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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.
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%I #17 Oct 13 2023 10:02:21

%S 0,0,3,12,36,84,174,336,612,1044,1701

%N 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.

%C a(n) >= 3*A001752(n-2).

%e For n = 4, three of the a(4) = 36 squares are (in barycentric coordinates)

%e {(0,2,1,1),(1,1,0,2),(1,1,2,0),(2,0,1,1)},

%e {(0,0,2,2),(0,2,0,2),(2,0,2,0),(2,2,0,0)}, and

%e {(0,0,1,3),(0,1,0,3),(1,0,1,2),(1,1,0,2)}.

%e The other squares can be derived from these by translations or symmetries of the tetrahedron.

%Y Cf. A000292, A001752.

%Y Cf. A334581 (equilateral triangle), A334881 (cubic grid).

%K nonn,more

%O 0,3

%A _Peter Kagey_, May 14 2020