OFFSET
5,2
COMMENTS
In other words, the number of (n - 5)-dissections of an n-gon modulo the dihedral action.
Equivalently, the number of two-dimensional faces of the (n-3)-dimensional associahedron modulo the dihedral action.
The dissection will always be composed of either 1 pentagon and n-5 triangles or 2 quadrilaterals and n-6 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 = 5..200
D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270 [math.CO], 2012.
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.
FORMULA
See Maple program.
MAPLE
C:=n->binomial(2*n, n)/(n+1);
T32:=proc(n) local t1; global C;
if n mod 2 = 0 then
t1 := (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));
if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;
if n mod 2 = 0 then t1:=t1+((3*(n-4)*(n-1))/(16*(n-3)))*C(n/2-1) fi;
else
t1 := (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));
if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;
if n mod 2 = 1 then t1:=t1+((n^2-2*n-11)/(8*(n-4)))*C((n-3)/2) fi;
fi;
t1; end;
[seq(T32(n), n=5..40)];
MATHEMATICA
c = CatalanNumber;
T32[n_] := Module[{t1}, If[EvenQ[n], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n*(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5)*c[n/5-1]]; If[EvenQ[n], t1 = t1 + ((3*(n-4)*(n-1))/(16*(n-3)))*c[n/2-1]], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n *(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5) * c[n/5-1]]; If[OddQ[n], t1 = t1 + ((n^2 - 2*n - 11)/(8*(n-4)))*c[(n-3)/2]]]; t1];
Table[T32[n], {n, 5, 40}] (* Jean-François Alcover, Dec 11 2017, translated from Maple *)
PROG
(PARI) \\ See A295419 for DissectionsModDihedral()
{ my(v=DissectionsModDihedral(apply(i->if(i>=3&&i<=5, y^(i-3) + O(y^3)), [1..30]))); apply(p->polcoeff(p, 2), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
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