A352533 Irregular table read by rows: T(n,k) is the number of regions formed after k diagonals, with k>=0, are drawn between vertices of a regular n-gon, with n>=3, when each vertex is fully connected to all other vertices in counterclockwise order before the next vertex, in counterclockwise order, is chosen.

1, 1, 2, 4, 1, 2, 3, 5, 8, 11, 1, 2, 3, 4, 6, 9, 13, 16, 20, 24, 1, 2, 3, 4, 5, 7, 10, 14, 19, 22, 27, 34, 38, 45, 50, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 29, 34, 39, 46, 50, 56, 62, 67, 74, 80, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 37, 42, 49, 58, 69, 73, 80, 90, 103, 108, 117, 130, 136, 147, 154

Offset: 3

Author

Scott R. Shannon, Mar 19 2022

Keywords

• nonn
• tabf

Comments

To create the diagonals between the vertices of the regular n-gon a random starting vertex is first chosen. This vertex is then connected to all other vertices where those vertices are chosen in a counterclockwise direction around the n-gon from the starting vertex. Once all those diagonals are drawn the next neighboring vertex, in a counterclockwise direction from the starting vertex, is chosen. This vertex is then connected to all other vertices in the same way. This method is repeated until all vertices are connected by diagonals. The sequence gives the number of regions inside the n-gon after each such diagonal is drawn.

Examples

			The table begins:
1;
1,2,4;
1,2,3,5,8,11;
1,2,3,4,6,9,13,16,20,24;
1,2,3,4,5,7,10,14,19,22,27,34,38,45,50;
1,2,3,4,5,6,8,11,15,20,26,29,34,39,46,50,56,62,67,74,80;
1,2,3,4,5,6,7,9,12,16,21,27,34,37,42,49,58,69,73,80,90,103,108,117,130,136,147,154;
1,2,3,4,5,6,7,8,10,13,17,22,28,35,43,46,51,58,65,75,86,90,97,107,116,130,135,143, \
      154,164,170,180,194,201,212,220;
1,2,3,4,5,6,7,8,9,11,14,18,23,29,36,44,53,56,61,68,77,88,101,116,120,127,137,150, \
      166,185,190,199,212,229,250,256,267,283,304,311,324,343,351,366,375;
.
.
See the linked file for the table up to n=100. See the linked images for examples of the 7-gon.
		

Links

• Scott R. Shannon, Table for n=3..100.
• Scott R. Shannon, Image for the 7-gon after first vertex is connected to all other vertices. Total regions = 5.
• Scott R. Shannon, Image for the 7-gon after second vertex is connected to all other vertices. Total regions = 19.
• Scott R. Shannon, Image for the 7-gon after third vertex is connected to all other vertices. Total regions = 34.
• Scott R. Shannon, Image for the 7-gon after fourth vertex is connected to all other vertices. Total regions = 45.
• Scott R. Shannon, Image for the 7-gon after fifth vertex is connected to all other vertices. Total regions = 50.

Crossrefs

Cf. A007678, A350000, A344857.

Formula

The last term in each row n = A007678(n).