# Quadratic Cyclic Sequences

## TL;DR

This paper investigates quadratic cyclic sequences linked to cyclotomic polynomials and plane walks, revealing unique phenomena for certain angles and algebraic numbers, expanding understanding of cyclic structures and their algebraic properties.

## Contribution

It introduces new relations between quadratic difference relations, cyclotomic polynomials, and plane walks, highlighting non-symmetric phenomena for specific angles and algebraic numbers.

## Key findings

- Non-symmetric phenomena occur for n ≥ 12 when the turn angle is 2π/n.
- Examples involve algebraic numbers of modulus one that are not roots of unity.
- Connections established between cyclic sequences, cyclotomic polynomials, and planar walks.

## Abstract

We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn either left or right through a fixed angle. In the case when this angle is $2 \pi /n$, then non-symmetric phenomena occurs for $n\geq 12$. Examples arise from algebraic numbers of modulus one which are not $n$'th roots of unity.