A351045 Irregular table read by rows: row n gives the number of edges with k facing edges for a regular n-gon with all diagonals drawn, with n>=3 and k>=2.

3, 4, 0, 4, 5, 0, 10, 0, 5, 6, 0, 18, 12, 6, 7, 0, 28, 14, 21, 14, 7, 8, 0, 56, 48, 24, 9, 0, 54, 54, 72, 72, 18, 0, 9, 10, 0, 80, 160, 120, 20, 11, 0, 88, 154, 198, 198, 55, 0, 0, 0, 11, 12, 0, 240, 336, 168, 13, 0, 130, 260, 507, 390, 91, 104, 0, 0, 0, 0, 13, 14, 0, 266, 616, 644, 140, 42

Offset: 3

Author

Scott R. Shannon and N. J. A. Sloane, Jan 30 2022

Keywords

• nonn
• tabf

Comments

The number of facing edges for a given edge is the number of other edges in the one (for edges on the outside of the n-gon) or two polygons that the edge forms a part of. For example, for an edge shared between two adjoined triangles the number of facing edges is four, as it faces two edges in each of the two triangles it forms a part of.

All edges that are on the outside of the n-gon have two facing edges as any such edge belongs to only one (interior) triangle. Thus T(n,2) = n. For odd n the central created n-gon, see A342222, is surrounded by triangles, thus the edges that form this central n-gon have (n-1)+(3-1) = n+1 facing edges, thus T(n,n+1) >= n.

For all n-gons with even n, or odd n if the central n-gon is ignored, the maximum k for which row(n,k) > 0 is unknown, although it is clearly related to the maximum sided cell for all n-gons; see A349784.

Examples

			A hexagon with all diagonals drawn has six edges (those on the outside of the hexagon) which form one side of a single triangle and thus face two edges, eighteen edges that adjoin two triangles and thus face four edges, twelve edges that adjoin a triangle and a quadrilateral and thus face five edges, and six edges that adjoin two quadrilaterals and thus face six edges. Thus the row for n = 6 is [6, 0, 18, 12, 6]. See the attached image.
The table begins:
3;
4, 0, 4;
5, 0, 10, 0, 5;
6, 0, 18, 12, 6;
7, 0, 28, 14, 21, 14, 7;
8, 0, 56, 48, 24;
9, 0, 54, 54, 72, 72, 18, 0, 9;
10, 0, 80, 160, 120, 20;
11, 0, 88, 154, 198, 198, 55, 0, 0, 0, 11;
12, 0, 240, 336, 168;
13, 0, 130, 260, 507, 390, 91, 104, 0, 0, 0, 0, 13;
14, 0, 266, 616, 644, 140, 42;
15, 0, 180, 600, 945, 630, 435, 0, 15, 0, 0, 0, 0, 0, 15;
16, 0, 448, 1056, 960, 576, 32;
17, 0, 238, 816, 1853, 1224, 425, 272, 34, 0, 0, 0, 0, 0, 0, 0, 17;
18, 0, 900, 1836, 1314, 108, 144;
19, 0, 304, 1520, 2717, 2128, 798, 304, 95, 0, 19, 0, 0, 0, 0, 0, 0, 0, 19;
20, 0, 1000, 2120, 3280, 1600, 100, 240;
21, 0, 378, 2352, 4494, 3276, 1365, 252, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21;
22, 0, 1056, 3828, 5258, 1716, 374, 396, 132;
.
.
See the linked file for the table n = 3..100.
		

Links

• Scott R. Shannon, Table for n = 3..100.
• Scott R. Shannon, Image of the edges for n = 6.
• Scott R. Shannon, Image of the edges for n = 9.
• Scott R. Shannon, Image of the edges for n = 15.
• Scott R. Shannon, Image of the edges for n = 25.
• Scott R. Shannon, Image of the edges for n = 30.

Crossrefs

Cf. A135565, A342222, A349784, A344899, A342236, A342222, A342236.

Formula

Sum of row n = A135565(n).

T(n,2) = n.

T(n,n+1) >= n for odd n.