A351129 Number of regions in a regular n-gon with all diagonals drawn whose edges all have the same number of facing edges.

1, 0, 1, 0, 1, 8, 1, 0, 1, 132, 66, 56, 46, 144, 171, 576, 305, 620, 652, 616, 852, 1296, 1376, 1482, 1891, 1820, 2379, 4530, 3163, 3328, 3532, 4046, 4656, 4896, 6661, 6460, 7411, 7560, 9595, 11676, 10923, 13552, 10936, 13294, 14806, 17232, 17935, 17200, 20452, 20540, 24964, 27270

Offset: 3

Author

Scott R. Shannon and N. J. A. Sloane, Feb 02 2022

Keywords

• nonn

Comments

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

Examples

			a(5) = 1. A pentagon with all diagonals drawn contains a central pentagon which is surrounded by five other triangles and therefore all its edges have a facing edge count of 6. See the attached image.
a(8) = 8. An octagon with all diagonals drawn contains eight central triangles all of which are surrounded by three other triangles and therefore all their edges have a facing edge count of 4. See the attached image.
a(15) = 46. A 15-gon with all diagonals drawn contains one central 15-gon which is surrounded by triangles, thirty quadrilaterals which are surrounded by other quadrilaterals, and fifteen triangles which are surrounded by pentagons. This gives a total of forty-six regions whose edges all have the same facing edge count. See the attached image.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 3..100
• Scott R. Shannon, Image for n = 5. In this and other images the regions with edges with the same facing edge count are highlighted in the corresponding edge color.
• Scott R. Shannon, Image for n = 8.
• Scott R. Shannon, Image for n = 12.
• Scott R. Shannon, Image for n = 15.
• Scott R. Shannon, Image for n = 18.
• Scott R. Shannon, Image for n = 24.

Crossrefs

Cf. A351045, A135565, A007678, A342222, A349784, A344899, A342236.