A166084 Maximal volume of a closed box created by using at most n voxels as the boundary, skipping values of n for which the volume is the same as for n-1 and also skipping values of n for which volume(n)-n <= volume(n-1)-(n-1). In other words, the empty enclosed space must increase.

8, 75, 84, 90, 105, 108, 120, 126, 135, 140, 144, 147, 160, 168, 180, 192, 196, 210, 216, 224, 240, 245, 252, 256, 270, 280, 288, 300, 315, 320, 336, 350, 360, 378, 384, 400, 405, 420, 432, 440, 450, 462, 480, 486, 495, 500, 504, 528, 540, 560, 567, 576, 594

Offset

1,1

Comments

For example, a 3 X 3 X 3 box can be created by using top and bottom plates of 3 X 3 X 1 voxels, and using 8 voxels to connect them, totaling 26 voxels.

Links

Table of n, a(n) for n=1..53.

Li-yao Xia, Examples of (N, Volume(N)), N<1000

Formula

For each N (starting at 8), calculate the max Volume(N)=w*h*d such that (N <= (w*h*d - (w-1)*(h-1)*(d-1)). Keep only those N for which Volume(N)>Volume(N-1) and Volume(N)-N > Volume(N-1)-(N-1). The minimum box is 2 X 2 X 2 voxels to prevent overlapping voxels (multiple voxels occupying the same location in space) or degenerate cases.

Example

N Volume
8 8
26 27
34 36
42 45
44 48
54 60
56 64
66 75
68 80
80 96

Prog

(Java)
int lastMax = 0;
int lastGain = -1;
for (int voxels = 8; voxels <= 1000; 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;
      }
    }
  }
  if (lastMax < max)
  {
    lastMax = max;
    int gain = max - voxels;
    if (lastGain < gain)
    {
      lastGain = gain;
      System.out.println("%e A166084 " + voxels + " " + max);
    }
  }
}

Crossrefs

Cf. A166082 (full sequence without conditions), A166083 (sequence with only the Volume(N)>Volume(N-1) condition).

Sequence in context: A357208 A163970 A116251 * A123661 A288487 A123003

Adjacent sequences: A166081 A166082 A166083 * A166085 A166086 A166087

Keyword

nonn

Author

Mark Jeronimus, Oct 06 2009, Dec 01 2009

Extensions

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

Long example moved to a file and code indented by Li-yao Xia, Nov 02 2014

Status

approved