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

0, 0, 0, 0, 7, 24, 63, 110, 242, 360, 650, 812, 1425, 1680, 2737, 2718, 4788, 5200, 7812, 8272, 12075, 11328, 17875, 18486, 25542, 26264, 35438, 29070, 47957, 48800, 63525, 64362, 82600, 77940, 105672, 106552, 133263, 134200, 165927, 149478, 204250, 205128, 248850, 249596, 300377

Offset: 3

Author

Martin Renner, Feb 07 2013

Keywords

• nonn

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.MG/9508209.
• Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.

Crossrefs

Cf. A006561, A146213, A211383.

Programs

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+i):
                    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 (not y.is_zero()): S.add(x/y)
    return len(S)  # Robin Visser, Jul 29 2024

Formula

a(n) = 1/24*n*(n-3)*(n-5)*(2*n-11) for n odd

Extensions

a(17)-a(29) from Martin Renner, Feb 24 2013

More terms from Robin Visser, Jul 29 2024