  We study the many-body electronic structure of the stoichiometric and
electron-doped trilayer nickelate Pr$_4$Ni$_3$O$_8$ in comparison to that of
the stoichiometric and hole-doped infinite layer nickelate NdNiO$_2$ within the
framework of density functional plus dynamical mean field theory, noting that
Pr$_4$Ni$_3$O$_8$ has the same nominal carrier concentration as NdNiO$_2$ doped
to a level of 1/3 holes/Ni. We find that the correlated Ni-$3d$ shells of both
of these low valence nickelates have similar many-body configurations with
correlations dominated by the $d_{x^2-y^2}$ orbital. Additionally, when
compared at the same nominal carrier concentration, the materials exhibit
similar many-body electronic structures, self energies, and correlation
strengths. Compared to cuprates, these materials are closer to the Mott-Hubbard
regime due to their larger charge transfer energies. Moreover, doping involves
the charge reservoir provided by the rare earth $5d$ electrons, as opposed to
cuprates where it is realized via the oxygen $2p$ electrons.
