  Two of the possibilities for the formation of low-mass ($M_{\star}\lesssim
0.5\,M_{\odot}$) hydrogen-deficient white dwarfs are the occurrence of a
very-late thermal pulse after the asymptotic giant-branch phase or a late
helium-flash onset in an almost stripped core of a red giant star. We aim to
asses the potential of asteroseismology to distinguish between the hot flasher
and the very-late thermal pulse scenarios for the formation of low-mass
hydrogen-deficient white dwarfs. We compute the evolution of low-mass
hydrogen-deficient white dwarfs from the zero-age main sequence in the context
of the two evolutionary scenarios. We explore the pulsation properties of the
resulting models for effective temperatures characterizing the instability
strip of pulsating helium-rich white dwarfs. We find that there are significant
differences in the periods and in the period spacings associated with low
radial-order ($k\lesssim 10$) gravity modes for white-dwarf models evolving
within the instability strip of the hydrogen-deficient white dwarfs. The
measurement of the period spacings for pulsation modes with periods shorter
than $\sim500\,$s may be used to distinguish between the two scenarios.
Moreover, period-to-period asteroseismic fits of low-mass pulsating
hydrogen-deficient white dwarfs can help to determine their evolutionary
history.
