A383908 Number of generalized polyforms with n cells on the snub trihexagonal tiling.

1, 3, 3, 7, 23, 69, 228, 766, 2642, 9309, 33382, 120629, 439752, 1613135, 5953061, 22075011, 82204128, 307213215, 1151820825, 4330858682, 16326297768, 61690058385

Offset

0,2

Comments

A generalized polyform on the snub trihexagonal tiling with n-cells is a collection of n faces connected edgewise. Two polyforms are considered the same they are related by an isometry (translation and/or rotation) of the snub trihexagonal tiling.

Links

Table of n, a(n) for n=0..21.

Bert Dobbelaere, Peter Kagey, Drake Thomas, and Andrés R. Vindas-Meléndez, Building with Blocks: Enumerating Polyforms on Tilings, arXiv:2602.23301 [math.CO], 2026. See p. 9.

Peter Kagey, Illustration of a(1)-a(4).

Wikipedia, Snub trihexagonal tiling

Example

For n=1, the a(1) = 3 generalized polyforms are the three types of faces: hexagons, hexagon-adjacent triangles, and hexagon-nonadjacent triangles.
For n=2, the a(2) = 3 generalized polyforms are
(1) a hexagon with a hexagon-adjacent triangle,
(2) a hexagon-adjacent triangle with a hexagon-nonadjacent triangle, and
(3) two hexagon-adjacent triangles.

Crossrefs

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square), A344211 (rhombitrihexagonal), A344213 (truncated trihexagonal).

Sequence in context: A262375 A232368 A333924 * A173321 A191498 A065747

Adjacent sequences: A383905 A383906 A383907 * A383909 A383910 A383911

Keyword

nonn,more,hard

Author

Peter Kagey, May 14 2025

Extensions

a(12)-a(21) from Bert Dobbelaere, Jun 05 2025

Status

approved