%I M2687 #42 Nov 29 2023 06:58:41

%S 1,1,3,7,24,74,259,891,3176,11326,40942,148646,543515,1996212,7367075,

%T 27294355,101501266,378701686,1417263770,5318762098,20011847548,

%U 75473144396,285267393358,1080432637662,4099856060808,15585106611244,59343290815356

%N Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation and reflection.

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

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

%C The dissection will always be composed of one quadrilateral and n-4 triangles. - _Andrew Howroyd_, Nov 24 2017

%C 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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A003449/b003449.txt">Table of n, a(n) for n = 4..200</a>

%H D. Bowman and A. Regev, <a href="http://arxiv.org/abs/1209.6270">Counting symmetry classes of dissections of a convex regular polygon</a>, arXiv:1209.6270 [math.CO], 2012.

%H P. Lisonek, <a href="http://dx.doi.org/10.1006/jsco.1995.1066">Closed forms for the number of polygon dissections</a>, Journal of Symbolic Computation 20 (1995), 595-601.

%H Ronald C. Read, <a href="http://dx.doi.org/10.1007/BF03031688">On general dissections of a polygon</a>, Aequat. math. 18 (1978) 370-388.

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

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

%p if n mod 2 = 0 then

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

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

%p else

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

%p fi;

%p t1; end;

%p [seq(T30(n),n=4..40)]; # _N. J. A. Sloane_, Dec 28 2012

%t c = CatalanNumber;

%t 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];

%t Table[T30[n], {n, 4, 40}] (* _Jean-François Alcover_, Dec 14 2017, after _N. J. A. Sloane_ *)

%o (PARI) \\ See A295419 for DissectionsModDihedral()

%o { 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

%Y A diagonal of A295634.

%Y Cf. A003450, A295419.

%K nonn

%O 4,3

%A _N. J. A. Sloane_

%E Entry revised (following Bowman and Regev) by _N. J. A. Sloane_, Dec 28 2012

%E Name clarified by _Andrew Howroyd_, Nov 24 2017