The discovery of superconductivity in Sr-doped NdNiO2 is a crucial breakthrough in the long pursuit for nickel oxide materials with electronic and magnetic properties similar to those of the cuprates. NdNiO2 is the infinite-layer member of a family of square-planar nickelates with general chemical formula Rn+1NinO2n+2 (R = La, Pr, Nd, n=2,3,...∞). In this Rapid Communication, we investigate superconductivity in the trilayer member of this series (R4Ni3O8) using a combination of first-principles and t−J model calculations. R4Ni3O8 compounds resemble cuprates more than RNiO2 materials in that only Ni-dx2−y2 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 R4Ni3O8 compounds is considerably stronger with a maximum gap about four times larger than that in Sr0.2Nd0.8NiO2.