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

3,3

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

Formula

a(n) = (1/8)*n*(n-3)*(n^2-8*n+19) for n odd.

a(n) = A006561(n) + A211382(n).

Prog

(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

Crossrefs

Cf. A006561, A146212, A211382.

Sequence in context: A111009 A012172 A316536 * A066873 A105262 A298234

Adjacent sequences: A211380 A211381 A211382 * A211384 A211385 A211386

Keyword

nonn

Author

Martin Renner, Feb 07 2013

Extensions

More terms from Robin Visser, Jul 29 2024

Status

approved