A397216 a(n) is the number of distinct triplets of triangles chosen from n^2 unit triangles of a triangular grid with their centers collinear up to rotation.

0, 0, 2, 12, 40, 102, 220, 424, 746, 1228, 1920, 2876, 4166, 5852, 8012, 10742, 14138, 18304, 23340, 29382, 36546, 44976, 54840, 66264, 79426, 94500, 111680, 131164, 153138, 177806, 205398, 236164, 270324, 308132, 349846, 395752, 446148, 501324, 561534, 627088

Offset

1,3

Comments

Oblique coordinate system is defined taking x axis and y axis along two edges of the triangular grid of unit triangles n^2. Side length of a unit triangle is taken as 3 units to obtain integer coordinates for the centroids of all unit triangles. Each triangle in the grid is uniquely denoted by coordinates of its centroid. The centroid of a triangle is referred as the center of that triangle.

Three centers (x_1,y_1),(x_2,y_2) and (x_3,y_3) are collinear if (y_1-y_2)*(x_2-x_3) = (x_1-x_2)*(y_2-y_3).

Links

Sean A. Irvine, Table of n, a(n) for n = 1..100

Janaka Rodrigo, Python code for A397216

Example

n = 4 has the following distinct triplets of triangles:
 {(4,1), (7,1), (10,1)},
 {(1,10), (4,7), (10,1)},
 {(1,10), (7,4), (10,1)},
 {(4,4), (8,2), (10,1)},
 {(2,5), (8,2), (10,1)},
 {(4,7), (7,4), (10,1)},
 {(2,5), (4,4), (10,1)},
 {(4,7), (5,5), (7,1)},
 {(4,1), (4,4), (4,7)},
 {(4,1), (5,2), (7,4)},
 {(2,8), (5,5), (8,2)},
 {(2,8), (4,4), (5,2)}.
Therefore a(4) = 12.

Crossrefs

Cf. A396285, A396467, A396468, A397215.

Sequence in context: A290131 A008911 A005719 * A143126 A118417 A069144

Adjacent sequences: A397209 A397212 A397215 * A397217 A397218 A397219

Keyword

nonn,new

Author

Janaka Rodrigo, Jun 19 2026

Extensions

More terms from Sean A. Irvine, Jul 05 2026

Status

approved