  The discovery of superconductivity in Sr-doped NdNiO$_{2}$ is a crucial
breakthrough in the long pursuit for nickel oxide materials with electronic and
magnetic properties similar to those of the cuprates. NdNiO$_2$ is the
infinite-layer member of a family of square-planar nickelates with general
chemical formula R$_{n+1}$Ni$_n$O$_{2n+2}$ (R = La, Pr, Nd, $n= 2, 3, ...
\infty$). In this letter, we investigate superconductivity in the trilayer
member of this series (R$_4$Ni$_3$O$_8$) using a combination of
first-principles and $t-J$ model calculations. R$_4$Ni$_3$O$_8$ compounds
resemble cuprates more than RNiO$_2$ materials in that only
Ni-$d_{x^{2}-y^{2}}$ bands cross the Fermi level, they exhibit a largely
reduced charge transfer energy, and as a consequence superexchange interactions
are significantly enhanced. We find that the superconducting instability in
doped R$_4$Ni$_3$O$_8$ compounds is considerably stronger with a maximum gap
about four times larger than that in Sr$_{0.2}$Nd$_{0.8}$NiO$_2$.
