A197459 Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

1, 1, 3, 4, 12, 27, 78, 208, 635, 1859, 5726, 17526, 54620, 170479, 536714, 1694567, 5376764, 17110286, 54631302, 174879997, 561229678, 1805022806, 5817191196, 18781911278, 60744460580

Offset: 1

Author

Joseph Myers, Oct 15 2011

Keywords

• hard
• nonn
• more

Comments

[3.6.3.6] refers to the face configuration of the rhombille tiling. - Peter Kagey, Mar 01 2020

If we draw the short diagonals of each tile in the rhombille tiling, we get a subset of edges of the regular hexagonal grid; two edges are adjacent if and only if the corresponding rhombi are adjacent. These are polyedges where angles are constrained to 120 degrees. So there is a 1-to-1 correspondence with the subset of polyedges counted in A159867 after removing polyedges with angles of 60 and/or 180 degrees. - Joseph Myers, Jul 12 2020

These are also known as polytwigs, by association with their representation as polyedges. - Aaron N. Siegel, May 15 2022

References

• Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Links

• Abaroth's World, Illustration of a(1) through a(6)
• Peter Kagey, Illustration for A197459(4) = 4.
• R. J. Mathar, OEIS A197459
• Wikipedia, Rhombille tiling

Crossrefs

Cf. A019988, A159867, A197460, A197461.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

Extensions

a(18)-a(22) from Aaron N. Siegel, May 15 2022

a(23)-a(25) from Bert Dobbelaere, Jun 04 2025