A038119 -id:A038119 - cp's OEIS frontend

A385271 Number of face-connected components of square pyramidal cells in the hexakis cubic honeycomb up to translation, rotation, and reflection of the honeycomb.

1, 1, 2, 3, 9, 17, 60, 166, 606, 2106, 8046, 30801, 122442, 491539, 2007571, 8272122, 34408439, 144084776, 607112043, 2571118048, 10938419260, 46720437135

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The "hexakis cubic honeycomb" is also called the "pyramidille" and is dual to the truncated cubic honeycomb.
The square pyramidal cells are similar to the convex hull of (0,0,0), (2,0,0), (0,2,0) (2,2,0), and (1,1,1).

Peter Kagey, Animation illustrating the a(5)=17 connected 5-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Tetragonal disphenoid honeycomb, see section "Hexakis cubic honeycomb."

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).

A385272 Number of face-connected components of phyllic disphenoidal cells in the phyllic disphenoidal honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 4, 13, 38, 141, 515, 2043, 8176, 33706, 140471, 593705, 2531933, 10893811, 47202599, 205843902, 902644191, 3977976135, 17609163491

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The "phyllic disphenoidal honeycomb" is also called the "eighth pyramidille," and its dual is the omnitruncated cubic honeycomb.
The phyllic disphenoidal cells are similar to the convex hull of (0,0,0), (1,0,0), (1,1,0), and (1,1,1).

Peter Kagey, Animation illustrating the a(5)=38 connected 5-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Tetragonal disphenoid honeycomb, see section "Phyllic disphenoidal honeycomb."

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).

A385273 Number of face-connected components of polyhedral cells in the quarter oblate octahedrille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 6, 24, 114, 647, 3883, 24605, 159837, 1060450, 7137627, 48624639, 334475495, 2319909330, 16205238283

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The quarter oblate octahedrille is dual to the cantellated cubic honeycomb.
The cells of the quarter oblate octahedrille are similar to the convex hull of (0,0,0), (1,0,0), (0, 1, 0), (1,1,1), and (1,1,-1).

Peter Kagey, Animation illustrating the a(4)=24 connected 4-cell components.
Wikipedia, Architectonic and catoptric tessellation

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).

A385274 Number of face-connected components of rhombic pyramidal cells in the rhombic pyramidal honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 4, 14, 43, 197, 850, 4154, 20371, 103405, 530355, 2760533, 14499363, 76842876, 410164367, 2203491401, 11903591737

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The "rhombic pyramidal honeycomb" is also called the "half oblate octahedrille" and is dual to the cantic cubic honeycomb, which is also called the "truncated tetraoctahedrille" or the "truncated tetrahedral-octahedral honeycomb"
The rhombic pyramidal cells are similar to the convex hull of (0,0,0), (1,1,1), (1,1,-1), (0,2,0), and (2,0,0).

Peter Kagey, Animation illustrating the a(4)=14 connected 4-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Rhombic dodecahedral honeycomb, see section "Rhombic pyramidal honeycomb."

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).

A385275 Number of face-connected components of irregular pyramidal cells in the square quarter pyramidille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 3, 6, 26, 92, 441, 2025, 10141, 51131, 264938, 1387761, 7364492, 39433242, 212959457, 1158325878, 6341136682, 34911146404

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The dual of the square quarter pyramidille is the runcitruncated cubic honeycomb.
Each irregular pyramidal cell is similar to the convex hull of (0,0,0), (0,0,1), (0,1,0), (0,1,1), and (1,1,1).

Peter Kagey, Animation illustrating the a(4)=26 connected 4-cell components.
Wikipedia, Architectonic and catoptric tessellation

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).

A385276 Number of face-connected components of trapezo-rhombic dodecahedral cells in the trapezo-rhombic dodecahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 9, 57, 460, 4641, 50353, 575375, 6754382, 80887484, 982952256, 12087512169

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

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

The trapezo-rhombic dodecahedral honeycomb is dual to the gyrated tetrahedral-octahedral honeycomb.

