The title compounds, [AnO2(18-crown-6)]n+, An = U, Np, and Pu and n = 1 and 2, as well as the related (experimentally observed) complex [UO2(dicyclohexyl-18-crown-6)]2+ are studied using relativistic density functional theory (DFT). Different relativistic methods (large-core and small-core effective core potentials, all-electron scalar four-component) and two flavors of approximate DFT (B3LYP and PBE) are used. Calculated bond lengths agree well with the available experimental data for the NpV complex, while larger differences for the UVI complexes appear to be related to the large uncertainties in the experimental data. The axial AnO bonds are found to be weaker and longer than in the corresponding penta-aquo complexes, though still of partial triple-bond character. The AnO bond lengths and strengths decrease along the actinide series, consistent with the actinide contraction. Gas-phase binding energies calculated for the penta-aquo complexes and crown-ether complexes of the actinides studied, as well as ligand-exchange energies, show that there is no intrinsic preference, or "better fit", for actinyl(V) cations as compared to actinyl(VI) ones. Rather, the ability of NpO2+ (NpV) to form in-cavity 18-crown-6 complexes in water, which is impossible for UO22+, is traced to solvation effects in polar solvents. Thus, the experimentally observed stabilization of the pentavalent oxidation state as compared to the hexavalent one is due to the effective screening of the charge provided by the macrocycle, and this leads to destabilization of the AnVI crown complexes relative to their AnV counterparts.