cp's OEIS Frontend

It appears that for the same sequence of angles there is one solution with all straight edges and one with all diagonal edges, so all terms are even.

For n even, the symmetry line passes through two opposite vertices, or cuts through two opposite edges. For n odd it passes through one vertex and cuts the opposite edge. That explains the even-odd irregularity in the sequence.

The embedding must map edges in the graph onto either horizontal or vertical grid lines of length 1 or diagonals of length sqrt(2). Vertices in the graph must map onto lattice points, and of course must preserve the incidence structure of the graph. A square in the lattice may have both diagonals present - their intersection does not count as an incidence.

Configurations differing only a rotation or reflection are not counted as different.

The resulting figures are variously called 'polysticks', 'polyedges' or 'polyforms'.