A101365 In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly five diagonals intersect.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 180, 0, 0, 0, 0, 0, 216, 0, 0, 0, 0, 0, 546, 0, 0, 0, 0, 0, 336, 0, 0, 0, 0, 0, 648, 0, 0, 0, 0, 0, 720, 0, 0, 0, 0, 0, 990, 0, 0, 0, 0, 0, 936, 0, 0, 0, 0, 0, 1404, 0, 0, 0, 0, 0, 2352, 0, 0, 0, 0, 0, 1890, 0, 0, 0, 0

Offset: 3

Author

Graeme McRae, Dec 26 2004, revised Feb 23 2008

Keywords

• nonn

Comments

When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.

When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet except the center.

When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet except the center.

I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet and no interior point other than the center where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points (all points interior excluding the center)."

Examples

			a(18)=54 because inside a regular 18-gon there are 54 points (3 on each radius) where exactly five diagonals intersect.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 3..10000 (terms 3..210 from Graeme McRae)
• Sequences formed by drawing all diagonals in regular polygon

Crossrefs

A column of A292105.

Cf. A006561, A007678.

Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon.

Cf. A006561: number of intersections of diagonals in the interior of regular n-gon.

Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon.

Cf. A101364: number of 4-way intersections in the interior of a regular n-gon.

Cf. A137938: number of 4-way intersections in the interior of a regular 6n-gon.

Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon.