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

%I A146213 #14 Jun 11 2025 16:24:16
%S A146213 0,0,5,18,49,88,198,300,550,588,1235,1414,2415,2576,4284,3708,7068,
%T A146213 7280,11025,11330,16445,14592,23650,23972,32994,33208,44863,34830,
%U A146213 59675,59808,77880,78064,99960,92844,126429,126426,157833,157560,194750
%N A146213 Number of intersection points outside the n-gon of all lines through pairs of vertices of a regular n-gon.
%C A146213 These are a subset of the points counted in A146212.
%C A146213 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.]
%C A146213 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
%H A146213 Jon E. Schoenfield, <a href="/A146213/b146213.txt">Table of n, a(n) for n = 3..100</a>
%F A146213 a(n) = A146212(n) - A007569(n).
%Y A146213 Cf. A007569, A146212.
%K A146213 nonn
%O A146213 3,3
%A A146213 _T. D. Noe_, Oct 28 2008
%E A146213 More terms from _Jon E. Schoenfield_, Nov 11 2008
%E A146213 Definition clarified by _N. J. A. Sloane_, Jun 11 2025