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

%I A003453 M2542 #74 Jan 19 2021 11:48:00
%S A003453 1,3,6,11,17,26,36,50,65,85,106,133,161,196,232,276,321,375,430,495,
%T A003453 561,638,716,806,897,1001,1106,1225,1345,1480,1616,1768,1921,2091,
%U A003453 2262,2451,2641,2850,3060,3290,3521,3773,4026
%N A003453 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation and reflection.
%C A003453 In other words, the number of 2-dissections of an n-gon modulo the dihedral action.
%C A003453 _John W. Layman_ observes that this appears to be the alternating sum transform (PSumSIGN) of A005744.
%C A003453 Row 2 of the convolution array A213847. - _Clark Kimberling_, Jul 05 2012
%C A003453 Number of nonisomorphic outer planar graphs of order n >= 3 and size n+2. - _Christian Barrientos_ and _Sarah Minion_, Feb 27 2018
%D A003453 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003453 T. D. Noe, <a href="/A003453/b003453.txt">Table of n, a(n) for n=5..1000</a>
%H A003453 Douglas Bowman and Alon Regev, <a href="http://arxiv.org/abs/1209.6270">Counting symmetry classes of dissections of a convex regular polygon</a>, arXiv preprint arXiv:1209.6270 [math.CO], 2012. See Theorem 5(2).
%H A003453 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 A003453 Petr Lisonek, <a href="http://dx.doi.org/10.1016/j.jcta.2006.06.013">Combinatorial families enumerated by quasi-polynomials</a>, Journal of Combinatorial Theory, Series A, Volume 114, Issue 4, May 2007, Pages 619-630.
%H A003453 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.
%H A003453 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A003453 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A003453 G.f.: (1+x-x^2) / ((1-x)^4*(1+x)^2).
%F A003453 See also the Maple code.
%F A003453 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
%F A003453 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
%F A003453 a(n) = (1/2) * Sum_{i=1..n-4} floor((i+1)*(n-i-2)/2). - _Wesley Ivan Hurt_, May 07 2016
%p A003453 T52:= proc(n)
%p A003453 if n mod 2 = 0 then (n-4)*(n-2)*(n+3)/24;
%p A003453 else (n-3)*(n^2-13)/24; fi end;
%p A003453 [seq(T52(n),n=5..80)]; # _N. J. A. Sloane_, Dec 28 2012
%t A003453 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 *)
%o A003453 (PARI) \\ See A295419 for DissectionsModDihedral()
%o A003453 { my(v=DissectionsModDihedral(apply(i->y + O(y^4), [1..40]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ _Andrew Howroyd_, Nov 24 2017
%Y A003453 Column 3 of A295634.
%Y A003453 Cf. A005744, A213847, A295419.
%K A003453 nonn,easy,nice
%O A003453 5,2
%A A003453 _N. J. A. Sloane_, _Simon Plouffe_
%E A003453 Entry revised (following Bowman and Regev) by _N. J. A. Sloane_, Dec 28 2012
%E A003453 Name clarified by _Andrew Howroyd_, Nov 24 2017