A344857 -id:A344857 - cp's OEIS frontend

A349421 Numbers k such that a regular k-gon, when its vertices are connected by infinite lines, creates polygons outside the k-gon with more sides than any polygon inside the k-gon, excluding the central k-sided polygon for odd values of k.

14, 19, 29, 32, 33, 39, 52

Offset: 1

Scott R. Shannon, Nov 17 2021

As a regular k-gon with an odd number of sides always creates a k-sided polygon at the center of the k-gon when its vertices are connected by lines (see A342222), this polygon is excluded when considering the polygons inside the k-gon with the maximum number of sides.

If the next term exists it is greater than 100.

			14 is a term as when a regular 14-gon's vertices are connected by infinite lines fourteen 6-gons are created outside the vertices while the maximum-sided polygons created inside are 5-gons. See the 14-gon image in A344857.
19 is a term as when a regular 19-gon's vertices are connected by infinite lines nineteen 10-gons are created outside the vertices while the maximum-sided polygons created inside, excluding the central 19-gon, are 8-gons. See the 19-gon image in A344857.
32 is a term as when a regular 32-gon's vertices are connected by infinite lines sixty-four 8-gons are created outside the vertices while the maximum-sided polygons created inside are 7-gons. See the linked image.

Scott R. Shannon, Image for the 32-gon.

Cf. A344938, A344857, A344311, A007678, A342222.

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.

Original entry on oeis.org

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

Scott R. Shannon, Apr 07 2022

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.

			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.

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.

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

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

cp's OEIS Frontend

A349421 Numbers k such that a regular k-gon, when its vertices are connected by infinite lines, creates polygons outside the k-gon with more sides than any polygon inside the k-gon, excluding the central k-sided polygon for odd values of k.

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