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).

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

Views

Author

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

Keywords

Links

Crossrefs

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

Extensions

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

A038169 Number of "connected animals" formed from n triakis truncated tetrahedra connected along hexagonal faces in the triakis truncated tetrahedral honeycomb, allowing translations, rotations, and reflections of the honeycomb.

Original entry on oeis.org

1, 1, 1, 3, 7, 24, 88, 385, 1713, 8112, 38869, 190128, 938357
Offset: 1

Views

Author

Achim Flammenkamp

Keywords

Comments

Previous name was 'Number of "connected animals" formed from n tricapped truncated tetrahedra in the diamond lattice, allowing translation and rotations of the lattice and reflections.' - Peter Kagey, May 30 2025

References

  • A. T. Balaban and Paul von R. Schleyer, "Graph theoretical enumeration of polymantanes", Tetrahedron, (1978), vol. 34, pp. 3599-3609. See Page 3605.

Links

Crossrefs

Cf. A000162, A038119, A038168, A038170, A038171, A038172, A038173, A038174.

Extensions

Name changed by Peter Kagey, May 30 2025

A300812 Irregular triangle T(n,c) read by rows: the number of clusters of n spheres centered on f.c.c. lattice sites with c contacts.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 0, 0, 0, 13, 4, 2, 1, 0, 0, 0, 0, 75, 35, 16, 3, 2, 0, 0, 0, 0, 0, 557, 384, 184, 54, 24, 5, 2, 1, 0, 0, 0, 0, 0, 0, 4808, 4230, 2354, 834, 355, 104, 37, 9, 2, 1, 0, 0, 0, 0, 0, 0, 0, 44334, 47328, 30517, 13081, 5716, 2083, 749, 253, 70, 20, 4, 3
Offset: 1

Views

Author

R. J. Mathar, Mar 13 2018

Keywords

Comments

Each of the n spheres is centered at a different vertex of the face-centered-cubic lattice.
"Clusters of spheres" means that the (simple, undirected) graph, which has edges for each "bond" where two spheres touch, is connected. The number of contacts equals the number of edges in the graph.
T(n,c) counts "free" clusters, which means that clusters are considered distinct if they cannot be mapped onto each other by one of the 48 symmetries of the octahedral group plus shifts along the directions of the face centers.
Since there are 12 nearest neighbors in the f.c.c. lattice, the degree of each vertex in the contact graph is in the interval [1,12].
T(n,c) = 0 for c < n: the minimum number of contacts is realized with chains of spheres, where the two spheres at the ends have degree 1 and the others degree 2.
The row length of row n of the triangle should equal 1+A214813(n) ... but does not for n=5, see the comment in A214813.

Examples

			The values T(n,c) start with n=1 sphere for 0 <= c contacts as:
  1
  0 1
  0 0 3 1
  0 0 0 13 4 2 1
  0 0 0 0 75 35 16 3 2
  0 0 0 0 0 557 384 184 54 24 5 2 1
  0 0 0 0 0 0 4808 4230 2354 834 355 104 37 9 2 1
  0 0 0 0 0 0 0 44334 47328 30517 13081 5716 2083 749 253 70 20 4 3
T(2,1) = 1 because there is one cluster with two spheres which touch each other at one point: (0,0,0), (1/2,0,1/2).
T(3,2) = 3 counts three spheres in three different geometries: (i) (0,0,0), (1/2,1/2,0), (1,0,1), linear, (ii) (0,0,0), (1/2,1/2,0), (1,1,0) with 90-degree bond angle, (iii) (0,0,0), (1/2,1/2,0), (1,1/2,1/2) with 120-degree bond angle.
T(3,3) = 1 counts the planar triangular canonball base arrangement: (0,0,0), (1/2,1/2,0), (1/2,0,1/2).

Links

Crossrefs

Cf. A038173 (row sums), A214813.
Previous Showing 11-20 of 25 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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