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

1,2

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

Formula

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

Example

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.

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 *)

Prog

(Scilab) // See Murthy link.

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).

Sequence in context: A364840 A190507 A087298 * A211642 A127958 A000414

Adjacent sequences: A237704 A237705 A237706 * A237708 A237709 A237710

Keyword

nonn

Author

Rajan Murthy, Feb 11 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

Status

approved