The Automata Properties Of The Differences Sequences Of The Paper-folding Sequences

Automatic sequence is between periodic sequence and random sequence. It can be generated by a finite state machine sequences. For a periodic sequence, it’s too strong regu-larity and its structure is too simple. For a random sequence, its structure is too complicated to study. Therefore, the study of automatic machine sequence is important. How to find it’s good nature is worth of our in-depth study. In mathematics and other fields, automatic sequence is playing a more and more important role. This makes people giving more in-terest on them. Thue-Morse sequence and Rudin-Shapiro sequence are classic automatic sequences.In this paper, we mainly studied the automatic properties of the regular paperfolding sequence and its difference sequences. We know that a sequence is automatic sequence, and its difference sequences are also automatic sequences. We mainly studied the automa-ta properties of the differences sequences of{Δm/0}m≥0,{Δmf1}m≥0,…,{Am/fn}m≥0in the paper. After introduced the related definition of automatic sequence, we raised the method of proving κ-automatic sequences. By calculating the recurrence relations of the sequences, we got the result that their κ-kernel are finite. So the difference sequences order are2-automatic sequences in model2.After calculated the recurrence relations of the subsequences K2(f) of the sequences f, we got the result that K2(f) is finite by mathematical induction, so the sequence f is2-automatic sequence. Then the difference sequences Δf, Δ2f,…, Δmf,…are still2-automatic sequences. What’s more, we got the conclusion that the difference sequences {Δmfn}m≥0are2-automatic sequences. For one thing, we calculated the64recurrence relations of the sequences{Δm/n}m≥0by Lucas’Theorem, and got the13formulas which are obtained by it. Then we had the conclusion that its κ-kernel are finite by mathematical induction, then the sequences are2-automatic sequences.