login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.
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
KEYWORD
nonn
AUTHOR
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 23 20:33 EDT 2024. Contains 371916 sequences. (Running on oeis4.)