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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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%I #64 Jun 09 2021 02:38:16

%S 0,0,3,28,116,340,847,1832,3570,6440,10889,17518,26966,40002,57601,

%T 80868,111186,150032,199147,260456,336080,428290,539709,673130,831436,

%U 1018154,1237155,1492352,1787780,2129250,2521323,2969584,3479302,4056636,4707661,5438808

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

%C 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.

%H Peter Kagey, <a href="/A338791/b338791.txt">Table of n, a(n) for n = 0..100</a>, based on the b-files for A098928, A103158, and A178797.

%H Eugen J. Ionascu and Andrei Markov, <a href="http://dx.doi.org/10.1016/j.jnt.2010.07.008">Platonic solids in Z^3</a>, Journal of Number Theory, Volume 131, Issue 1, January 2011, Pages 138-145.

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

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

%K nonn

%O 0,3

%A _Peter Kagey_, Dec 05 2020