cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

1, 1, 2, 3, 9, 17, 60, 166, 606, 2106, 8046, 30801, 122442, 491539, 2007571, 8272122, 34408439, 144084776, 607112043, 2571118048, 10938419260, 46720437135
Offset: 0

Views

Author

Peter Kagey and Bert Dobbelaere, Jun 25 2025

Keywords

Comments

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).

Links

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).

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

Views

Author

Peter Kagey and Bert Dobbelaere, Jun 25 2025

Keywords

Comments

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).

Links

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).

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

Views

Author

Peter Kagey and Bert Dobbelaere, Jun 25 2025

Keywords

Comments

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).

Links

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).

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

Views

Author

Peter Kagey and Bert Dobbelaere, Jun 25 2025

Keywords

Comments

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).

Links

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).

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

Views

Author

Peter Kagey and Bert Dobbelaere, Jun 25 2025

Keywords

Comments

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).

Links

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).

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

Views

Author

Peter Kagey and Bert Dobbelaere, Jun 25 2025

Keywords

Comments

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.

Links

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).

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

Views

Author

Peter Kagey and Bert Dobbelaere, Jun 25 2025

Keywords

Comments

These are "free polyforms" because they are counted up to rotation and reflection.
The triangular prismatic honeycomb is dual to the hexagonal prismatic honeycomb.

Links

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).

A365655 Number of fixed n-polyominoids, allowing right-angled connections only ("hard" polyominoids).

Original entry on oeis.org

3, 12, 68, 438, 3054, 22417, 170610, 1334316
Offset: 1

Views

Author

Pontus von Brömssen, Sep 17 2023

Keywords

Comments

Apparently, the definition of "hard" polyominoid in the Wikipedia article differs from the definition used here. Here, two squares are allowed to meet in a straight 180-degree connection provided that the structure be connected through right-angled ("hard") connections only; see A365654 for further details.

Links

Crossrefs

Cf. A075678 (polyominoids), A365654 (free).
13th and 17th row of A366767.

A366338 Number of free (4,2)-polyominoids with n cells, allowing right-angled (or hard) connections only.

Original entry on oeis.org

1, 1, 8, 44, 509, 7091
Offset: 1

Views

Author

Pontus von Brömssen, Oct 07 2023

Keywords

Comments

Two squares sharing an edge have a right-angled connection if they do not lie in the same plane.
Connections that are not right-angled (flat connections) may occur, but the polyominoids considered here must be connected through right-angled connections only.

Links

Crossrefs

45th row of A366766.
Cf. A075679, A365654, A366334, A366339 (fixed), A366340.

A366340 Number of free (4,3)-polyominoids with n cells, allowing right-angled (or hard) connections only.

Original entry on oeis.org

1, 1, 5, 19, 123, 954, 9324
Offset: 1

Views

Author

Pontus von Brömssen, Oct 07 2023

Keywords

Comments

Two cubes sharing a face have a right-angled connection if they do not lie in the same 3-dimensional affine subspace.
Connections that are not right-angled (flat connections) may occur, but the polyominoids considered here must be connected through right-angled connections only.
Also, number of free polysticks in 4 dimensions with right-angled connections.

Links

Crossrefs

41st and 77th row of A366766.
Cf. A075679, A365654, A366336, A366338, A366341 (fixed).
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