polygons - Point lattice leading to triangle lattice

My main purpose is to eventually generate a triangle lattice from one original point, let's say the origin. So I want to start with the origin and generate 6 points around it, which are the vertices of an hexagon of side with length 1. Then, I would like to have those new six points to do the same and generate six other points (I know there will be repetitions). Then, I want those points to from a triangle lattice, but I am new to mathematica.

So far I constructed a function that generates the vertices of an hexagon:

h[x_, y_] := Point[Table[{Cos[2 Pi k/6] + x, Sin[2 Pi k/6] + y}, {k, 6}]]

Now i am looking to generate the other poits from my new ones.