The Prouhet-Thue-Morse Sequence is well-known for its complex nature and is known for its being minimal and non-repative, and has long been known in the fields of Combinatorics, Number Theory and Theoretical Computer Science. This paper goes back to this

The Prouhet-Thue-Morse Sequence is well-known for its complex nature and is known for its being minimal and non-repative, and has long been known in the fields of Combinatorics, Number Theory and Theoretical Computer Science. This paper goes back to this sequence and revisits it from the perspective of the current-day computational problem, offers a unified structure comprising 21 specific definitions, and introduces some variations to the multi-agent system with arbitrary integer bases. We show that these definitions are equivalent using rigorous mathematical proof, and investigate qualities of these extensions such as periodicity, aperiodicity, and complexity. We also discuss the computing efficiency of both definitions, and provide complexity analysis that is practical in offering advice on how to apply in the real world. This paper is the foundation for the use of the Prouhet-Thue-Morse Sequence in several problems of multi-agent systems such as the resource distribution, the algorithm design as well as synthetic data generation.