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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 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: A337370 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 January 21 21:20 EST 2022. Contains 350480 sequences. (Running on oeis4.)