  Ardila and Block used tropical results of Brugalle and Mikhalkin to count
nodal curves on a certain family of toric surfaces. Building on a linearity
result of the first author, we revisit their work in the context of the
Goettsche-Yau-Zaslow formula for counting nodal curves on arbitrary smooth
surfaces, addressing several questions they raised by proving stronger versions
of their main theorems. In the process, we give new combinatorial formulas for
the coefficients arising in the Goettsche-Yau-Zaslow formulas, and give
correction terms arising from rational double points in the relevant family of
toric surfaces.
