A166082 Maximal volume of a closed box created by using at most n voxels as the boundary.

8, 8, 8, 8, 12, 12, 12, 12, 16, 16, 18, 18, 20, 20, 20, 20, 24, 24, 27, 27, 28, 28, 30, 30, 32, 32, 36, 36, 36, 36, 36, 36, 40, 40, 45, 45, 48, 48, 48, 48, 48, 48, 54, 54, 54, 54, 60, 60, 64, 64, 64, 64, 64, 64, 64, 64, 72, 72, 75, 75, 80, 80, 80, 80, 80, 80, 84, 84, 84, 84, 90

Offset

8,1

Comments

For example, a 3x3x3 box can be created by using top and bottom plates of 3x3x1 voxels, and using 8 voxels to connect them, totaling 26 voxels.

Links

Table of n, a(n) for n=8..78.

Formula

For a given value of N (at least 8), calculate the max Volume(N)=w*h*d such that (N <= (w*h*d - (w-1)*(h-1)*(d-1)). The minimum box is 2x2x2 voxels to prevent overlapping voxels (multiple voxels occupying the same location in space) or degenerate cases.

Prog

(Java) // input: int voxels int max = 0;
for (int depth = voxels / 4; depth >= 2; depth--)
{
    for (int width = voxels / (2 * depth); width >= 2; width--)
    {
        int remaining = voxels - 2 * width * depth;
        int height = 2 + remaining / (2 * ((width - 1) + (depth - 1)));
        int volume = width * depth * height;
        if (max < volume)
        {
            max = volume;
        }
    }
}

Crossrefs

Cf. A166083 (Sequence with only the Volume(N)>Volume(N-1) condition), A166084 (Sequence where the enclosed empty space must increase)

Sequence in context: A165926 A216865 A216412 * A145446 A248762 A186985

Adjacent sequences: A166079 A166080 A166081 * A166083 A166084 A166085

Keyword

base,nonn

Author

Mark Jeronimus, Oct 06 2009, Dec 01 2009

Extensions

Minor edits by N. J. A. Sloane, Dec 05 2009

Status

approved