We derive two new equations of state for linear, flexible hard-chain fluids based on Wertheim's Thermodynamic Perturbation Theory (TPT) of third order. The 4-particle distribution function in the third order term is first approximated solely by 2- and 3-particle distribution functions and is then refined by adjusting an expression to results of molecular simulations. The prediction of the compressibility factor of the third-order expansions show mildly better agreement compared to the first-order (TPT1) and second-order theories (TPT2). We validate the theories with new results of molecular simulation for chains with v = 4, 8, 16, and 64 spherical segments. For the second virial coefficient we find deviations between the third-order theory and the simulation results. These deviations can not be explained by limited order of the TPT-formalism, as the convergence with increasing order shows. We thus conclude, that the single-chain approximation is not sufficient to describe the low-density behavior of longer chains. (C) 2015 Elsevier B.V. All rights reserved.