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Search: id:A003441
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| A003441 |
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Number of dissections of a polygon.
(Formerly M2840)
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+0
3
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| 1, 1, 3, 10, 30, 99, 335, 1144, 3978, 14000, 49742, 178296, 643856, 2340135, 8554275, 31429068, 115997970, 429874830, 1598952498, 5967382200, 22338765540, 83859016956, 315614844558, 1190680751376, 4501802224520, 17055399281284
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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It would be nice to have a more precise description.
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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).
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
P. Lisonek, Closed forms for the number of polygon dissections. Journal of Symbolic Computation 20 (1995), 595-601.
R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.
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FORMULA
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a(n) = number of necklaces of n-1 white beads and n+2 black beads. a(n) = binomial[2n+1, n-1]/(2n+1) + 2/3 C[(n-1)/3] where C is the Catalan number A000108 (assumed to be 0 for nonintegral argument). G.f.: ( ((1-Sqrt[1-4x])/2)^3 + (1-Sqrt[1-4x^3]) )/(3x^2).
Numbers so far suggest that two trisections of sequence agree with those of A050181. - R. Stephan, Mar 28 2004
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MAPLE
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[seq(combstruct[count]([C, {C=Cycle(BT, card=3), BT=Union(Z, Prod(BT, BT))}], size=n), n=0..12)];
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CROSSREFS
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Sequence in context: A014531 A062107 A033113 this_sequence A136841 A136846 A004663
Adjacent sequences: A003438 A003439 A003440 this_sequence A003442 A003443 A003444
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
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