The Tribonacci and ABC Representations of Numbers are Equivalent

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.