Codimension growth of two-dimensional algebras

Let F be a field of characteristic zero and let A be a two-dimensional non-associative algebra over F. We prove that the sequence c_n(A), n=1, 2, . . . , of codimensions of A is either bounded by n + 1 or grows exponentially as 2^n. We also construct a family of two-dimensional algebras indexed by rational numbers with distinct T-ideals of polynomial identities and whose codimension sequence is n + 1, n ≥ 2.