directions

[vdmcc]

number of 3D integer direction vectors (a,b,c) towards grid points at
integer distance (2n-1)
with a >= b >= c >= 0. A vector is a direction vector if its non-zero
components
have their GCD = 1.

remark : no direction vectors can exist at even distances (easy to prove).

{1,1,1,1,2,2,2,2,3,3,4,3,3,5,4,4,7,4,5,6,6,6,6,6,7,9,7,6,11,8,8,12,7,9,12,9,
  10,10,12,10,14,11,9,14,12,12,16,10,13,19,13,13,16,14,14,18,15,12,18,16,17,

21,13,16,23,17,20,18,18,18,24,18,15,28,19,19,25,16,20,26,24,21,24,21,20,31,

22,20,31,23,23,30,19,25,36,24,25,24,25,25,35,28,21,36,31,27,36,22,32,37,25,
  28,30,29,29,48,30,24,40,30,31,41,28,30,43,32,36,32}

same with a, b and c >=0  (all vectors in first octant = all permutations)
{3,3,6,6,9,9,12,12,15,15,24,18,18,27,24,24,36,24,30,36,33,33,36,36,42,48,42,

36,60,45,48,72,42,51,72,54,57,60,72,60,81,63,54,84,69,72,96,60,75,108,78,78,
  96,81,84,108,87,72,108,96,99,120,78,96,132,99,120,108,105,105,144,108,90,

168,114,114,144,96,120,156,144,123,144,126,120,180,132,120,180,135,138,180,

114,144,216,144,147,144,150,150,204,168,126,216,180,159,216,132,192,216,150,
  168,180,171,174,288,177,144,240,180,183,243,168,180,252,189,216,192}

same over all integers. (all directions in 3D = all signed permutations)
{6,24,24,48,72,72,72,96,96,120,192,144,120,216,168,192,288,192,216,288,240,

264,288,288,336,384,312,288,480,360,360,576,288,408,576,432,432,480,576,480,

648,504,384,672,528,576,768,480,576,864,600,624,768,648,648,864,672,576,864,

768,792,960,600,768,1056,792,960,864,816,840,1152,864,672,1344,888,912,1152,
  768,936,1248,1152,984,1152,1008,936,1440,1032,960,1440,1080,1080,1440,864,
  1152,1728,1152,1152,1152,1176,1200,1632,1344,960,1728,1440,1272,1728,1056,

1536,1728,1152,1344,1440,1368,1368,2304,1392,1152,1920,1440,1440,1944,1344,
  1440,2016,1512,1728,1536}
example:
the 4 directions towards points at distance 29 : n=15 are (signed
permutations of)
[24,16,3],[24,12,11],[21,20,0],[21,16,12]
the vector [29,0,0] is no direction : multiple of [1,0,0]

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