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A166083 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. 3
8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 45, 48, 54, 60, 64, 72, 75, 80, 84, 90, 96, 100, 105, 112, 120, 125, 128, 140, 144, 150, 160, 168, 175, 180, 192, 200, 210, 216, 225, 240, 245, 252, 256, 270, 280, 288, 294, 300, 315, 324, 336, 343, 350, 360, 378 (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..57.

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). 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

12 12

16 16

18 18

20 20

24 24

26 27

28 28

30 30

32 32

PROG

(Java)

int lastMax = 0;

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;

   System.out.println(voxels + " " + max);

  }

}

CROSSREFS

Cf. A166082 (full sequence without conditions), A166084 (sequence where the enclosed empty space must increase).

Sequence in context: A111087 A341610 A337373 * A326111 A091375 A282671

Adjacent sequences:  A166080 A166081 A166082 * A166084 A166085 A166086

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

STATUS

approved

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Last modified November 30 19:02 EST 2021. Contains 349424 sequences. (Running on oeis4.)