A295622 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals rooted at a cell up to rotation.

3, 11, 24, 46, 75, 117, 168, 236, 315, 415, 528, 666, 819, 1001, 1200, 1432, 1683, 1971, 2280, 2630, 3003, 3421, 3864, 4356, 4875, 5447, 6048, 6706, 7395, 8145, 8928, 9776, 10659, 11611, 12600, 13662, 14763, 15941, 17160, 18460, 19803, 21231, 22704, 24266

Offset: 5

Author

Andrew Howroyd, Nov 24 2017

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 5..500
• P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
• Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.

Crossrefs

Cf. A003442, A003451, A003452, A003453.

Programs

PARI

\\ See A003442 for DissectionsModCyclicRooted()
{ my(v=DissectionsModCyclicRooted(apply(i->y + O(y^4), [1..40]))); apply(p->polcoeff(p, 3), v[5..#v]) }

Formula

Conjectures from Colin Barker, Nov 25 2017: (Start)

G.f.: x^5*(3 + 5*x - x^2 - x^3) / ((1 - x)^4*(1 + x)^2).

a(n) = (n-4)*(-5 + (-1)^n - 4*n + 2*n^2) / 8 for n>4.

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>10.

(End)

a(n) = Sum_{k=0..n-5} f(k), where f(n) = Sum_{k=0..n} (3 + lcm(k, 2)) (conjecture). - Jon Maiga, Nov 28 2018