A146212 Number of intersection points of all lines through pairs of vertices of a regular n-gon.

3, 5, 15, 37, 91, 145, 333, 471, 891, 901, 1963, 2185, 3795, 3969, 6681, 5563, 10963, 11141, 17031, 17293, 25323, 21913, 36325, 36479, 50571, 50485, 68643, 51661, 91171, 90753, 118833, 118355, 152355, 139861, 192511, 191445, 240123, 238481

Offset: 3

Author

T. D. Noe, Oct 28 2008

Keywords

• nice
• nonn

Comments

This includes intersection points outside of the n-gon. Note that for odd n, n divides a(n); for even n, n divides a(n)-1. For odd n, it appears that a(n)=n*(n^3-7*n^2+15*n-1)/8.

That formula for odd n is correct: see the Sidorenko link. - N. J. A. Sloane, Sep 12 2021

Examples

			a(5)=15 because there are 5 points inside the pentagon, 5 points on the pentagon and five points outside of the pentagon.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 3..100
• T. D. Noe, Pentagon Illustrated
• J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
• Scott R. Shannon, Image for n = 3. In this and other images the dots showing the regular n-gon's vertices are slightly larger and circled with white for clarity. The dot color key is at the top-left of the image.
• Scott R. Shannon, Image for n = 4.
• Scott R. Shannon, Image for n = 5.
• Scott R. Shannon, Image for n = 6.
• Scott R. Shannon, Image for n = 7.
• Scott R. Shannon, Image for n = 8.
• Scott R. Shannon, Image for n = 9.
• Scott R. Shannon, Image for n = 10.
• Scott R. Shannon, Image for n = 11.
• Scott R. Shannon, Image for n = 12.
• Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.

Crossrefs

Cf. A006561, A007569, A146213.

Bisection: A347319, A347321.

Formula

There is a formula for odd n: see Comment section and the Sidorenko link. - N. J. A. Sloane, Sep 12 2021

Extensions

More terms from Jon E. Schoenfield, Nov 10 2008

Definition clarified by N. J. A. Sloane, Jun 06 2025