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Volume (multiplied by 3) of polyhedron formed by points (i,j,k) in Z^3 with i^2+j^2+k^2 = n.
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%I #7 Aug 04 2018 06:38:28

%S 0,4,20,24,32,96,136,0,160,272,244,344,192,404,516,0,256,712,744,616,

%T 768,988,644,0,1088,1156,1484,1272,0,1740,1628,0,1280,1992,2092,2088,

%U 2176,1360,2608,0,1952,3032,2476,2408,2752,3344,3136,0,1536,3692,3948

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

%C 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)

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

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

%Y Cf. A065089, A004215.

%K nonn,nice

%O 0,2

%A _Wouter Meeussen_, Dec 10 2001