A211383 Number of intersections of diagonals in the interior and exterior of a regular n-gon.
0, 1, 5, 13, 42, 73, 189, 271, 572, 661, 1365, 1569, 2790, 3057, 5117, 4555, 8664, 9041, 13797, 14213, 20930, 18625, 30525, 30967, 43092, 43513, 59189, 45871, 79422, 79713, 104445, 104619, 134960, 124921, 171717, 171533, 215514, 215081, 267197, 234319, 327660, 326569, 397845, 396337
Offset: 3
Keywords
Links
- Robin Visser, Table of n, a(n) for n = 3..95
- Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: 10.1137/S0895480195281246; arXiv: math/9508209 [math.MG], 1995-2006.
- Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals
Programs
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Sage
def a(n): K = CyclotomicField(n); z = K.gen(); S = set() for i in range(n): for j in range(i+2, n): for k in range(j+1, n): for l in range(k+2, n+j): x = (z^(i-j)-z^(j-i))*(z^l-z^k)-(z^(k-l)-z^(l-k))*(z^j-z^i) y = (z^-j-z^-i)*(z^l-z^k)-(z^-l-z^-k)*(z^j-z^i) if (l!=n+i) and (not y.is_zero()): S.add(x/y) return len(S) # Robin Visser, Jul 29 2024
Extensions
More terms from Robin Visser, Jul 29 2024