An optimal locating-dominating set in the infinite triangular grid

Assume that G=(V,E)G=(V,E) is an undirected graph, and C⊆VC⊆V. For every v∈Vv∈V, we denote by I(v)I(v) the set of all elements of C   that are within distance one from vv. If all the sets I(v)I(v) for v∈V⧹Cv∈V⧹C are non-empty, and pairwise different, then C   is called a locating-dominating set. The smallest possible density of a locating-dominating set in the infinite triangular grid is shown to be 1357.