A353876 - OEIS

%I #27 Jan 04 2024 14:30:12

%S 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,

%T 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,

%U 0,11,12,24,144,24,60,0,36,0,1,0,13,78,39,130,260,91,65,26,0,0,13

%N 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.

%C 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.

%C 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.

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

%H Scott R. Shannon, <a href="/A353876/a353876.png">Image of the 7-gon</a>. In this and other images the vertex color is based on the surrounding polygon edge count shown in the key.

%H Scott R. Shannon, <a href="/A353876/a353876_1.png">Image of the 8-gon</a>.

%H Scott R. Shannon, <a href="/A353876/a353876_2.png">Image of the 9-gon</a>.

%H Scott R. Shannon, <a href="/A353876/a353876_3.png">Image of the 10-gon</a>.

%H Scott R. Shannon, <a href="/A353876/a353876_4.png">Image of the 12-gon</a>.

%H Scott R. Shannon, <a href="/A353876/a353876.txt">Table for n=3..100</a>.

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

%e 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].

%e The table begins:

%e 0;

%e 1;

%e 0, 0, 5;

%e 0, 6, 6, 0, 0, 0, 0, 0, 1;

%e 0, 7, 0, 14, 0, 7, 7;

%e 0, 8, 24, 8, 9;

%e 0, 9, 18, 18, 0, 63, 0, 18;

%e 0, 10, 70, 30, 20, 10, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

%e 0, 11, 44, 33, 55, 143, 11, 22, 0, 11;

%e 12, 24, 144, 24, 60, 0, 36, 0, 1;

%e 0, 13, 78, 39, 130, 260, 91, 65, 26, 0, 0, 13;

%e 0, 14, 182, 196, 168, 126, 56, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \

%e                                                                        0, 0, 1;

%e 0, 15, 120, 90, 345, 525, 135, 105, 15, 0, 0, 0, 0, 15;

%e 0, 32, 256, 240, 480, 224, 96, 16, 32, 0, 0, 0, 1;

%e .

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

%K nonn,tabf

%O 3,5

%A _Scott R. Shannon_, May 09 2022