A135565 -id:A135565 - cp's OEIS frontend

A350718 Number of regions in a regular n-gon with all diagonals drawn whose edges all have a different number of facing edges.

0, 0, 0, 0, 0, 0, 0, 0, 44, 0, 130, 84, 180, 128, 374, 180, 418, 440, 714, 704, 1104, 624, 1750, 1976, 2484, 2744, 3190, 2880, 3658, 4416, 5280, 6188, 7000, 7128, 8214, 8892, 10296, 10560, 13120, 14028, 16082, 15928, 22140, 20332, 22466, 26112, 27538, 29200, 36924, 36504, 35934, 40284, 41140

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Feb 03 2022

nonn

See A351045 for details of an edge's count of facing edges in an n-gon with all diagonals drawn.

For n = 3 to n = 80 the regions with edges all with a different number of facing edges are all triangles or quadrilaterals. The 81-gon is the first n-gon to contain pentagons with this property. The largest number of edges possible for such regions is unknown.

			a(11) = 44. The 11-gon contains forty-four triangles whose three edges all have a different number of facing edges. This is the first n-gon to contain such regions. See the attached image.

Scott R. Shannon, Table of n, a(n) for n = 3..140
Scott R. Shannon, Image for n = 11. This is the first n-gon to contain regions whose edges all have a different facing edge count. In this and other images such regions are highlighted in gray.
Scott R. Shannon, Image for n = 13.
Scott R. Shannon, Image for n = 18.
Scott R. Shannon, Image for n = 81. This is zoomed-in on one of the pentagons whose edges all have a different facing edge count: 6,7,8,9,10.

Cf. A351045, A351129, A135565, A007678, A342222, A349784.

A354133 Irregular table read by rows: for each interior cell of a regular n-gon with all diagonals drawn remove all its edges and then count the number of sides in the resulting polygon; row n gives the number of resulting k-sided polygons, for k >= 6, for all interior cells.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 1, 0, 12, 0, 0, 6, 0, 14, 7, 0, 0, 0, 14, 0, 8, 8, 32, 8, 0, 24, 0, 36, 9, 0, 9, 0, 18, 36, 36, 0, 0, 0, 1, 0, 60, 20, 0, 100, 0, 30, 0, 66, 11, 0, 33, 0, 143, 0, 66, 22, 22, 0, 0, 0, 0, 0, 1, 48, 144, 48, 72, 60, 48, 12, 0, 104, 13, 0, 39, 52, 208, 78, 156, 26, 78, 0, 13, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 3

Scott R. Shannon, May 18 2022

An interior cell is one that has no edges that form the outside of the n-gon, i.e., all of its edges are shared with another cell. The number of such cells is A007678(n) - n = A191101(n).
The minimum number of sides in the created k-gons is 6 - this corresponds to a triangle that is adjoined to three other triangles. Only n-gons with an even number of sides can contain these triangles as their vertices must be formed by the intersection of three or more diagonals; only even-sided polygons contain such vertices.
Numerous patterns appear in the terms. For odd n >= 13 there is always one 2n-sided polygon which is created by the central n-sided polygon being surrounded by n triangles, thus row(n,2n) = 1. These n triangles themselves are adjoined to the central n-gon and two 4-gons so they create an n-1 + 3 + 3 = (n+5)-sided polygon, thus row(n,n+5) = n.
Almost all even-n n-gons contain triangles surrounded by three other triangles and therefore have values for k=6. The exceptions for n >= 6 up to the 140-gon are n=6,10,14,22,26,46,50,58,70. It is plausible that the 70-gon is the last even-n polygon not to contain such triangles.
Ignoring the central n-gon and its surrounding triangles for odd-sided n-gons, the largest possible created k-gon is unknown. It is likely related to the maximum number of sides of any cell, see A349784, which is also unknown. For n <= 140 the largest created k-gon is a 34-gon which surrounds a 14-sided cell in the 132-gon. See the linked image.
Up to the 36-gon the most commonly created k-sided polygon is shared between k values of 8 to 13 inclusive. The 36-gon has the 11-gon as the most commonly created, but from the 37-gon up to at least the 140-gon the 12-gon becomes the most common. The distribution of k-gons for the larger n values becomes quite uniform and it is therefore possible that the 12-gon is the most commonly created polygon for all n-gons for n >= 37.

			The 8-gon contains eight triangles that adjoin three triangles and thus create a 6-gon, thirty-two triangles that adjoin two triangles and one quadrilateral and thus create a 7-gon, eight triangles that adjoin one triangle and two quadrilaterals and thus create an 8-gon, and twenty-four quadrilaterals that adjoin two triangles and two quadrilaterals and thus create a 10-gon. Therefore row 8 is [8,32,8,0,24].
The table begins:
0;
0;
0, 0, 5, 0, 1;
0, 12, 0, 0, 6;
0, 14, 7, 0, 0, 0, 14, 0, 8;
8, 32, 8, 0, 24;
0, 36, 9, 0, 9, 0, 18, 36, 36, 0, 0, 0, 1;
0, 60, 20, 0, 100, 0, 30;
0, 66, 11, 0, 33, 0, 143, 0, 66, 22, 22, 0, 0, 0, 0, 0, 1;
48, 144, 48, 72, 60, 48, 12;
0, 104, 13, 0, 39, 52, 208, 78, 156, 26, 78, 0, 13, 0, 0, 0, 0, 0, 0, 0, 1;
0, 140, 126, 140, 196, 112, 140, 28, 56;
0, 150, 15, 0, 60, 180, 465, 150, 210, 60, 135, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, \
                                                                       0, 0, 0, 1;
32, 256, 144, 192, 240, 352, 240, 160, 32, 0, 32;
.
See the linked file for the table n = 3..120.

Scott R. Shannon, Table for n = 3..120.
Scott R. Shannon, Image for the 8-gon. In this and other images the color of the interior cells is based on the number of edges in the surrounding k-gon given in the key.
Scott R. Shannon, Image for the 11-gon.
Scott R. Shannon, Image for the 12-gon.
Scott R. Shannon, Image for the 17-gon.
Scott R. Shannon, Image for the 18-gon.
Scott R. Shannon, Image for the 26-gon. This is one of the few even-sided n-gons that does not contain triangles adjoined to three other triangles.
Scott R. Shannon, Image for the 36-gon.
Scott R. Shannon, Image showing a close-up of a 14-sided cell in the 132-gon. This creates a 34-sided k-gon.

Cf. A007678, A191101, A349784, A353876, A007569, A135565, A351045.

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A350718 Number of regions in a regular n-gon with all diagonals drawn whose edges all have a different number of facing edges.

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A354133 Irregular table read by rows: for each interior cell of a regular n-gon with all diagonals drawn remove all its edges and then count the number of sides in the resulting polygon; row n gives the number of resulting k-sided polygons, for k >= 6, for all interior cells.

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A301748 Number of faces in the n-polygon diagonal intersection graph.

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