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

5,1

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.

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

Prog

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

Crossrefs

Cf. A003442, A003451, A003452, A003453.

Sequence in context: A293404 A293414 A212252 * A294415 A141595 A112051

Adjacent sequences: A295619 A295620 A295621 * A295623 A295624 A295625

Keyword

nonn

Author

Andrew Howroyd, Nov 24 2017

Status

approved