A338791 a(n) is the number of Platonic solids in three dimensions with all vertices (x,y,z) in the set {1,2,...,n}^3.

0, 0, 3, 28, 116, 340, 847, 1832, 3570, 6440, 10889, 17518, 26966, 40002, 57601, 80868, 111186, 150032, 199147, 260456, 336080, 428290, 539709, 673130, 831436, 1018154, 1237155, 1492352, 1787780, 2129250, 2521323, 2969584, 3479302, 4056636, 4707661, 5438808

Offset

0,3

Comments

Dodecahedra and icosahedra with integer coordinates cannot be formed in Euclidean space (of any dimension) because pentagons with integer coordinates cannot be formed in Euclidean space, and both polyhedra contain a subset of vertices that form a pentagon. Therefore, this sequence counts the regular tetrahedra, cubes, and octahedra in the bounded cubic lattice.

Links

Peter Kagey, Table of n, a(n) for n = 0..100, based on the b-files for A098928, A103158, and A178797.

Eugen J. Ionascu and Andrei Markov, Platonic solids in Z^3, Journal of Number Theory, Volume 131, Issue 1, January 2011, Pages 138-145.

Formula

a(n) = A098928(n) + 2*A103158(n-1) + A178797(n-1) for n >= 2.

Crossrefs

Cf. A098928 (cubes), A103158 (tetrahedra), A178797 (octahedra), A338323 (regular polygons).

Sequence in context: A165393 A107651 A239057 * A100019 A053132 A316390

Adjacent sequences: A338788 A338789 A338790 * A338792 A338793 A338794

Keyword

nonn

Author

Peter Kagey, Dec 05 2020

Status

approved