A178797 -id:A178797 - cp's OEIS frontend

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

Peter Kagey, Dec 05 2020

nonn

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.

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.

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

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

cp's OEIS Frontend

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.

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