The Tribonacci and ABC Representations of Numbers are Equivalent

TL;DR

This paper proves the equivalence between the tribonacci number representation and the ABC sequence representation of positive integers, using auxiliary representations and a labeled tribonacci tree.

Contribution

It establishes a formal bijection between two unique integer representations and analyzes the sequences in relation to the tribonacci word.

Findings

The representations are shown to be equivalent through a direct bijection.

Auxiliary representations facilitate the proof of equivalence.

A systematic study of the A, B, C sequences in relation to the tribonacci word is provided.

Abstract

It is shown that the unique representation of positive integers in terms of tribonacci numbers and the unique representation in terms of iterated A, B and C sequences defined from the tribonacci word are equivalent. Two auxiliary representations are introduced to prove this bijection. It will be established directly on a node and edge labeled tribonacci tree as well as formally. A systematic study of the A, B and C sequences in terms of the tribonacci word is also presented.