A340688 Irregular table read by rows: Take a concave circular triangle with all diagonals drawn, as in A340685. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

1, 12, 22, 3, 3, 66, 36, 67, 108, 12, 222, 186, 48, 6, 265, 465, 132, 6, 582, 786, 174, 48, 732, 1905, 324, 76, 3, 6, 1410, 2268, 558, 156, 6, 1704, 3732, 861, 223, 18, 3, 2778, 4242, 1260, 324, 42, 3369, 6540, 1872, 409, 42, 24, 4896, 7302, 2502, 540, 72, 24, 6138, 10467, 3306, 907, 99, 30

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Jan 16 2021

Keywords

• nonn
• tabf

Comments

See A340685 for images of the regions and A340686 for images of the vertices.

Examples

			A concave circular triangle with 1 point dividing its edges, n = 2, contains 12 triangles and no other n-gons, so the second row is [12]. A concave circular triangle with 2 points dividing its edges, n = 3, contains 22 triangles, 3 quadrilaterals, 3 pentagons and no other n-gons, so the third row is [22, 3, 3].
The table begins:
1;
12;
22, 3, 3;
66, 36;
67, 108, 12;
222, 186, 48, 6;
265, 465, 132, 6;
582, 786, 174, 48;
732, 1905, 324, 76, 3, 6;
1410, 2268, 558, 156, 6;
1704, 3732, 861, 223, 18, 3;
2778, 4242, 1260, 324, 42;
3369, 6540, 1872, 409, 42, 24;
4896, 7302, 2502, 540, 72, 24;
6138, 10467, 3306, 907, 99, 30;
8364, 12522, 4566, 1020, 120, 18;
10132, 16149, 5439, 1410, 288, 57, 0, 3;
13398, 19308, 6870, 1962, 252, 30, 12;
16029, 23082, 8859, 2422, 336, 90, 3;
20682, 29658, 10800, 2976, 528, 66;
		

Links

• Scott R. Shannon, Image of the regions for n = 20.
• Wikipedia, Circular triangle.

Crossrefs

Cf. A340685 (regions), A340686 (vertices), A340687 (edges), A340614, A007678, A092867.