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A003449 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation and reflection.
(Formerly M2687)
4
1, 1, 3, 7, 24, 74, 259, 891, 3176, 11326, 40942, 148646, 543515, 1996212, 7367075, 27294355, 101501266, 378701686, 1417263770, 5318762098, 20011847548, 75473144396, 285267393358, 1080432637662, 4099856060808, 15585106611244, 59343290815356 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,3

COMMENTS

In other word, the number of almost-triangulations of an n-gon modulo the dihedral action.

Equivalently, the number of edges of the (n-3)-dimensional associahedron modulo the dihedral action.

The dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 24 2017

See Theorem 30 of Bowman and Regev (although there appears to be a typo in the formula - see Maple code below). - N. J. A. Sloane, Dec 28 2012

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

D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270, 2012.

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.

MAPLE

C:=n->binomial(2*n, n)/(n+1);

T30:=proc(n) local t1; global C;

if n mod 2 = 0 then

t1:=(1/4-(3/(4*n)))*C(n-2) + (3/8)*C(n/2-1) + (1-3/n)*C(n/2-2);

if n mod 4 = 0 then t1:=t1+C(n/4-1)/4 fi;

else

t1:=(1/4-(3/(4*n)))*C(n-2) + (1/2)*C((n-3)/2);

fi;

t1; end;

[seq(T30(n), n=4..40)]; # N. J. A. Sloane, Dec 28 2012

MATHEMATICA

c = CatalanNumber;

T30[n_] := Module[{t1}, If[EvenQ[n], t1 = (1/4 - (3/(4*n)))*c[n - 2] + (3/8)*c[n/2 - 1] + (1 - 3/n)*c[n/2 - 2]; If[Mod[n, 4] == 0, t1 = t1 + c[n/4 - 1]/4], t1 = (1/4 - (3/(4*n)))*c[n-2] + (1/2)*c[(n-3)/2]]; t1];

Table[T30[n], {n, 4, 40}] (* Jean-Fran├žois Alcover, Dec 14 2017, after N. J. A. Sloane *)

PROG

(PARI) \\ See A295419 for DissectionsModDihedral()

{ my(v=DissectionsModDihedral(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 24 2017

CROSSREFS

A diagonal of A295634.

Cf. A003450, A295419.

Sequence in context: A148716 A148717 A148718 * A258308 A148719 A138541

Adjacent sequences:  A003446 A003447 A003448 * A003450 A003451 A003452

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012

Name clarified by Andrew Howroyd, Nov 24 2017

STATUS

approved

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Last modified March 26 18:36 EDT 2019. Contains 321511 sequences. (Running on oeis4.)