OFFSET
3,5
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.
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.
FORMULA
Sum of terms in row n = A007569(n) - n.
EXAMPLE
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;
.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Scott R. Shannon, May 09 2022
STATUS
approved