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%I #36 Jun 07 2022 11:12:43
%S 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,
%T 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,
%U 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
%N 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.
%C 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).
%C 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.
%C 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.
%C 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.
%C 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.
%C 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.
%H Scott R. Shannon, <a href="/A354133/a354133.txt">Table for n = 3..120</a>.
%H Scott R. Shannon, <a href="/A354133/a354133.png">Image for the 8-gon</a>. 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.
%H Scott R. Shannon, <a href="/A354133/a354133_3.png">Image for the 11-gon</a>.
%H Scott R. Shannon, <a href="/A354133/a354133_1.png">Image for the 12-gon</a>.
%H Scott R. Shannon, <a href="/A354133/a354133_4.png">Image for the 17-gon</a>.
%H Scott R. Shannon, <a href="/A354133/a354133_2.png">Image for the 18-gon</a>.
%H Scott R. Shannon, <a href="/A354133/a354133.jpg">Image for the 26-gon</a>. This is one of the few even-sided n-gons that does not contain triangles adjoined to three other triangles.
%H Scott R. Shannon, <a href="/A354133/a354133_1.jpg">Image for the 36-gon</a>.
%H Scott R. Shannon, <a href="/A354133/a354133_2.jpg">Image showing a close-up of a 14-sided cell in the 132-gon</a>. This creates a 34-sided k-gon.
%e 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].
%e The table begins:
%e 0;
%e 0;
%e 0, 0, 5, 0, 1;
%e 0, 12, 0, 0, 6;
%e 0, 14, 7, 0, 0, 0, 14, 0, 8;
%e 8, 32, 8, 0, 24;
%e 0, 36, 9, 0, 9, 0, 18, 36, 36, 0, 0, 0, 1;
%e 0, 60, 20, 0, 100, 0, 30;
%e 0, 66, 11, 0, 33, 0, 143, 0, 66, 22, 22, 0, 0, 0, 0, 0, 1;
%e 48, 144, 48, 72, 60, 48, 12;
%e 0, 104, 13, 0, 39, 52, 208, 78, 156, 26, 78, 0, 13, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 140, 126, 140, 196, 112, 140, 28, 56;
%e 0, 150, 15, 0, 60, 180, 465, 150, 210, 60, 135, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, \
%e 0, 0, 0, 1;
%e 32, 256, 144, 192, 240, 352, 240, 160, 32, 0, 32;
%e .
%e See the linked file for the table n = 3..120.
%Y Cf. A007678, A191101, A349784, A353876, A007569, A135565, A351045.
%K nonn,tabf
%O 3,5
%A _Scott R. Shannon_, May 18 2022
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