1,2

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)."

Sequences formed by drawing all diagonals in regular polygon

a(3)=54 because there are 54 points in the interior of an 18-gon at which exactly four diagonals intersect.

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

Adjacent sequences: A137935 A137936 A137937 this_sequence A137939 A137940 A137941

Sequence in context: A005549 A124858 A126399 this_sequence A009726 A009653 A133001

nonn

Graeme McRae (g_m(AT)mcraefamily.com), Feb 23 2008