A003442 Number of nonequivalent dissections of an n-gon into (n-3) polygons by nonintersecting diagonals rooted at a cell up to rotation.

1, 2, 11, 48, 208, 858, 3507, 14144, 56698, 226100, 898942, 3565920, 14124496, 55887930, 220985795, 873396480, 3450940830, 13633173180, 53855628554, 212750148000, 840496068160, 3320817060132, 13122294166126, 51860761615488

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of dissections of regular n-gon into n-3 polygons without reflection and rooted at a cell. - Sean A. Irvine, May 05 2015

The conditions imposed mean that the dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 23 2017

Examples

			Case n=5: A pentagon can be dissected into 1 quadrilateral and 1 triangle. Either one of these can be chosen as the root cell so a(n)=2. - _Andrew Howroyd_, Nov 23 2017
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrew Howroyd, Table of n, a(n) for n = 4..200
• 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.
• Andrey Zabolotskiy, Illustration for n = 4,5,6

Crossrefs

Cf. A003443, A003454, A220881, A295622.

Programs

PARI

DissectionsModCyclicRooted(v)={my(n=#v);
my(q=vector(n)); q[1]=serreverse(x-sum(i=3,#v,x^i*v[i])/x + O(x*x^n));
for(i=2, n, q[i]=q[i-1]*q[1]);
my(vars=variables(q[1]));
my(u(m,r)=substvec(q[r]+O(x^(n\m+1)), vars, apply(t->t^m,vars)));
my(p=O(x*x^n) + sum(i=3, #v, my(c=v[i]); if(c, c*sumdiv(i, d, eulerphi(d)*u(d,i/d))/i)));
vector(n,i,polcoeff(p,i))}
{ my(v=DissectionsModCyclicRooted(apply(i->if(i>=3&&i<=4,y^(i-3) + O(y^2)), [1..25]))); apply(p->polcoeff(p,1), v[4..#v]) } \\ Andrew Howroyd, Nov 22 2017

Extensions

More terms from Sean A. Irvine, May 05 2015

Name clarified by Andrew Howroyd, Nov 22 2017