A353876 Irregular table read by rows: for each internal vertex of a regular n-gon with all diagonals drawn remove all the edges connected directly to the vertex and then count the number of sides in the polygon that surrounds it; row n gives the number of resulting k-sided polygons, for k>=4, for all internal vertices.

0, 1, 0, 0, 5, 0, 6, 6, 0, 0, 0, 0, 0, 1, 0, 7, 0, 14, 0, 7, 7, 0, 8, 24, 8, 9, 0, 9, 18, 18, 0, 63, 0, 18, 0, 10, 70, 30, 20, 10, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 11, 44, 33, 55, 143, 11, 22, 0, 11, 12, 24, 144, 24, 60, 0, 36, 0, 1, 0, 13, 78, 39, 130, 260, 91, 65, 26, 0, 0, 13

Offset: 3

Author

Scott R. Shannon, May 09 2022

Keywords

• nonn
• tabf

Comments

Numerous patterns are found in the values of the k-gons for different n. For example for n = 4*m + 2, with m>=1, there is one maximum sided k-gon with 2*n edges. For n = 4*m, with m>=3, there is one maximum sided k-gon with n edges. For odd n, where n>=11, there is n maximum sided k-gons with n+2 edges.

The 8-gon appears to be unique in that there is 9 maximum sided k-gons, k=8, which is not 1 or a multiple of 8.

Only a limit number of even-n n-gons have vertex-surrounding polygons with 4 edges, the minimum possible value. See A353991.

Examples

			The 7-gon has seven internal vertices surrounded by polygons with 5 edges, fourteen internal vertices surrounded by polygons with 7 edges, seven internal vertices surrounded by polygons with 9 edges, and seven internal vertices surrounded by polygons with 10 edges, so row 7 is [0, 7, 0, 14, 0, 7, 7].
The table begins:
0;
1;
0, 0, 5;
0, 6, 6, 0, 0, 0, 0, 0, 1;
0, 7, 0, 14, 0, 7, 7;
0, 8, 24, 8, 9;
0, 9, 18, 18, 0, 63, 0, 18;
0, 10, 70, 30, 20, 10, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 11, 44, 33, 55, 143, 11, 22, 0, 11;
12, 24, 144, 24, 60, 0, 36, 0, 1;
0, 13, 78, 39, 130, 260, 91, 65, 26, 0, 0, 13;
0, 14, 182, 196, 168, 126, 56, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \
                                                                       0, 0, 1;
0, 15, 120, 90, 345, 525, 135, 105, 15, 0, 0, 0, 0, 15;
0, 32, 256, 240, 480, 224, 96, 16, 32, 0, 0, 0, 1;
.
		

Links

• Scott R. Shannon, Image of the 7-gon. In this and other images the vertex color is based on the surrounding polygon edge count shown in the key.
• Scott R. Shannon, Image of the 8-gon.
• Scott R. Shannon, Image of the 9-gon.
• Scott R. Shannon, Image of the 10-gon.
• Scott R. Shannon, Image of the 12-gon.
• Scott R. Shannon, Table for n=3..100.

Crossrefs

Cf. A353991, A007569, A007678, A135565, A351045.

Formula

Sum of terms in row n = A007569(n) - n.