers $d$. In our search for new patterns we run into a Red Wall and exotic numeration systems.