Quadratic automaton algebras and intermediate growth

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.
Comment: To appear in Journal of Cobinatorial Algebra