On the ubiquity of the ruler sequence

TL;DR

This paper explores the pervasive ruler sequence across various mathematical contexts, introducing four new problems where the sequence emerges through recursive duplication, enhancing understanding of its properties and potential applications.

Contribution

It presents four novel problems involving the ruler sequence, demonstrating its recursive and self-containing properties in diverse mathematical structures.

Findings

The ruler sequence appears in demographic automata, Cantor sets, polygon duplication, and Feigenbaum sequences.

Recursive duplication methods reveal new insights into the sequence's properties.

These representations may inspire further research in discrete mathematics.

Abstract

The ruler function or the Gros sequence is a classical infinite integer sequence that is underlying some interesting mathematical problems. In this paper, we provide four new problems containing this type of sequence: (i) a demographic discrete dynamical automata, (ii) the middle interval Cantor set, (iii) the construction by duplication of polygons and (iv) the horizontal visibility sequence at the accumulation point of the Feigenbaum cascade. In all of them, the infinte sequence is obtained by a recursive procedure of duplication. The properties of the ruler sequence, in particular, those relating to recursiveness and self-containing, are used to get a deeper understanding of these four problems. These new representations of the ruler sequence could inspire new studies in the field of discrete mathematics.