A146213 Number of intersection points outside the n-gon of all lines through pairs of vertices of a regular n-gon.

0, 0, 5, 18, 49, 88, 198, 300, 550, 588, 1235, 1414, 2415, 2576, 4284, 3708, 7068, 7280, 11025, 11330, 16445, 14592, 23650, 23972, 32994, 33208, 44863, 34830, 59675, 59808, 77880, 78064, 99960, 92844, 126429, 126426, 157833, 157560, 194750

Offset: 3

Author

T. D. Noe, Oct 28 2008

Keywords

• nonn

Comments

These are a subset of the points counted in A146212.

Note that n divides a(n). For odd n, it appears that a(n)=n(2n^3-15n^2+34n-21)/24. [Corrected by N. J. A. Sloane, Jun 11 2025. See next line.]

Theorem: For odd n >= 3, a(n) = n*(n-1)*(n-3)*(2*n-7)/24. Proof. We now know that for n odd >= 3, A146212(n) = n*(n^3-7*n^2+15*n-1)/8. Subtracting n + binomial(n,4) (see A007569), we get the stated formula. - N. J. A. Sloane, Jun 11 2025

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 3..100

Crossrefs

Cf. A007569, A146212.

Formula

a(n) = A146212(n) - A007569(n).

Extensions

More terms from Jon E. Schoenfield, Nov 11 2008

Definition clarified by N. J. A. Sloane, Jun 11 2025