A003447 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals rooted at a cell up to rotation and reflection.

1, 2, 7, 26, 108, 434, 1765, 7086, 28384, 113092, 449582, 1783092, 7062611, 27944394, 110494113, 436699670, 1725474562, 6816591452, 26927828642, 106375090796, 420248084468, 1660408588852, 6561147261682, 25930381015756, 102496390643352, 405212762977544

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of dissections of regular n-gon into n-3 polygons with reflection and rooted at a cell. - Sean A. Irvine, May 13 2015

The dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 24 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.

Crossrefs

Cf. A003448, A003452, A003456.

Programs

PARI

DissectionsModDihedralRooted(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(R=sum(i=1, (#v-1)\2, v[2*i+1]*u(2, i)), Q=sum(i=2, #v\2, v[2*i]*u(2, i-1)), T=sum(i=3, #v, my(c=v[i]); if(c, c*sumdiv(i, d, eulerphi(d)*u(d, i/d))/i)));
my(p=O(x*x^n) + (R*(x+R)/(1-Q) + Q*(u(2,1)+(x+R)^2/(1-Q)^2)/2 + T)/2);
vector(n, i, polcoeff(p, i))}
my(v=DissectionsModDihedralRooted(apply(i->if(i>=3&&i<=4,y^(i-3)+O(y^2)),[1..25]))); apply(p->polcoeff(p,1), v[4..#v])

Extensions

More terms from Sean A. Irvine, May 13 2015

Name clarified by Andrew Howroyd, Nov 24 2017