2-Free Tetranacci Sequences

TL;DR

This paper introduces a variant of the Tetranacci sequence involving iterative division by two, provides an algorithm for special initial sequences, and uses probabilistic modeling and computational data to analyze their unboundedness.

Contribution

It presents a novel sequence variant, an algorithm for constructing specific initial sequences, and a probabilistic model supported by computational evidence.

Findings

Most sequences are unbounded according to the model

Algorithm successfully constructs initially division-poor sequences

Computational data supports the probabilistic assumptions

Abstract

We consider a variant on the Tetranacci sequence, where one adds the previous four terms, then divides the sum by two until the result is odd. We give an algorithm for constructing "initially division-poor" sequences, where over an initial segment one divides by two only once for each term. We develop a probabilistic model that suggests that "most" sequences are unbounded, and provide computational data to support the underlying assumptions of the model.