Peter Kagey, Animation illustrating the a(4)=57 connected 4-cell components.
Wikipedia, Rhombic dodecahedral honeycomb, see section "Trapezo-rhombic dodecahedral honeycomb."
Wikipedia, Trapezo-rhombic dodecahedron

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).

A385277 Number of face-connected components of triangular prismatic cells in the triangular prismatic honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 3, 11, 30, 137, 606, 3243, 17681, 101718, 593931, 3532385, 21220273, 128680158, 785895888, 4830179751, 29847223514

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

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

The triangular prismatic honeycomb is dual to the hexagonal prismatic honeycomb.

Peter Kagey, Animation illustrating the a(5)=30 connected 5-cell components.
Wikipedia, Triangular prismatic honeycomb

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).

A038168 Number of "connected animals" formed from n tricapped truncated tetrahedra in the diamond lattice, allowing translation and rotations of the lattice.

Original entry on oeis.org

1, 1, 1, 4, 10, 39, 160, 726, 3344, 16004, 77323, 379117, 1874540

Offset: 1

Achim Flammenkamp

nonn

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.

Cf. A000162, A038119, A038169-A038174.

A038174 Number of "polyspheres", or "connected animals" formed from n rhombic dodecahedra (or edge-connected cubes) in the f.c.c. lattice, allowing translation and rotations of the lattice, reflections and 180 deg. rotations about a 3-fold symmetry axis of the lattice.

Original entry on oeis.org

1, 1, 4, 25, 210, 2209, 24651, 284768, 3360995, 40328652, 490455189

Offset: 1

Achim Flammenkamp, Torsten Sillke (TORSTEN.SILLKE(AT)LHSYSTEMS.COM)

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
Ishino Keiichiro, Polysphere, Puzzle will be played, 2008.
T. Sillke, Notations for polyspheres
Index entries for sequences related to f.c.c. lattice

Cf. A000162, A038119, A038168, A038169, A038170, A038171, A038172, A038173.

a(9) and a(10) from Achim Flammenkamp Feb 15 1999

a(11) from Ishino Keiichiro's website added by Andrey Zabolotskiy, Mar 03 2023

A038171 Number of "connected animals" formed from n 6-gon connected truncated octahedra (or corner connected cubes) in the b.c.c. lattice, allowing translation and rotations of the lattice and reflections.

Original entry on oeis.org

1, 1, 3, 12, 61, 407, 3226, 28335, 262091, 2501168, 24328920, 239931556

Offset: 1

Achim Flammenkamp

S. T. Coffin, Puzzling World of Polyhedral Dissections, Oxford Univ. Press, 1991.
Index entries for sequences related to b.c.c. lattice

Cf. A000162, A038119, A038168-A038174.

35th row of A366766.

a(10) and a(11) from Joerg Arndt and Márk Péter Légrádi, Apr 30 2023

cp's OEIS Frontend

A385271 Number of face-connected components of square pyramidal cells in the hexakis cubic honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385272 Number of face-connected components of phyllic disphenoidal cells in the phyllic disphenoidal honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385273 Number of face-connected components of polyhedral cells in the quarter oblate octahedrille up to translation, rotation, and reflection of the honeycomb.

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A385274 Number of face-connected components of rhombic pyramidal cells in the rhombic pyramidal honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385275 Number of face-connected components of irregular pyramidal cells in the square quarter pyramidille up to translation, rotation, and reflection of the honeycomb.

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A385276 Number of face-connected components of trapezo-rhombic dodecahedral cells in the trapezo-rhombic dodecahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385277 Number of face-connected components of triangular prismatic cells in the triangular prismatic honeycomb up to translation, rotation, and reflection of the honeycomb.

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A038168 Number of "connected animals" formed from n tricapped truncated tetrahedra in the diamond lattice, allowing translation and rotations of the lattice.

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A038174 Number of "polyspheres", or "connected animals" formed from n rhombic dodecahedra (or edge-connected cubes) in the f.c.c. lattice, allowing translation and rotations of the lattice, reflections and 180 deg. rotations about a 3-fold symmetry axis of the lattice.

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A038171 Number of "connected animals" formed from n 6-gon connected truncated octahedra (or corner connected cubes) in the b.c.c. lattice, allowing translation and rotations of the lattice and reflections.

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