A003453 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation and reflection.

1, 3, 6, 11, 17, 26, 36, 50, 65, 85, 106, 133, 161, 196, 232, 276, 321, 375, 430, 495, 561, 638, 716, 806, 897, 1001, 1106, 1225, 1345, 1480, 1616, 1768, 1921, 2091, 2262, 2451, 2641, 2850, 3060, 3290, 3521, 3773, 4026

Offset: 5

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy
• nice

Comments

In other words, the number of 2-dissections of an n-gon modulo the dihedral action.

John W. Layman observes that this appears to be the alternating sum transform (PSumSIGN) of A005744.

Row 2 of the convolution array A213847. - Clark Kimberling, Jul 05 2012

Number of nonisomorphic outer planar graphs of order n >= 3 and size n+2. - Christian Barrientos and Sarah Minion, Feb 27 2018

References

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

Links

• T. D. Noe, Table of n, a(n) for n=5..1000
• Douglas Bowman and Alon Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv preprint arXiv:1209.6270 [math.CO], 2012. See Theorem 5(2).
• P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
• Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, Journal of Combinatorial Theory, Series A, Volume 114, Issue 4, May 2007, Pages 619-630.
• Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.
• N. J. A. Sloane, Transforms
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Column 3 of A295634.

Cf. A005744, A213847, A295419.

Programs

Maple

T52:= proc(n)
if n mod 2 = 0 then (n-4)*(n-2)*(n+3)/24;
else (n-3)*(n^2-13)/24; fi end;
[seq(T52(n),n=5..80)]; # N. J. A. Sloane, Dec 28 2012

Mathematica

nd[n_]:=If[EvenQ[n],(n-4)(n-2) (n+3)/24,(n-3) (n^2-13)/24]; Array[nd,50,5] (* or *) LinearRecurrence[{2,1,-4,1,2,-1},{1,3,6,11,17,26},50] (* Harvey P. Dale, Jan 28 2013 *)

PARI

\\ See A295419 for DissectionsModDihedral()
{ my(v=DissectionsModDihedral(apply(i->y + O(y^4), [1..40]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017

Formula

G.f.: (1+x-x^2) / ((1-x)^4*(1+x)^2).

See also the Maple code.

a(5)=1, a(6)=3, a(7)=6, a(8)=11, a(9)=17, a(10)=26, a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a (n-6). - Harvey P. Dale, Jan 28 2013

a(n) = (2*n^3-6*n^2-23*n+63+3*(n-5)*(-1)^n)/48, for n>=5. - Luce ETIENNE, Apr 07 2015

a(n) = (1/2) * Sum_{i=1..n-4} floor((i+1)*(n-i-2)/2). - Wesley Ivan Hurt, May 07 2016

Extensions

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

Name clarified by Andrew Howroyd, Nov 24 2017