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

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

Offset

0,3

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.

Also, 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).

Also, number of n-colorings of the vertices of the truncated cube (equivalently faces of the triakis octahedron) up to rotational octahedral symmetry.

Also, number of n-colorings of the rhombicuboctahedron (equivalently faces of the deltoidal icositetrahedron) up to rotational octahedral symmetry. - Peter Kagey, Jun 12 2026

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Peter Kagey and William Keehn, Escher's Cubes: Tiling the Faces of Polyhedra, arXiv:2606.26140 [math.GM], 2026. See pp. 5, 21, 23, 25.

Wikipedia, Pentagonal icositetrahedron

Wikipedia, Snub cube

Formula

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

Asymptotically, a(n) ~ n^24/24.

Mathematica

A378475[n_] := n^6*(n^18 + 9*n^6 + 8*n^2 + 6)/24; Array[A378475, 15, 0] (* Paolo Xausa, Jun 16 2026 *)

Crossrefs

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

Sequence in context: A393635 A204496 A332850 * A183835 A273969 A203834

Adjacent sequences: A378472 A378473 A378474 * A378476 A378477 A378478

Keyword

nonn,easy,changed

Author

Peter Kagey, Nov 27 2024

Status

approved