OFFSET
4,2
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
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
EXAMPLE
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
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, May 05 2015
Name clarified by Andrew Howroyd, Nov 22 2017
STATUS
approved