Automatic Sequence - Examples

Examples

The following sequences are automatic:

  • Thue-Morse sequence: take E = A = {0, 1}, e = 0, π = id, and σ such that σ(0) = 01, σ(1) = 10; we get the fixpoint 01101001100101101001011001101001..., which is in fact the Thue-Morse word. The n-th term is the parity of the base 2 representation of n and the sequence is thus 2-automatic. The 2-kernel consists of the sequence itself and its complement. The associated power series T(z) satisfies
over the field F2(z).
  • Rudin–Shapiro sequence
  • Baum–Sweet sequence
  • Regular paperfolding sequence and a general paperfolding sequence with a periodic sequence of folds
  • The period-doubling sequence, defined by the parity of the power of 2 dividing n; it is the fixed point of the morphism 0 → 01, 1 → 00.

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