

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 n1 and also skipping values of n for which volume(n)n <= volume(n1)(n1). 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
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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.
Liyao 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  (w1)*(h1)*(d1)). Keep only those N for which Volume(N)>Volume(N1) and Volume(N)N > Volume(N1)(N1). 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(N1) 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 Liyao Xia, Nov 02 2014


STATUS

approved



