cp's OEIS Frontend

A211382 Number of intersections of diagonals in the exterior of a regular n-gon.

%I A211382 #22 Feb 16 2025 08:33:17
%S A211382 0,0,0,0,7,24,63,110,242,360,650,812,1425,1680,2737,2718,4788,5200,
%T A211382 7812,8272,12075,11328,17875,18486,25542,26264,35438,29070,47957,
%U A211382 48800,63525,64362,82600,77940,105672,106552,133263,134200,165927,149478,204250,205128,248850,249596,300377
%N A211382 Number of intersections of diagonals in the exterior of a regular n-gon.
%H A211382 Robin Visser, <a href="/A211382/b211382.txt">Table of n, a(n) for n = 3..95</a>
%H A211382 Bjorn Poonen and Michael Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">The Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: <a href="http://dx.doi.org/10.1137/S0895480195281246">10.1137/S0895480195281246</a>; arXiv: <a href="http://arXiv.org/abs/math.MG/9508209">math.MG/9508209</a>.
%H A211382 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RegularPolygonDivisionbyDiagonals.html">Regular Polygon Division by Diagonals</a>.
%F A211382 a(n) = 1/24*n*(n-3)*(n-5)*(2*n-11) for n odd
%o A211382 (Sage)
%o A211382 def a(n):
%o A211382     K = CyclotomicField(n); z = K.gen(); S = set()
%o A211382     for i in range(n):
%o A211382         for j in range(i+2, n):
%o A211382             for k in range(j+1, n):
%o A211382                 for l in range(k+2, n+i):
%o A211382                     x = (z^(i-j)-z^(j-i))*(z^l-z^k)-(z^(k-l)-z^(l-k))*(z^j-z^i)
%o A211382                     y = (z^-j-z^-i)*(z^l-z^k)-(z^-l-z^-k)*(z^j-z^i)
%o A211382                     if (not y.is_zero()): S.add(x/y)
%o A211382     return len(S)  # _Robin Visser_, Jul 29 2024
%Y A211382 Cf. A006561, A146213, A211383.
%K A211382 nonn
%O A211382 3,5
%A A211382 _Martin Renner_, Feb 07 2013
%E A211382 a(17)-a(29) from _Martin Renner_, Feb 24 2013
%E A211382 More terms from _Robin Visser_, Jul 29 2024