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A003451
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Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.
(Formerly M3330)
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4
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1, 4, 8, 16, 25, 40, 56, 80, 105, 140, 176, 224, 273, 336, 400, 480, 561, 660, 760, 880, 1001, 1144, 1288, 1456, 1625, 1820, 2016, 2240, 2465, 2720, 2976, 3264, 3553, 3876, 4200, 4560, 4921, 5320, 5720, 6160, 6601, 7084, 7568, 8096, 8625, 9200, 9776, 10400
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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 cyclic 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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Table of n, a(n) for n=5..52.
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.
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FORMULA
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G.f.: x^5 * (1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2).
See also the Maple code for an explicit formula.
a(n) = A006584(n+3) - A027656(n). - Yosu Yurramendi, Aug 07 2008
a(n) = (n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24, for n>=5. - Luce ETIENNE, Apr 04 2015
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MAPLE
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T51:= proc(n)
if n mod 2 = 0 then n*(n-2)*(n-4)/12;
else (n+1)*(n-3)*(n-4)/12; fi end;
[seq(T51(n), n=5..80)]; # N. J. A. Sloane, Dec 28 2012
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MATHEMATICA
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Table[((n - 4) (2 n^2 - 4 n - 3 (1 - (-1)^n)) / 24), {n, 5, 60}] (* Vincenzo Librandi, Apr 05 2015 *)
CoefficientList[Series[(1+2*x-x^2)/((1-x)^4*(1+x)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 05 2015 *)
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PROG
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(PARI) Vec((1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ Michel Marcus, Apr 04 2015
(PARI) \\ See A295495 for DissectionsModCyclic()
{ my(v=DissectionsModCyclic(apply(i->y, [1..30]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017
(Magma) [(n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24: n in [5..60]]; // Vincenzo Librandi, Apr 05 2015
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CROSSREFS
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Column 3 of A295633.
Cf. A003453, A006584, A027656.
Sequence in context: A022560 A290190 A193452 * A277029 A013934 A167189
Adjacent sequences: A003448 A003449 A003450 * A003452 A003453 A003454
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012
First formula adapted to offset by Vaclav Kotesovec, Apr 05 2015
Name clarified by Andrew Howroyd, Nov 25 2017
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STATUS
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approved
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