The connection between the Noyes-CFS modified effective-range formula for triplet S-wave n-p scattering and the N/D method is reviewed. Two- and three-pole approximations are compared with the exact numerical solution of the coupled integral equations of this method. Analytic iteration is performed to second order, yielding new approximations which retain much of the structure of the exact solution, and provide very accurate numerical results, as well as a new shape dependent version of the effective-range expansion. It is found that the one-pion-exchange cut plus two closely spaced effective interaction poles can reproduce the shape dependence given by realistic nucleon-nucleon potentials.