We determine thresholds $p_c$ for random site percolation on a triangular lattice for neighbourhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN) and next-next-next-next-nearest (5NN) neighbours, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [M. E. J. Newman and R. M. Ziff, Physical Review 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. [N. Bastas, K. Kosmidis, P. Giazitzidis, and M. Maragakis, Physical Review E 90, 062101 (2014)], of estimating thresholds from low statistics data. The estimated values of percolation thresholds are $p_c(text{4NN})=0.192410(43)$, $p_c(text{3NN+2NN})=0.232008(38)$, $p_c(text{5NN+4NN})=0.140286(5)$, $p_c(text{3NN+2NN+NN})=0.215484(19)$, $p_c(text{5NN+4NN+NN})=0.131792(58)$, $p_c(text{5NN+4NN+3NN+2NN})=0.117579(41)$, $p_c(text{5NN+4NN+3NN+2NN+NN})=0.115847(21)$. The method is tested on the standard case of site percolation on triangular lattice, where $p_c(text{NN})=p_c(text{2NN})=p_c(text{3NN})=p_c(text{5NN})=frac{1}{2}$ is recovered with five digits accuracy $p_c(text{NN})=0.500029(46)$ by averaging over one thousand lattice realisations only.
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