Using the stationary perturbation theory the equations for small polaron energies are obtained. This polaron has small spatial dimensions - only three equal atoms of a linear chain. The initial Hamiltonian has the contributions related to electron hoppings between the atoms, interaction of electron of the central atom with its oscillations and anharmonic contributions to the energy of its oscillations of the third and the fourth orders. The obtained analytical expressions give evidence that the polaron states with the fully filled (or significantly filled) atomic orbitals have the lowest energies that is in agreement with the results of numerical calculations available in the literature.