Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences

TL;DR

This paper introduces TRIP-Stern sequences, higher-dimensional analogues of the Stern diatomic sequence derived from subdividing triangles, and explores their combinatorial properties and generalizations.

Contribution

It defines and analyzes TRIP-Stern sequences, extending the classical Stern sequence to multidimensional continued fractions through triangle subdivision algorithms.

Findings

Established combinatorial properties of TRIP-Stern sequences

Connected TRIP-Stern sequences to well-known sequences

Generalized TRIP-Stern sequences with analogous properties

Abstract

The Stern diatomic sequence is closely linked to continued fractions via the Gauss map on the unit interval, which in turn can be understood via systematic subdivisions of the unit interval. Higher dimensional analogues of continued fractions, called multidimensional continued fractions, can be produced through various subdivisions of a triangle. We define triangle partition-Stern sequences (TRIP-Stern sequences for short), higher-dimensional generalizations of the Stern diatomic sequence, from the method of subdividing a triangle via various triangle partition algorithms. We then explore several combinatorial results about TRIP-Stern sequences, which may be used to give rise to certain well-known sequences. We finish by generalizing TRIP-Stern sequences and presenting analogous results for these generalizations.