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A003453
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Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation and reflection.
(Formerly M2542)
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5
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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
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OFFSET
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5,2
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COMMENTS
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In other words, the number of 2-dissections of an n-gon modulo the dihedral action.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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