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A354133
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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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1
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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
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3,5
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COMMENTS
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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.
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LINKS
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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 26-gon. This is one of the few even-sided n-gons that does not contain triangles adjoined to three other triangles.
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EXAMPLE
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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;
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See the linked file for the table n = 3..120.
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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