

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
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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



FORMULA

a(n) = number of necklaces of n1 white beads and n+2 black beads. a(n) = binomial(2n+1, n1)/(2n+1) + (2/3)*C((n1)/3) where C is the Catalan number A000108 (assumed to be 0 for nonintegral argument). G.f.: ( ((1sqrt(14x))/2)^3 + (1sqrt(14x^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, n1], EulerPhi[#] Binomial[(2n+1)/#, (n1)/#]/ (2n+1)&];


PROG

(PARI) catalan(n) = binomial(2*n, n)/(n+1);
a(n) = binomial(2*n+1, n1)/(2*n+1) + 2/3*(if ((n1) % 3, 0, catalan((n1)/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


STATUS

approved



