A352883 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 in turn is connected to the vertex floor(n/2) vertices to its left, then floor(n/2)-1 vertices to its left, then floor(n/2)-2 vertices ... until all vertices are connected by diagonals.

1, 1, 2, 4, 1, 2, 4, 6, 8, 11, 1, 2, 4, 6, 8, 11, 14, 17, 20, 24, 1, 2, 4, 7, 10, 13, 17, 22, 25, 29, 33, 37, 41, 45, 50, 1, 2, 4, 6, 8, 11, 14, 18, 22, 26, 30, 35, 40, 44, 49, 54, 59, 64, 69, 74, 80, 1, 2, 4, 7, 11, 15, 19, 24, 30, 37, 42, 48, 55, 62, 69, 76, 83, 91, 100, 105, 111, 117, 123, 129, 135, 141, 147, 154

Offset

3,3

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 the vertex floor(n/2) vertices to its left. The left neighboring vertex of the starting vertex is then chosen and this is connected to the vertex floor(n/2) to its left. This process is continued until all vertices are connected by diagonals to the vertices floor(n/2) to their left. Note that when n is even only n/2 diagonals are drawn in this phase as further diagonals would just retrace the previous diagonal connections. The initial vertex is then chosen again and it is connected to the vertex floor(n/2)-1 to its left. Its left neighboring vertex is then connected to the vertex floor(n/2)-1 to its left, and so on. This process of connecting all vertices to those on their left by diagonals, where the step size decreases by one after each complete circuit of the n-gon until the step size is 2, is continued until all vertices are connected by diagonals. The sequence gives the number of regions inside the n-gon after each such diagonal is drawn.

Links

Table of n, a(n) for n=3..86.

Scott R. Shannon, Table for n=3..100.

Scott R. Shannon, Image for the 11-gon after the first five vertices are connected to the vertices floor(n/2) = 5 to their left. Total regions = 16.

Scott R. Shannon, Image for the 11-gon after all vertices are connected to the vertices floor(n/2) = 5 to their left. Total regions = 56.

Scott R. Shannon, Image for the 11-gon after all vertices are connected to the vertices floor(n/2)-1 = 4 to their left. Total regions = 166.

Scott R. Shannon, Image for the 11-gon after all vertices are connected to the vertices floor(n/2)-2 = 3 to their left. Total regions = 287.

Scott R. Shannon, Image for the 11-gon after all vertices are connected to the vertices floor(n/2)-3 = 2 to their left. Total regions = 375.

Formula

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

Example

The table begins:
1;
1,2,4;
1,2,4,6,8,11;
1,2,4,6,8,11,14,17,20,24;
1,2,4,7,10,13,17,22,25,29,33,37,41,45,50;
1,2,4,6,8,11,14,18,22,26,30,35,40,44,49,54,59,64,69,74,80;
1,2,4,7,11,15,19,24,30,37,42,48,55,62,69,76,83,91,100,105,111,117,123,129,135, \
  141,147,154;
1,2,4,6,8,10,14,19,24,30,36,42,48,55,62,70,77,84,92,100,108,116,124,132,141,150, \
  156,163,170,177,184,191,198,205,212,220;
1,2,4,7,11,16,21,26,32,39,47,56,63,71,80,90,100,110,120,130,141,153,166,175,185, \
  196,207,218,229,240,251,262,274,287,294,302,310,318,326,334,342,350,358,366,375;
.
.
See the linked file for the table up to n=100. See the linked images for examples of the 11-gon.

Crossrefs

Cf. A352866, A352533, A007678, A350000, A344857.

Sequence in context: A358431 A062039 A352866 * A035492 A101229 A057176

Adjacent sequences: A352880 A352881 A352882 * A352884 A352885 A352886

Keyword

nonn,tabf

Author

Scott R. Shannon, Apr 07 2022

Status

approved