Fibonacci Plays Billiards

TL;DR

This paper explores the construction of integer chains with adjacent sums of specific mathematical forms, utilizing billiard ball paths on polygonal tables to find such sequences and necklaces.

Contribution

It introduces a novel approach linking billiard ball paths to the generation of special integer chains with sums of particular forms.

Findings

Identifies conditions for forming Fibonacci, triangular, and other chains.

Demonstrates the use of billiard paths to discover these chains.

Provides examples of chains and necklaces with special sum properties.

Abstract

A chain is an ordering of the integers 1 to n such that adjacent pairs have sums of a particular form, such as squares, cubes, triangular numbers, pentagonal numbers, or Fibonacci numbers. For example 4 1 2 3 5 form a Fibonacci chain while 1 2 8 7 3 12 9 6 4 11 10 5 form a triangular chain. Since 1 + 5 is also triangular, this latter forms a triangular necklace. A search for such chains and necklaces can be facilitated by the use of paths of billiard balls on a rectangular or other polygonal billiard table.