"chiral" polyominoes

[Jonathan Post]

It is insufficient, in my opinion, to single out those polynomials which
lack bilateral symmetry.  After all, we have ones such as the S and Z which
have order 2 rotational symmetry, and the X with order 4 rotational
symmetry.

Coincidently, my son (Andrerw C. Post, responsible as coauthor) and I
yesterday were playing with something which is neither polyomino nor
polyplet (although they are subsets), but closely related:

Number of connected nonisomorphic shapes made of n identical squares in the
plane, connected edge-to-edge or corner-to-corner.

1, 1, 2, 8, 36, ?

Offset 0,3

Comment: the corner-to-corner connection need not have square's edges
aligned with X or Y axes.

Comment: the number which are not bilaterally symmetric is
0, 0, 0, 1, 27
and the a(n)  have had each pair counted as one; if each pair is counted as
2, then a(n) = 1, 1, 2, 9, 58, ...

Comment: the number with holes are 0, 0, 0, 1, 6, ...

One of the a(4) has a hole which is a concave quadrilateral.
If we use equilateral triangles, we have a new sequence based on shapes
which are neither polyiamonds (although they are a subset) nor animals in
the isometric grid.  Oddly, I first did the 3-dimensional analogue: A119602.

Number of connected nonisomorphic shapes made of n identical equilateral
trinagles in the plane, connected edge-to-edge or corner-to-corner.

a(n) = 1, 1, 2, 6, 21, ?

offset 0,3

Comment: the corner-to-corner connection need not have the triangle's edges
aligned with an isometric grid.
Comment: the number which are not bilaterally symmetric is
0, 0, 0, 1, 7
and the a(n)  have had each pair counted as one; if each pair is counted as
2, then a(n) = 1, 1, 2, 7, 28, ...

Comment: the number with holes are 0, 0, 0, 1, 4, ...
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