A197465 Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

1, 2, 2, 6, 8, 22, 42, 112, 252, 650, 1584, 4091, 10369, 26938, 69651, 182116, 476272, 1253067, 3302187, 8733551, 23142116, 61477564, 163612714, 436278921, 1165218495, 3117021788, 8349892686, 22397754046, 60153261611

Offset

1,2

Comments

See the link below for a definition of the tetrakis square tiling. When a square grid cell is divided into triangles, it must be divided dexter (\) or sinister (/) according to the parity of the grid cell.

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

Table of n, a(n) for n=1..29.

Peter Kagey, Example illustrating a(4) = 6.

Wikipedia, Tetrakis square tiling

Example

For n=3 there are 4 triaboloes.  Of these, 2 conform to the tetrakis grid.  Each of these 2 has a unique dissection into 6 tetrakis cells. - George Sicherman, Mar 25 2021

Crossrefs

Cf. A197466, A197467, A006074.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

Sequence in context: A052957 A383802 A275441 * A074933 A383372 A157253

Adjacent sequences: A197462 A197463 A197464 * A197466 A197467 A197468

Keyword

nonn,hard,more

Author

Joseph Myers, Oct 15 2011

Extensions

Name clarified by George Sicherman, Mar 25 2021

a(21)-a(26) from Aaron N. Siegel, May 18 2022

a(27)-a(29) from Bert Dobbelaere, Jun 04 2025

Status

approved