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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A144850 A163970 A116251 * A123661 A288487 A123003

Adjacent sequences:  A166081 A166082 A166083 * A166085 A166086 A166087

KEYWORD

nonn

AUTHOR

Mark Jeronimus (mark.jeronimus(AT)gmail.com), 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

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)