Quadratic cyclic sequences

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 $$n\ge 12$$ . Examples arise from algebraic integers of modulus one which are not n’th roots of unity.