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A003451 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.
(Formerly M3330)
4

%I M3330

%S 1,4,8,16,25,40,56,80,105,140,176,224,273,336,400,480,561,660,760,880,

%T 1001,1144,1288,1456,1625,1820,2016,2240,2465,2720,2976,3264,3553,

%U 3876,4200,4560,4921,5320,5720,6160,6601,7084,7568,8096,8625,9200,9776,10400

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

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

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

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

%F G.f.: x^5 * (1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2).

%F See also the Maple code for an explicit formula.

%F a(n) = A006584(n+3) - A027656(n). - _Yosu Yurramendi_, Aug 07 2008

%F a(n) = (n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24, for n>=5. - _Luce ETIENNE_, Apr 04 2015

%p T51:= proc(n)

%p if n mod 2 = 0 then n*(n-2)*(n-4)/12;

%p else (n+1)*(n-3)*(n-4)/12; fi end;

%p [seq(T51(n),n=5..80)]; # _N. J. A. Sloane_, Dec 28 2012

%t Table[((n - 4) (2 n^2 - 4 n - 3 (1 - (-1)^n)) / 24), {n, 5, 60}] (* _Vincenzo Librandi_, Apr 05 2015 *)

%t CoefficientList[Series[(1+2*x-x^2)/((1-x)^4*(1+x)^2),{x,0,20}],x] (* _Vaclav Kotesovec_, Apr 05 2015 *)

%o (PARI) Vec((1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ _Michel Marcus_, Apr 04 2015

%o (PARI) \\ See A295495 for DissectionsModCyclic()

%o { my(v=DissectionsModCyclic(apply(i->y, [1..30]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ _Andrew Howroyd_, Nov 24 2017

%o (MAGMA) [(n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24: n in [5..60]]; // _Vincenzo Librandi_, Apr 05 2015

%Y Column 3 of A295633.

%Y Cf. A003453, A006584, A027656.

%K nonn

%O 5,2

%A _N. J. A. Sloane_

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

%E First formula adapted to offset by _Vaclav Kotesovec_, Apr 05 2015

%E Name clarified by _Andrew Howroyd_, Nov 25 2017

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Last modified March 20 23:22 EDT 2019. Contains 321354 sequences. (Running on oeis4.)