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A063549 Smallest number of crossing-free matchings on n points in the plane. 5
1, 1, 3, 2, 10, 5, 35, 14, 126, 42, 462, 132, 1716, 429, 6435, 1430, 24310, 4862, 92378, 16796, 352716, 58786, 1352078, 208012, 5200300, 742900, 20058300, 2674440, 77558760, 9694845, 300540195, 35357670, 1166803110, 129644790, 4537567650 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
a(n) = Catalan(n/2) if n is even else n*Catalan((n-1)/2) (see Garcia reference). The same as A057977. - Vladeta Jovovic, Mar 20 2010
LINKS
O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
A. Garcia, M. Noy, and J. Tejel, Lower bounds on the number of crossing-free subgraphs of K_N, Comput. Geom., 16 (2000), pp. 211-221.
FORMULA
(n+2)*a(n) -n*a(n-1) +4*(-2*n+1)*a(n-2) +4*(n-1)*a(n-3) +16*(n-3)*a(n-4)=0. - R. J. Mathar, Jun 13 2013
Sum_{n>=1} 1/a(n) = 5/3 + 8*Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022
MAPLE
# See A057977 for an implementation based on ballot numbers. Peter Luschny, Feb 23 2019
MATHEMATICA
a[n_?EvenQ] := CatalanNumber[n/2]; a[n_?OddQ] := n*CatalanNumber[(n-1)/2]; Table[a[n], {n, 3, 35}] (* Jean-François Alcover, Feb 03 2012, after Vladeta Jovovic *)
CROSSREFS
Sequence in context: A371220 A318280 A057977 * A071653 A227631 A246830
KEYWORD
nonn,nice,easy
AUTHOR
N. J. A. Sloane, Aug 14 2001
EXTENSIONS
More terms from Jean-François Alcover, Feb 03 2012
a(1) = a(2) = 1 inserted and added Garcia reference from Nathaniel Johnston, Nov 17 2014
STATUS
approved

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Last modified April 23 03:30 EDT 2024. Contains 371906 sequences. (Running on oeis4.)