A358951 Irregular table read by rows: T(n,k) = number of k-gons, k >= 3, formed inside a triangle with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,m)/A006843(n,m), m = 1..A005728(n).

1, 12, 180, 42, 6, 810, 576, 72, 6, 6786, 4932, 744, 48, 6, 13662, 12522, 2568, 258, 12, 72582, 64932, 14376, 1632, 36, 6, 164484, 155088, 38688, 5958, 414, 18, 439524, 422370, 114804, 18462, 1392, 120, 750108, 749928, 211518, 35226, 3336, 204, 6, 2265462, 2240994, 647184, 109602, 10230, 666, 18

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

• nonn
• tabf

Comments

The number of points along each edge is given by A005728(n).

Examples

			The table begins:
1;
12;
180, 42, 6;
810, 576, 72, 6;
6786, 4932, 744, 48, 6;
13662, 12522, 2568, 258, 12;
72582, 64932, 14376, 1632, 36, 6;
164484, 155088, 38688, 5958, 414, 18;
439524, 422370, 114804, 18462, 1392, 120;
750108, 749928, 211518, 35226, 3336, 204, 6;
2265462, 2240994, 647184, 109602, 10230, 666, 18;
3263436, 3312270, 990072, 176172, 18294, 1188, 66;
.
.
		

Links

• Scott R. Shannon, Image for n = 7.
• Wikipedia, Farey sequence.

Crossrefs

Cf. A358948 (regions), A358949 (vertices), A358950 (edges), A358889, A006842, A006843, A005728, A358882.

Formula

Sum of row n = A358948(n).