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

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

Offset: 0

Author

Peter Kagey, Nov 27 2024

Keywords

• nonn
• easy

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

• Wikipedia, Deltoidal hexecontahedron
• Wikipedia, Pentakis dodecahedron
• Wikipedia, Rhombicosidodecahedron
• Wikipedia, Triakis icosahedron
• Wikipedia, Truncated dodecahedron
• Wikipedia, Truncated icosahedron

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