In this paper we compute the square lattice random sites percolation thresholds in case when sites from the 4-th and the 5-th coordination shells are included for neighbourhood. The obtained results support earlier claims, that (a) the coordination number and the space dimension are insufficient for building universal formulae for percolation thresholds and (b) that percolation threshold may not decrease monotonically with lattice site coordination number.

PACS numbers: 64.60.Ak, 05.70.Fh