  The pairing temperature of superconducting thin films is expected to display,
within the Bardeen-Cooper-Schrieffer theory, oscillations as a function of the
film thickness. We show that the pattern of these oscillations switches between
two different periodicities at a density-dependent value of the superconducting
coupling. The transition is most abrupt in the anti-adiabatic regime, where the
Fermi energy is less than the Debye energy. To support our numerical data, we
provide new analytical expressions for the chemical potential and the pairing
temperature as a function of thickness, which only differ from the exact
solution at weak coupling by exponentially-small corrections.
