Multiple Point Control

Dear All,

I am trying to make a surface with equilateral triangles and I divided the surface into grid and after exploding and subdividing surface got vertices. Now I want to make equilateral triangle by using certain points. How can I select points which I need in one go from grid?
How can I do that?
I used 2 cull pattern(True, False) & other one (False, True) and got result points which are not ok in the upper line.

is there any other way of doing it ?

Snapshots are attached.