  We obtain a formula for the degrees of the varieties parameterizing complex
algebraic curves of any divisor class and genus on P^2_6, the projective plane
blown-up at 6 generic points. Moreover, the formula computes the degrees of the
varieties parameterizing curves on P^2_6 which additionally satisfy certain
tangency conditions to a fixed exceptional divisor on P^2_6. Our formula
contains as special cases the degrees of the analogous varieties parameterizing
curves on P^2_q, for q=1,...,5, and on the quadric (P^1)^2. It is an extension
of the Caporaso-Harris formula counting curves of any degree and genus in the
projective plane, and it can be viewed as an alternative to the Vakil formula,
which gives an indirect way to count curves of any genus on P^2_6.
