We report site percolation thresholds for square lattice with neighbor bonds at various increasing ranges. Using Monte Carlo techniques we found that nearest neighbors (NN), next-nearest neighbors (NNN), next-next-nearest neighbors (4N), and fifth-nearest neighbors (6N) yield the same pc = 0.592... . The fourth-nearest neighbors (5N) give pc = 0.298... . This equality is proved to be mathematically exact using symmetry argument. We then consider combinations of various kinds of neighborhoods with (NN+NNN), (NN+4N), (NN+NNN+4N), and (NN+5N). The calculated associated thresholds are respectively pc = 0.407..., 0.337..., 0.288..., and 0.234... . The existing Galam-Mauger universal formula for percolation thresholds does not reproduce the data showing dimension and coordination number are not sufficient to build a universal law which extends to complex lattices.