A331911 Triangle read by rows: Take an equilateral triangle with all diagonals drawn, as in A092867. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+2 and where n is the number of equal parts each side is divided into.

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

Offset: 1

Author

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

Keywords

• nonn
• more
• tabl

Examples

			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.
		

Links

• 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

Crossrefs

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