An open-shell restricted Hartree-Fock singles and doubles coupled-cluster method including perturbative triple excitations, CCSD(T), that does not include any spin contamination for the correlation energies, is computationally formulated and implemented. The new open-shell CCSD(T) program is applied in conjunction with large atomic natural-orbital basis sets to calculate the energy difference between the linear and bent isomers of C3+. With a [5s4p3d2flg] basis set, the CCSD(T) method predicts the bent (cyclic) 2B2 state to lie 6.8 kcal/mol lower than the linear 2-SIGMA-u+ species. The importance of correlation effects included at the CCSD(T) level of theory is analyzed in connection to the existence of symmetry-broken Hartree-Fock solutions for this problem.