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. (Formerly M1772)

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

4,2

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.

Prog

(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])

Crossrefs

Cf. A003448, A003452, A003456.

Sequence in context: A006603 A080244 A124542 * A150569 A150570 A150571

Adjacent sequences: A003444 A003445 A003446 * A003448 A003449 A003450

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, May 13 2015

Name clarified by Andrew Howroyd, Nov 24 2017

Status

approved