A065984 Volume (multiplied by 3) of polyhedron formed by points (i,j,k) in Z^3 with i^2+j^2+k^2 = n.

0, 4, 20, 24, 32, 96, 136, 0, 160, 272, 244, 344, 192, 404, 516, 0, 256, 712, 744, 616, 768, 988, 644, 0, 1088, 1156, 1484, 1272, 0, 1740, 1628, 0, 1280, 1992, 2092, 2088, 2176, 1360, 2608, 0, 1952, 3032, 2476, 2408, 2752, 3344, 3136, 0, 1536, 3692, 3948

Offset: 0

Author

Wouter Meeussen, Dec 10 2001

Keywords

• nonn
• nice

Comments

The polytopes with zero volume occur when n = {0,7,15,23,28,31,39,47,55,60,63, ...} = A004215. See A065089 for the subset of these polyhedra with extra condition that all points should be at an integer distance from the origin; a(n^2) = A065089(n)

Examples

			a(5) = 96 because all permutations of {0, +/- 1, +/- 2} produce a polytope with volume 96/3, so the triple volume is 96.
		

Crossrefs

Cf. A065089, A004215.

Programs

Mathematica

polyhedra = Flatten[ forms/@SumOfSquaresRepresentations[ 3, # ], 1 ]&/@(Range[ 1, 36 ]); HullVolume[ #, ConvexHull3D[ # ] ]&/@polyhedra; (* Cf. A065089 *)