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A003442 Number of nonequivalent dissections of an n-gon into (n-3) polygons by nonintersecting diagonals rooted at a cell up to rotation.
(Formerly M2002)
5
1, 2, 11, 48, 208, 858, 3507, 14144, 56698, 226100, 898942, 3565920, 14124496, 55887930, 220985795, 873396480, 3450940830, 13633173180, 53855628554, 212750148000, 840496068160, 3320817060132, 13122294166126, 51860761615488 (list; graph; refs; listen; history; text; internal format)
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-1 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.

C. R. 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

Cf. A003443, A003454, A220881, A295622.

Sequence in context: A019005 A112288 A192699 * A054894 A270662 A139475

Adjacent sequences:  A003439 A003440 A003441 * A003443 A003444 A003445

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Sean A. Irvine, May 05 2015

Name clarified by Andrew Howroyd, Nov 22 2017

STATUS

approved

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Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)