Using the coupled cluster method (CCM), we present a simple accurate calculation for the energies of the ground- and first excited states of the linear E x e Jahn-Teller and pseudo-Jahn-Teller Hamiltonians. From the solution of a single transcendental equation, we obtain results with a maximal error of 1.2%. These results are notably better than previous results obtained both via the CCM and other many-body approximations.