1,2

Hugo Pfoertner, Intersections of diagonals in polygons of triangular shape.

Scott R. Shannon, Triangle regions for n = 2.

Scott R. Shannon, Triangle regions for n = 3.

Scott R. Shannon, Triangle regions for n = 4.

Scott R. Shannon, Triangle regions for n = 5.

Scott R. Shannon, Triangle regions for n = 6.

Scott R. Shannon, Triangle regions for n = 7.

Scott R. Shannon, Triangle regions for n = 8.

Scott R. Shannon, Triangle regions for n = 9.

Scott R. Shannon, Triangle regions for n = 10.

Scott R. Shannon, Triangle regions for n = 11.

Scott R. Shannon, Triangle regions for n = 12.

Scott R. Shannon, Triangle regions for n = 13.

Scott R. Shannon, Triangle regions for n = 14.

Scott R. Shannon, Triangle regions for n = 9, random distance-based coloring.

Scott R. Shannon, Triangle regions for n = 12, random distance-based coloring

An equilateral triangle with no other point along its edges, n = 1, contains 1 triangle so the first row is [1]. An equilateral triangle with 1 point dividing its edges, n = 2, contains 12 triangles and no other n-gons, so the second row is [12,0]. An equilateral triangle with 2 points dividing its edges, n = 3, contains 48 triangles, 24 quadrilaterals and 3 pentagons, so the third row is [48,24,3].

Triangle begins:

1

12,0

48,24,3

162,90,0,0

378,306,15,16,0

774,696,84,18,0,0

1470,1383,219,37,0,0,0

2604,2382,600,78,6,6,0,0

4224,4089,771,177,24,6,0,0,0

6624,6186,1470,234,42,0,0,0,0,0

9738,9486,2307,498,48,0,0,3,0,1,0

14010,13548,3984,816,144,0,0,0,0,0,0,0

19248,19224,5007,1102,156,18,0,0,0,0,0,0,0

26208,26142,8634,1668,192,24,0,0,0,0,0,0,0,0

The row sums are A092867.

Cf. A092867, A092866, A092866, A274586, A331451, A331452, A331907, A331909

Sequence in context: A225951 A333577 A278711 * A307841 A257949 A375664

Adjacent sequences: A331908 A331909 A331910 * A331912 A331913 A331914

nonn,more,tabl

Scott R. Shannon and N. J. A. Sloane, Feb 01 2020

approved