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.

Previous Showing 11-17 of 17 results.

A378474 The number of n-colorings of the vertices of the truncated cuboctahedron up to rotation and reflection.

Original entry on oeis.org

0, 1, 5864068667776, 1661800897546646288751, 1650586719047285117763813376, 74014868308343792955106160546875, 467755368903219944377426648894114176, 764653504526960946768130306131125170501, 464598858302721315450530067459906444722176
Offset: 0

Views

Author

Peter Kagey, Nov 27 2024

Keywords

Comments

Equivalently, the number of n-colorings of the faces of the disdyakis dodecahedron, which is the polyhedral dual of the truncated cuboctahedron.
Colorings are counted up to the full octahedral group of order 48.

Links

Crossrefs

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378475, A378476, A378477, A378478.

Formula

a(n) = (1/48)*(n^48 + 19*n^24 + 8*n^16 + 12*n^12 + 8*n^8).
Asymptotically, a(n) ~ n^48/48.

A378475 The number of n-colorings of the vertices of the snub cube up to rotation.

Original entry on oeis.org

0, 1, 700688, 11768099013, 11728130343936, 2483526957328125, 197432556580265616, 7982551312716034313, 196765270145344012288, 3323601794975613468921, 41666666667041700250000, 410405528159827444816781, 3312368633477962187301888, 22616698765607508420521013
Offset: 0

Views

Author

Peter Kagey, Nov 27 2024

Keywords

Comments

Equivalently, the number of n-colorings of the faces of the pentagonal icositetrahedron, which is the polyhedral dual of the snub cube.
Colorings are counted up to the rotational octahedral symmetry group of order 24.
This is also:
1) The number of n-colorings of the vertices of the truncated octahedron (equivalently faces of the tetrakis hexahedron) up to rotational octahedral symmetry (alternatively full tetrahedral symmetry).
2) The number of n-colorings of the vertices of the truncated cube (equivalently faces of the triakis octahedron) up to rotational octahedral symmetry.

Links

Crossrefs

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378474, A378476, A378477, A378478.

Formula

a(n) = (1/24)*(n^24 + 9*n^12 + 8*n^8 + 6*n^6).
Asymptotically, a(n) ~ n^24/24.

A378476 The number of n-colorings of the vertices of the truncated dodecahedron up to rotation and reflection.

Original entry on oeis.org

0, 1, 9607679885269312, 353259652293727442874919719, 11076899964874301400431118585745408, 7228014483236696229750911410649667971875, 407280649839077145745380578110103790290896704, 4233515506163528044351709372473136729199352546645
Offset: 0

Views

Author

Peter Kagey, Nov 27 2024

Keywords

Comments

Equivalently,
1) the number of n-colorings of the faces of the triakis icosahedron, which is the polyhedral dual of the truncated dodecahedron.
2) the number of n-colorings of the faces of the pentakis dodecahedron, or n-colorings of the vertices of the truncated icosahedron, its polyhedral dual.
3) the number of n-colorings of the faces of the deltoidal hexecontahedron, or n-colorings of the vertices of the rhombicosidodecahedron, its polyhedral dual.
Colorings are counted up to the full icosahedral symmetry group of order 120.

Links

Crossrefs

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378474, A378475, A378477, A378478.

Formula

a(n) = (1/120)*(n^60 + 15*n^32 + 16*n^30 + 20*n^20 + 24*n^12 + 20*n^10 + 24*n^6).
Asymptotically, a(n) ~ n^60/120.

Extensions

a(0) = 0 prepended by Georg Fischer, Apr 16 2025

A378477 The number of n-colorings of the vertices of the truncated icosidodecahedron up to rotation and reflection.

Original entry on oeis.org

0, 1, 11076899964874299238703297447907328, 14975085832620260086776498590197757887552760437584786915, 14723725539819869413194145839524321308612931385268246121155792029614080, 6269303204385533375833261531851976948366440371233447120478861810030555725146484375
Offset: 0

Views

Author

Peter Kagey, Nov 27 2024

Keywords

Comments

Equivalently, the number of n-colorings of the faces of the disdyakis triacontahedron, which is the polyhedral dual of the truncated octahedron.
Colorings are counted up to the full icosahedral symmetry group of order 120.

Links

Crossrefs

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378474, A378475, A378476, A378478.

Formula

a(n) = 1/120*(n^120 + 31*n^60 + 20*n^40 + 24*n^24 + 20*n^20 + 24*n^12).
Asymptotically, a(n) ~ n^120/120

A378478 The number of n-colorings of the vertices of the snub dodecahedron up to rotation.

Original entry on oeis.org

0, 1, 19215358678900736, 706519304586988199183738259, 22153799929748598169960860333637632, 14456028966473392453665534687042333984375, 814561299678154291488767806377392301451223040, 8467031012327056088703142262372040966699399765293
Offset: 0

Views

Author

Peter Kagey, Nov 27 2024

Keywords

Comments

Equivalently, the number of n-colorings of the faces of the pentagonal hexecontahedron, which is the polyhedral dual of the snub dodecahedron.
Colorings are counted up to the rotational icosahedral symmetry group of order 60.

Links

Crossrefs

Cf. A000332, A060530, A128766, A199406, A252704, A252705, A274900, A337963, A378473, A378474, A378475, A378476, A378477.

Formula

a(n) = 1/60*(n^60 + 15*n^30 + 20*n^20 + 24*n^12).
Asymptotically, a(n) ~ n^60/60.

A333418 Irregular triangle: T(n,k) gives the number of ways to 2-color k edges of the n-cube up to rotation and reflection, with 0 <= k <= A001787(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 9, 18, 24, 30, 24, 18, 9, 4, 1, 1, 1, 1, 6, 24, 140, 604, 2596, 9143
Offset: 1

Views

Author

Peter Kagey, Mar 20 2020

Keywords

Comments

Conjecture: All rows are unimodal (increasing, then decreasing).
Each row is a palindrome.
A333333 is analogous with the restriction that the colorings must be connected.

Examples

			Table begins:
n\k| 0  1   2   3    4    5     6     7   8  9 10 11 12 ...
---+-------------------------------------------------------
  1| 1, 1;
  2| 1, 1,  2,  1,   1;
  3| 1, 1,  4,  9,  18,  24,   30,   24, 18, 9, 4, 1, 1;
  4| 1, 1,  6, 24, 140, 604, 2596, 9143, ...
  5| 1, 1,  8, 50, 608, ...
  6| 1, 1, 10, 89, ...

Links

Crossrefs

Row sums are A333444.
Cf. A001787, A060530, A199406, A331359, A333444, A333333.

Formula

T(n,k) >= ceiling(binomial(A001787(n),k)/A000165(n)).

A306194 Non-isomorphic colorings of the edges of a cube using at most n colors under rotational symmetries and permutations of the colors.

Original entry on oeis.org

1, 114, 3891, 29854, 87981, 143797, 170335, 177160, 178153, 178243, 178248, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249, 178249
Offset: 1

Views

Author

Marko Riedel, Jan 28 2019

Keywords

Comments

This uses Power Group Enumeration (PGE). The sequence ceases to grow once it reaches a(12) = 178249 because at most twelve colors can be represented and in a coloring with additional colors beyond the initial twelve the new colors can be permuted out and replaced by colors from the initial set, forming part of an orbit that has already been counted.

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973.

Links

Crossrefs

Cf. A060530.
Previous Showing 11-17 of 17 results.

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

Content is available under The OEIS End-User License Agreement.