A334881
Number of squares in 3-dimensional space whose four vertices have coordinates (x,y,z) in the set {1,...,n}.
Original entry on oeis.org
0, 0, 6, 54, 240, 810, 2274, 5304, 10752, 19992, 34854, 57774, 91200, 139338, 206394, 296832, 417120, 575556, 779238, 1037514, 1359792, 1760694, 2251362, 2845140, 3554976, 4404876, 5416278, 6605946, 7996896, 9621678, 11500962, 13667772, 16143552, 18973608, 22190406
Offset: 0
For n = 5, one such square has vertex set {(2,1,1), (5,4,1), (4,5,5), (1,2,5)}.
a(13)-a(20) from
Peter Kagey, Jul 29 2020 via Mathematics Stack Exchange link
Terms a(21) and beyond from
Zachary Kaplan, Sep 01 2020, via Mathematics Stack Exchange link
A334386
a(n) is the number of ways to choose 3 points in a size n tetrahedral grid in such a way that the three points form an equilateral triangle that touches all four sides of the tetrahedron.
Original entry on oeis.org
0, 0, 4, 8, 12, 16, 32, 36, 28, 32, 60, 100, 80, 84, 64, 80, 96, 88, 116, 132, 172, 188, 144, 208, 128, 228, 112, 188, 156, 268, 212, 312, 196, 224, 288, 328, 296, 324, 232, 344, 324, 412, 260, 384, 244, 512, 420, 364, 296, 492, 316, 452, 432, 556, 404, 588
Offset: 0
For n = 6 there are 28 equilateral triangles that touch all of the sides of the six-vertex-per-side tetahedron. In barycentric coordinates, these come in four equivalence classes:
{(0, 0, 0, 1), (0, 0, 1, 0), ( 0, 1, 0, 0)},
{(0, 0, 1/5, 4/5), (0, 1/5, 4/5, 0), ( 0, 4/5, 0, 1/5)},
{(0, 0, 2/5, 3/5), (0, 2/5, 3/5, 0), ( 0, 3/5, 0, 2/5)}, and
{(0, 0, 2/5, 3/5), (0, 3/5, 2/5, 0), (3/5, 1/5, 0, 1/5)},
where two triangles are considered equivalent if the coordinates of one are permutations of the other.
The equivalence classes contain 4, 8, 8, and 8 elements respectively.
A334891
Number of ways to choose 4 points that form a square from the A000292(n) points in a regular tetrahedral grid where each side has n vertices.
Original entry on oeis.org
0, 0, 3, 12, 36, 84, 174, 336, 612, 1044, 1701
Offset: 0
For n = 4, three of the a(4) = 36 squares are (in barycentric coordinates)
{(0,2,1,1),(1,1,0,2),(1,1,2,0),(2,0,1,1)},
{(0,0,2,2),(0,2,0,2),(2,0,2,0),(2,2,0,0)}, and
{(0,0,1,3),(0,1,0,3),(1,0,1,2),(1,1,0,2)}.
The other squares can be derived from these by translations or symmetries of the tetrahedron.
A342353
Number of ways to choose 3 points that form an equilateral triangle from the A001845(n) points in a regular octahedral grid of size n.
Original entry on oeis.org
0, 8, 80, 416, 1512, 4216, 10000, 21256
Offset: 0
For n = 1, the a(1) = 8 equilateral triangles are given by the convex hulls of {(+-1,0,0),(0,+-1,0),(0,0,+-1)}.
Cf.
A102698 and
A334581 are analogous for the cubic grid and tetrahedral grid respectively.
Showing 1-4 of 4 results.
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