A065089 Volume (multiplied by 3) of polyhedron formed by points (i,j,k) in Z^3 with i^2+j^2+k^2 = n^2.

0, 4, 32, 272, 256, 1156, 2176, 3692, 2048, 8496, 9248, 15196, 17408, 22324, 29536, 39820, 16384, 56144, 67968, 79252, 73984, 111956, 121568, 143176, 139264, 184852, 178592, 238884, 236288, 285940, 318560, 358004, 131072, 435396, 449152

Offset

0,2

Comments

For n=2^k, a(n)=4 n^3 because A016727(2^k) = SumOfSquaresRepresentations[3,(2^k)^2] contains only {0,0,2^k}.

This is why a(16) and a(32) are visibly so much smaller than their neighbors when you look at the graph. [Jonathan Vos Post, Apr 22 2011]

Links

Table of n, a(n) for n=0..34.

Example

a(2) = 32 because the volume of the polyhedron formed by all integer points at distance 2 from the origin, {{-2, 0, 0}, {0, -2, 0}, {0, 0, -2}, {0, 0, 2}, {0, 2, 0}, {2, 0, 0}}, is 32/3.

Mathematica

forms[ z:{_Integer, _, _} ] := Union[ Flatten[ Permutations/@(Times[ z, # ]&/@Flatten[ Outer[ List, {1, -1}, {1, -1}, {1, -1} ], 2 ]), 1 ] ]; polyhedra=Flatten[ forms/@SumOfSquaresRepresentations[ 3, # ], 1 ]&/@(Range[ 1, 36 ]^2); HullVolume[ #, ConvexHull3D[ # ] ]&/@polyhedra;

Crossrefs

Cf. A016727.

Sequence in context: A363440 A009509 A036725 * A113329 A246818 A145710

Adjacent sequences: A065086 A065087 A065088 * A065090 A065091 A065092

Keyword

nonn

Author

Wouter Meeussen, Nov 10 2001

Status

approved