  We present an example of a quadratic algebra given by three generators and
three relations, which is automaton (the set of normal words forms a regular
language) and such that its ideal of relations does not possess a finite
Gr\"obner basis with respect to any choice of generators and any choice of a
well-ordering of monomials compatible with multiplication. This answers a
question of Ufnarovski.
  Another result is a simple example (4 generators and 7 relations) of a
quadratic algebra of intermediate growth.
