Abstract: Automatic sequences are sequences produced by automata, which can be seen as directed graphs with extra decoration. Most sequences arising in combinatorics are automatic when reduced modulo a prime power. Cayley graphs, on the other hand, are directed graphs obtained from finite groups with distinguished generators.

Following an observation by Rowland, we study those sequences which can be produced by an automaton which is a Cayley graph (with extra information). For 2-automatic sequences (for which the n-th term is a computed from the digits of n in base 2, essentially) the result is particularly satisfying: a given sequence comes from a Cayley graph if and only if it enjoys a certain symmetry, which we call self-similarity.

We give an application to the computation of certain rational fractions associated to automatic sequences.