A003441 Number of nonequivalent dissections of a polygon into n triangles by nonintersecting diagonals rooted at a cell up to rotation. (Formerly M2840)

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

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