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A350718
Number of regions in a regular n-gon with all diagonals drawn whose edges all have a different number of facing edges.
1
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, 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
KEYWORD
nonn
AUTHOR
STATUS
approved