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A349784
Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678), excluding the central n-sided cell for odd values of n.
5
3, 3, 4, 5, 4, 6, 5, 6, 4, 8, 5, 7, 6, 8, 7, 8, 8, 7, 8, 8, 6, 8, 8, 8, 8, 11, 6, 10, 7, 8, 8, 9, 8, 10, 8, 8, 12, 10, 8, 14, 10, 9, 8, 10, 10, 10, 10, 12, 8, 12, 10, 10, 10, 10, 10, 12, 9, 12, 10, 12, 10, 12, 9, 10, 10, 10, 10, 11, 10, 12, 10, 12, 12, 10, 10, 10, 10, 10, 10, 10, 10, 12, 10
OFFSET
4,1
COMMENTS
As a regular n-gon with an odd number of sides always creates an n-sided cell at its center when all its diagonals are drawn, see A342222, this n-sided cell is not considered for odd n.
Although the behavior of the sequence is unknown as n -> infinity, the data up to n = 765 implies the sequence is possibly bounded. In the range studied the 14-gon is the predominant maximum-sided cell for n > 300.
No n-gon is currently known that produces a cell with 17 sides or 19 sides and above, other than the corresponding central n-sided cell for odd values of n.
See A342222 and A342236 for images of the n-gons.
LINKS
Scott R. Shannon, Image showing a close-up of the 18-sided cell in the 708-gon. Zoom in to see the vertices marked as white dots around the 18-gon, shown in violet. The bottom three vertices are extremely close together -- if the image were expanded so that the outer two of these vertices were 1cm away from the inner vertex then the size of the entire 708-gon image would be slightly over 12.9 km in diameter.
EXAMPLE
a(4) = 3 as a regular 4-gon (square) creates four 3-gons (triangles) when all its diagonals are drawn.
a(5) = 3 as a regular 5-gon (pentagon) creates ten 3-gons when all its diagonals are drawn. Also created is a central 5-gon but this cell is not considered.
a(6) = 4 as a regular 6-gon (hexagon) creates eighteen 3-gons and six 4-gons when all its diagonals are drawn.
a(7) = 5 as a regular 7-gon (heptagon) creates thirty-five 3-gons, seven 4-gons and seven 5-gons when all its diagonals are drawn. Also created is a central 7-gon but this cell is not considered.
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved