A396761 a(n) is the number of distinct triplets of triangles chosen from the n^2 unit triangles of the triangular grid up to rotation and reflection.

0, 2, 20, 109, 416, 1249, 3167, 7091, 14433, 27246, 48395, 81746, 132376, 206804, 313240, 461855, 665072, 937875, 1298139, 1766981, 2369129, 3133312, 4092671, 5285188, 6754136, 8548550, 10723716, 13341681, 16471784, 20191205, 24585535, 29749367, 35786905, 42812594, 50951771, 60341334, 71130432, 83481176

Offset

1,2

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.

Any set of triplets fully obtainable by one of the following transformations produce reflections and are considered as duplicates.

(h,k) into (n-h-k,h)

(h,k) into (k,h)

(h,k) into (h,n-h-k)

Links

Janaka Rodrigo, Table of n, a(n) for n = 1..500

Janaka Rodrigo, Python code for A396761

Janaka Rodrigo, Illustration for n = 3

Formula

a(n) = (A396285(n) + (1/2)*(n-1)*n^2 + binomial(n,3)) / 2.

Example

n = 3 has 20 distinct sets of triplets of triangles which can be obtained by removing the following 10 reflections from the 30 sets of triplets in A396285(3):
 {(1,1), (1,7), (2,5)},
 {(1,1), (1,7), (4,1)},
 {(1,1), (2,2), (4,1)},
 {(1,1), (2,2), (5,2)},
 {(1,1), (2,5), (4,1)},
 {(1,1), (4,1), (4,4)},
 {(1,1), (4,1), (5,2)},
 {(1,1), (4,4), (5,2)},
 {(1,4), (2,5), (4,1)},
 {(1,4), (2,5), (5,2)}.
Therefore a(3) = 20.

Crossrefs

Cf. A396285, A396467, A396761, A396762.

Sequence in context: A052361 A001884 A028477 * A073077 A069537 A282818

Adjacent sequences: A396758 A396759 A396760 * A396762 A396763 A396764

Keyword

nonn,changed

Author

Janaka Rodrigo, Jun 04 2026

Status

approved