A350718 Number of regions in a regular n-gon with all diagonals drawn whose edges all have a different number of facing edges.

0, 0, 0, 0, 0, 0, 0, 0, 44, 0, 130, 84, 180, 128, 374, 180, 418, 440, 714, 704, 1104, 624, 1750, 1976, 2484, 2744, 3190, 2880, 3658, 4416, 5280, 6188, 7000, 7128, 8214, 8892, 10296, 10560, 13120, 14028, 16082, 15928, 22140, 20332, 22466, 26112, 27538, 29200, 36924, 36504, 35934, 40284, 41140

Offset

3,9

Comments

See A351045 for details of an edge's count of facing edges in an n-gon with all diagonals drawn.

For n = 3 to n = 80 the regions with edges all with a different number of facing edges are all triangles or quadrilaterals. The 81-gon is the first n-gon to contain pentagons with this property. The largest number of edges possible for such regions is unknown.

Links

Scott R. Shannon, Table of n, a(n) for n = 3..140

Scott R. Shannon, Image for n = 11. This is the first n-gon to contain regions whose edges all have a different facing edge count. In this and other images such regions are highlighted in gray.

Scott R. Shannon, Image for n = 13.

Scott R. Shannon, Image for n = 18.

Scott R. Shannon, Image for n = 81. This is zoomed-in on one of the pentagons whose edges all have a different facing edge count: 6,7,8,9,10.

Example

a(11) = 44. The 11-gon contains forty-four triangles whose three edges all have a different number of facing edges. This is the first n-gon to contain such regions. See the attached image.

Crossrefs

Cf. A351045, A351129, A135565, A007678, A342222, A349784.

Sequence in context: A138086 A023933 A271141 * A022075 A178986 A174730

Adjacent sequences: A350715 A350716 A350717 * A350719 A350720 A350721

Keyword

nonn

Author

Scott R. Shannon and N. J. A. Sloane, Feb 03 2022

Status

approved