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It is shown that the unique representation of positive integers in terms oftribonacci numbers and the unique representation in terms of iterated A, B andC sequences defined from the tribonacci word are equivalent. Two auxiliaryrepresentations are introduced to prove this bijection. It will be establisheddirectly on a node and edge labeled tribonacci tree as well as formally. Asystematic study of the A, B and C sequences in terms of the tribonacci word isalso presented.

The Tribonacci and ABC Representations of Numbers are Equivalent