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

1,2

Links

Table of n, a(n) for n=1..78.

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

Example

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.

Crossrefs

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

Keyword

nonn,more,tabl

Author

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

Status

approved