With the use of one-range addition theorems of Slater type orbitals (STOs) introduced by one of the authors, three-center nuclear attraction integrals containing Coulomb-Yukawa like correlated interaction potentials (C-CIPs and Y-CIPs) appearing in the Hartree-Fock-Roothaan (HFR) equations for molecules are evaluated. These integrals are expressed through the overlap integrals which depend on the frictional quantum number alpha, where -infinity < alpha <= 2. The convergence of the series is tested by calculating three-center nuclear attraction integrals of C-ClPs, Y-CIPs, and STOs for the arbitrary values of potential parameters and locations of orbitals. For rapid calculations of these integrals, we use the partial summations of some indices corresponding to progressively increasing upper limits appearing in the series expansion relations. Additionally, the binomial coefficients arising in the series are stored in the memory of the computer using their recurrence relation. The fast and accurate computation approach suggested in this work is demonstrated.