2automatic
2automatic, or 2-automatic sequences, are a class of sequences over a finite alphabet that are generated by finite automata reading the base-2 representations of the nonnegative integers. In practical terms, there exists a deterministic finite automaton with output (a DFAO) that, when fed the binary expansion of n (usually from most significant to least significant digit), outputs the nth term of the sequence. This framework places 2automatic sequences within the broader family of k-automatic sequences, with k = 2.
A standard example is the Thue–Morse sequence, where a_n is the parity of the number of 1s
2automatic sequences have applications in combinatorics on words, number theory, and theoretical computer science. They are
Historically, the concept was developed within the theory of automatic sequences, with prominent foundational work by
In summary, 2automatic sequences are finite-state generated sequences indexed by natural numbers via binary representations, exemplified