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A003444 Number of dissections of a polygon.
(Formerly M3455)
10
1, 4, 12, 43, 143, 504, 1768, 6310, 22610, 81752, 297160, 1086601, 3991995, 14732720, 54587280, 202997670, 757398510, 2834510744, 10637507400, 40023636310, 150946230006, 570534578704, 2160865067312, 8199711378716, 31170212479588, 118686578956272 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

COMMENTS

See A220881 for an essentially identical sequence, but with a different offset and a more precise definition. - N. J. A. Sloane, Dec 28 2012

Also number of necklaces of 2 colors with 2n beads and n-2 black ones. - Wouter Meeussen, Aug 03 2002

REFERENCES

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, Aequat. Math. 18 (1978), 370-388.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 4..201

D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270, 2012.

FORMULA

(1/(2n)) Sum_{d |(2n, k)} phi(d)*binomial(2n/d, k/d) with k=n-2 - Wouter Meeussen, Aug 03 2002

MATHEMATICA

Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#, (n-2)/# ] &)/@Intersection[Divisors[2n], Divisors[n-2]])/(2n), {n, 3, 32}]

CROSSREFS

Cf. A003445, A003450, A047996, A073020, A220881.

Sequence in context: A149354 A197659 A292287 * A220881 A149355 A149356

Adjacent sequences:  A003441 A003442 A003443 * A003445 A003446 A003447

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Wouter Meeussen, Aug 03 2002

STATUS

approved

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Last modified April 25 03:57 EDT 2019. Contains 322449 sequences. (Running on oeis4.)