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

%I A146212 #27 Jun 12 2025 13:11:45
%S A146212 3,5,15,37,91,145,333,471,891,901,1963,2185,3795,3969,6681,5563,10963,
%T A146212 11141,17031,17293,25323,21913,36325,36479,50571,50485,68643,51661,
%U A146212 91171,90753,118833,118355,152355,139861,192511,191445,240123,238481
%N A146212 Number of intersection points of all lines through pairs of vertices of a regular n-gon.
%C A146212 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.
%C A146212 That formula for odd n is correct: see the Sidorenko link. - _N. J. A. Sloane_, Sep 12 2021
%H A146212 Jon E. Schoenfield, <a href="/A146212/b146212.txt">Table of n, a(n) for n = 3..100</a>
%H A146212 T. D. Noe, <a href="http://www.sspectra.com/math/A146212.gif">Pentagon Illustrated</a>
%H A146212 J. F. Rigby, <a href="https://doi.org/10.1007/BF00147438">Multiple intersections of diagonals of regular polygons, and related topics</a>, Geom. Dedicata 9 (1980), 207-238.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212.png">Image for n = 3</a>. 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.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212_1.png">Image for n = 4</a>.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212_2.png">Image for n = 5</a>.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212_3.png">Image for n = 6</a>.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212_4.png">Image for n = 7</a>.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212_5.png">Image for n = 8</a>.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212_6.png">Image for n = 9</a>.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212_7.png">Image for n = 10</a>.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212_8.png">Image for n = 11</a>.
%H A146212 Scott R. Shannon, <a href="/A146212/a146212_9.png">Image for n = 12</a>.
%H A146212 Alexander Sidorenko, <a href="/A344857/a344857.txt">Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.</a>
%F A146212 There is a formula for odd n: see Comment section and the Sidorenko link. - _N. J. A. Sloane_, Sep 12 2021
%e A146212 a(5)=15 because there are 5 points inside the pentagon, 5 points on the pentagon and five points outside of the pentagon.
%Y A146212 Cf. A006561, A007569, A146213.
%Y A146212 Bisection: A347319, A347321.
%K A146212 nice,nonn
%O A146212 3,1
%A A146212 _T. D. Noe_, Oct 28 2008
%E A146212 More terms from _Jon E. Schoenfield_, Nov 10 2008
%E A146212 Definition clarified by _N. J. A. Sloane_, Jun 06 2025