2,3

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.

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.

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 points 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."

Graeme McRae (g_m(AT)mcraefamily.com), Feb 23 2008, Table of n, a(n) for n = 2..105

Sequences formed by drawing all diagonals in regular polygon

a(6)=60 because inside a regular 12-gon there are 60 points (4 on each radius and 1 midway between radii) where exactly three diagonals intersect.

Cf. A006561, A007678, A101364, A101365

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. 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

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

Sequence in context: A086169 A107816 A036835 this_sequence A003685 A066011 A007016

Adjacent sequences: A101360 A101361 A101362 this_sequence A101364 A101365 A101366

nonn

Graeme McRae (g_m(AT)mcraefamily.com), Dec 26 2004, revised Feb 23 2008, Feb 26 2008

There was some confusion about the precise definition of this sequence and some of the comments may still be wrong. The whole entry needs to be rechecked. - N. J. A. Sloane (njas(AT)research.att.com), Mar 17 2008