Okay, here's another clarification, which I hope will clear up the confusion I created with my original post: 1) I wrote
tiling a triangle of dots of side n with tiles consisting of either 3 dots in a row or 3 dots forming a triangle.
but what I should have written was
tiling a triangle of dots of side n with tiles, where EITHER all tiles consist of 3 dots in a row OR all tiles consist of 3 dots in a triangle.
As David Moulton points out, if one intermixes the two kinds of tiles (as the wording of my original post would permit), then such tilings exist for all values of n. But if intermixing is not allowed, things are more interesting. In the case where all tiles consist of 3 dots in a row, no such tiling exists for any positive value of n. In the case where all tiles consist of 3 dots forming a triangle, such a tiling exists precisely when n is congruent to 0, 2, 9, or 11 (mod 12). Jim Propp