A385278 Number of face-connected components of polyhedral cells in the triangular pyramidille up to translation, rotation, and reflection of the honeycomb.

1, 1, 3, 4, 16, 39, 152, 517, 2056, 8002, 32692, 134198, 561511, 2366909, 10075926, 43174057, 186208658, 807426463, 3518610508, 15400996653

Offset: 0

Author

Peter Kagey and Bert Dobbelaere, Jun 25 2025

Keywords

• nonn
• hard
• more

Comments

These are "free polyforms" because they are counted up to rotation and reflection.

The triangular pyramidille is dual to the cantitruncated cubic honeycomb.

The polyhedral cells are each 1/24 of a cube and are similar to the convex hull of (0,0,0), (2,0,0), (1,1,0), and (1,1,1).

Links

• Peter Kagey, Animation illustrating the a(5)=39 connected 5-cell components.
• Wikipedia, Architectonic and catoptric tessellation
• Wikipedia, Cubic honeycomb

Crossrefs

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).