A003444 Number of dissections of a polygon. (Formerly M3455)

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

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 [math.CO], 2012.

Formula

a(n) = (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: A197659 A360468 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