

A354133


Irregular table read by rows: for each interior cell of a regular ngon 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 ksided polygons, for k >= 6, for all interior cells.


1



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
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OFFSET

3,5


COMMENTS

An interior cell is one that has no edges that form the outside of the ngon, 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 kgons is 6  this corresponds to a triangle that is adjoined to three other triangles. Only ngons 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 evensided polygons contain such vertices.
Numerous patterns appear in the terms. For odd n >= 13 there is always one 2nsided polygon which is created by the central nsided polygon being surrounded by n triangles, thus row(n,2n) = 1. These n triangles themselves are adjoined to the central ngon and two 4gons so they create an n1 + 3 + 3 = (n+5)sided polygon, thus row(n,n+5) = n.
Almost all evenn ngons contain triangles surrounded by three other triangles and therefore have values for k=6. The exceptions for n >= 6 up to the 140gon are n=6,10,14,22,26,46,50,58,70. It is plausible that the 70gon is the last evenn polygon not to contain such triangles.
Ignoring the central ngon and its surrounding triangles for oddsided ngons, the largest possible created kgon 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 kgon is a 34gon which surrounds a 14sided cell in the 132gon. See the linked image.
Up to the 36gon the most commonly created ksided polygon is shared between k values of 8 to 13 inclusive. The 36gon has the 11gon as the most commonly created, but from the 37gon up to at least the 140gon the 12gon becomes the most common. The distribution of kgons for the larger n values becomes quite uniform and it is therefore possible that the 12gon is the most commonly created polygon for all ngons for n >= 37.


LINKS

Scott R. Shannon, Image for the 8gon. In this and other images the color of the interior cells is based on the number of edges in the surrounding kgon given in the key.
Scott R. Shannon, Image for the 26gon. This is one of the few evensided ngons that does not contain triangles adjoined to three other triangles.


EXAMPLE

The 8gon contains eight triangles that adjoin three triangles and thus create a 6gon, thirtytwo triangles that adjoin two triangles and one quadrilateral and thus create a 7gon, eight triangles that adjoin one triangle and two quadrilaterals and thus create an 8gon, and twentyfour quadrilaterals that adjoin two triangles and two quadrilaterals and thus create a 10gon. 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.


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



