Equilateral triangles in lattice cube

[Pfoertner, Hugo]

SeFans,

please throw away a part of the results below; the "alltri" and "obtuse"
figures are wrong starting at n=5.
The "needle"-test in my program is not sharp enough to distinguish between
nearly aligned and true aligned configs. I'll replace it by an exact area
computation of the candidate triangle. I think I can compute terms up to
n=25.

With a similar program (just 4 loops instead of 3) + test for equal edge
length I computed the sequence

2*number of regular tetrahedra that can be formed
using the points in an (n+1) X (n+1) X (n+1) lattice cube:

1,9,36,104,257,549,1058,1896,3199,5145,7926,11768,16967,23859,32846,44378,

Hugo Pfoertner


-----Original Message-----
From: Pfoertner, Hugo [mailto:Hugo.Pfoertner at muc.mtu.de] 
Sent: Monday, February 07, 2005 17:27
To: seqfan at ext.jussieu.fr
Subject: RE: Equilateral triangles in lattice cube

Why not also count other kinds of triangles? I've extended my program by a
few more counts.

All possible triangles, isosceles, equilateral, acute, obtuse, right:


 n=           4
 alltri:                315892 /4=   78973
 isoscl:                 33628 /4=   8407
 equila:                  1264 /8=   158
 acute:                 117424 /4=   29356
 obtuse:                135192 /4=   33798
 righta:                 63276 /12=   5273

 n=           5
 alltri:               1650592 /4=   412648  wrong, correct: 1650664 
 isoscl:                128008 /4=   32002
 equila:                  3448 /8=   431
 acute:                 648760 /4=   162190
 obtuse:                759792 /4=   189948  wrong, correct: 759864
 righta:                242040 /12=   20170

.....

Insufficient test:
=================

C check for 3 points in a line
          needle = abs(( sqrt(float(dma)) -
     &               sqrt(float(dmi)) - sqrt(float(dmid)) )) .lt. 0.01
          if ( .not. needle ) obtuse = obtuse + 1
        endif
        if ( needle ) then
C not counted as triangle
          alltri = alltri - 1