Site percolation thresholds on triangular lattice with complex neighborhoods

We determine thresholds _{p c} for random site percolation on a triangular lattice for neighborhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN), and next-next-next-next-nearest (5NN) neighbors, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, and 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [Phys. Rev. E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al. [Phys. Rev. E 90, 062101 (2014)], for estimating thresholds from low statistics data. The estimated values of percolation thresholds are _{p c} ( 4NN ) = 0.192 410 ( 43 ), _{p c} ( 3NN+2NN ) = 0.232 008 ( 38 ), _{p c} ( 5NN+4NN ) = 0.140 286 ( 5 ), _{p c} ( 3NN+2NN+NN ) = 0.215 484 ( 19 ), _{p c} ( 5NN+4NN+NN ) = 0.131 792 ( 58 ), _{p c} ( 5NN+4NN+3NN+2NN ) = 0.117 579 ( 41 ), and _{p c} ( 5NN+4NN+3NN+2NN+NN ) = 0.115 847 ( 21 ). The method is tested on the standard case of site percolation on the triangular lattice, where _{p c} ( NN ) = _{p c} ( 2NN ) = _{p c} ( 3NN ) = _{p c} ( 5NN ) = /1 2 is recovered with five digits accuracy _{p c} ( NN ) = 0.500 029 ( 46 ) by averaging over one thousand lattice realizations only.