Other properties and applications of automatic sequences
Automatic sequences and formal algebraic series

Automatic sequences or their generalizations (multi-dimensional automatic sequences, morphic sequences, q-regular sequences, etc.) have many properties, which mean they can be found in many branches of mathematics and theoretical computer science. We'll just take a quick look here, referring the reader to the bibliography for more information.

The first area we're going to discuss is number theory. We have seen that q-automatic sequences are exactly the coefficients of formal algebraic series over the field of rational fractions with coefficients in the q-element field. The question arises as to what kind of real numbers automatic sequences can be the "coefficients" of (i.e. the digits in a given integer numeration base). For example, the Prouhet-Thue-Morse sequence encountered above can be interpreted as the development in binary (or base 10, for example) of a real number θ :