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 A003441 Number of nonequivalent dissections of a polygon into n triangles by nonintersecting diagonals rooted at a cell up to rotation. (Formerly M2840) 4
 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; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Number of dissections of regular (n+2)-gon into n polygons without reflection and rooted at a cell. - Sean A. Irvine, May 05 2015 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4. F. Harary, E. M. Palmer, R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974 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, Preprint (1974) Ronald C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978) 370-388. FORMULA 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. - Ralf Stephan, Mar 28 2004 MAPLE [seq(combstruct[count]([C, {C=Cycle(BT, card=3), BT=Union(Z, Prod(BT, BT))}], size=n), n=0..12)]; MATHEMATICA a[n_] := DivisorSum[GCD[3, n-1], EulerPhi[#] Binomial[(2n+1)/#, (n-1)/#]/ (2n+1)&]; Array[a, 30] (* Jean-François Alcover, Jul 02 2018 *) PROG (PARI) catalan(n) = binomial(2*n, n)/(n+1); a(n) = binomial(2*n+1, n-1)/(2*n+1) + 2/3*(if ((n-1) % 3, 0, catalan((n-1)/3))); \\ Michel Marcus, Jan 23 2016 CROSSREFS Column k=3 of A295222. Sequence in context: A290718 A300421 A302289 * A136841 A136846 A004663 Adjacent sequences: A003438 A003439 A003440 * A003442 A003443 A003444 KEYWORD nonn AUTHOR N. J. A. Sloane EXTENSIONS More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003 Name edited by Andrew Howroyd, Nov 20 2017 STATUS approved

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