A237707 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, completely within the first octant of a sphere centered at the origin, ordered by increasing radius.

1, 4, 7, 10, 11, 17, 20, 23, 26, 32, 35, 38, 44, 48, 54, 60, 66, 69, 75, 78, 87, 96, 102, 105, 108, 114, 120, 121, 127, 133, 139, 145, 157, 163, 169, 178, 184, 196, 202, 214, 217, 220, 232, 238, 241, 244, 256, 263, 266, 278, 284, 296, 299, 308, 314, 329, 332

Offset: 1

Author

Rajan Murthy, Feb 11 2014

Keywords

• nonn

Examples

			When the radius of the sphere reaches 3^(1/2), one cube is completely within the sphere. When the radius reaches 6^(1/2), four cubes are completely within the sphere.
		

Links

• Rajan Murthy, Table of n, a(n) for n = 1..200
• Rajan Murthy, Table of n, a(n), and squared radius for n = 1..200
• Rajan Murthy, Scilab program for this sequence
• Charles R Greathouse IV, Illustration of this sequence

Crossrefs

The radii corresponding to the terms are given by the square roots of A000408 starting with squared radius 3.

Cf. A232499 (2-dimensional analog).

Partial sums of A014465 and A063691 (but then with repeated terms omitted).

Programs

Mathematica

(* Illustrates the sequence *)
Cube[x_,y_,z_]:=Cuboid[{x-1,y-1,z-1},{x,y,z}]
Cubes[r_]:=Cube@@#&/@Select[Flatten[Table[{x,y,z},{x,1,r},{y,1,r},{z,1,r}],2],Norm[#]<=r&]
Draw[r_]:=Graphics3D[Union[Cubes[r],{{Green, Opacity[0.3], Sphere[{0,0,0},r]}}],PlotRange->{{0,r},{0,r},{0,r}},ViewPoint->{r,3r/4,3r/5}];
Draw/@Sqrt/@{3,6,9,11,12,14} (* Charles R Greathouse IV, Mar 12 2014 *)

Scilab

// See Murthy link.

Formula

a(n) ~ (Pi*sqrt(30)/25)*n^(3/2). - Charles R Greathouse IV, Mar 14 2014

Extensions

Duplicate terms deleted by Rajan Murthy, Mar 06 2014

Terms a(36) and beyond added from b-file by Andrew Howroyd, Feb 27 2018