Retroelongated triangular tiling

The retroelongated triangular tiling, or retrat, is a uniform tiling of the Euclidean plane. 3 triangles and 2 squares join at each vertex of this tiling. It can be formed by inserting layers of squares between layers of the triangular tiling, in the opposite way to the elongated triangular tiling, such that the vertex figure pentagon becomes nonconvex. It can be considered a blend of infinitely many apeirogonal antiprisms and apeirogonal prisms.

Vertex coordinates
The vertices of an elongated triangular tiling of edge length 1 are given by

where i and j range over the integers.
 * $$\left(i\frac{2-\sqrt3}{2}\pm\frac12,\,j+\frac{i}{2}\right),$$