A137939 Number of 5-way intersections in the interior of a regular 6n-gon.

0, 0, 54, 24, 180, 216, 546, 336, 648, 720, 990, 936, 1404, 2352, 1890, 1824, 2448, 2592, 3078, 3720, 4284, 3960, 4554, 4464, 5400, 5616, 6318, 7896, 7308, 7560, 8370, 8256, 9504, 9792, 11550, 10584, 11988, 12312, 13338, 14640, 14760, 17640, 16254, 16104, 17820, 18216, 19458, 19296, 22344, 21600

Offset: 1

Author

Graeme McRae, 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(3) = 54 because there are 54 points in the interior of an 18-gon at which exactly five diagonals meet.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
• Sequences formed by drawing all diagonals in regular polygon

Crossrefs

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. A101365: number of 5-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.

Formula

a(n) = A101365(6*n). - Seiichi Manyama, Jul 20 2024

Extensions

More terms from Seiichi Manyama, Jul 20 2024