A146213 - OEIS

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

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OFFSET

3,3

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

P. Ryckelynck and L. Smoch, Simuorb: a new method for generating and describing the intersection points of clique-arrangements, arXiv:2509.25234 [math.GM], 2025. See p. 2.

FORMULA

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

CROSSREFS

Cf. A007569, A146212.

Sequence in context: A256539 A109363 A218214 * A344311 A176145 A270978

Adjacent sequences: A146210 A146211 A146212 * A146214 A146215 A146216

KEYWORD

nonn

AUTHOR

T. D. Noe, Oct 28 2008

EXTENSIONS

More terms from Jon E. Schoenfield, Nov 11 2008

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

STATUS

approved