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
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