cp's OEIS Frontend

A005315 Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.

%I A005315 M1862 #117 Aug 24 2025 11:35:29
%S A005315 1,1,2,8,42,262,1828,13820,110954,933458,8152860,73424650,678390116,
%T A005315 6405031050,61606881612,602188541928,5969806669034,59923200729046,
%U A005315 608188709574124,6234277838531806,64477712119584604,672265814872772972,7060941974458061392
%N A005315 Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.
%C A005315 There is a 1-to-1 correspondence between loops crossing a road 2n times and lines crossing a road 2n-1 times.
%D A005315 S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303.
%D A005315 S. K. Lando and A. K. Zvonkin, Meanders, Selecta Mathematica Sovietica, Vol. 11, Number 2, pp. 117-144, 1992.
%D A005315 A. Phillips, Simple Alternating Transit Mazes, preprint. Abridged version appeared as "La topologia dei labirinti," in M. Emmer, editor, L'Occhio di Horus: Itinerari nell'Imaginario Matematico. Istituto della Enciclopedia Italia, Rome, 1989, pp. 57-67.
%D A005315 V. R. Pratt, personal communication.
%D A005315 J. A. Reeds and L. A. Shepp, An upper bound on the meander constant, preprint, May 25, 1999. [Obtains upper bound of 13.01]
%D A005315 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005315 For additional references, see A005316.
%H A005315 Andrew Howroyd, <a href="/A005315/b005315.txt">Table of n, a(n) for n = 0..28</a> (first 24 terms from Iwan Jensen)
%H A005315 Oswin Aichholzer, Carlos Alegría Galicia, Irene Parada, Alexander Pilz, Javier Tejel, Csaba D. Tóth, Jorge Urrutia, and Birgit Vogtenhuber, <a href="http://www.ist.tugraz.at/files/publications/geometry/aappttuv-hmpcb-19.pdf">Hamiltonian meander paths and cycles on bichromatic point sets</a>, XVIII Spanish Meeting on Computational Geometry (Girona, 2019).
%H A005315 V. I. Arnol'd, <a href="https://doi.org/10.1007/BF00970265">A branched covering of CP^2->S^4, hyperbolicity and projective topology</a> [ Russian ], Sibir. Mat. Zhurn., 29 (No. 2, 1988), 36-47 = Siberian Math. J., 29 (1988), 717-725.
%H A005315 Roland Bacher, <a href="http://www-fourier.ujf-grenoble.fr/sites/default/files/ref_478.pdf">Meander algebras</a>
%H A005315 David Bevan, <a href="http://demonstrations.wolfram.com/RandomClosedMeanders">Random Closed Meanders</a> - _David Bevan_, Jun 25 2010
%H A005315 Alexander E. Black, Kevin Liu, Alex Mcdonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, <a href="https://arxiv.org/abs/2304.05318">Sampling planar tanglegrams and pairs of disjoint triangulations</a>, arXiv:2304.05318 [math.CO], 2023.
%H A005315 B. Bobier and J. Sawada, <a href="http://www.cis.uoguelph.ca/~sawada/papers/meander.pdf">A fast algorithm to generate open meandric systems and meanders</a>, Transactions on Algorithms, Vol. 6 No. 2 (2010) article #42, 12 pages.
%H A005315 P. Di Francesco, O. Golinelli and E. Guitter, <a href="http://arXiv.org/abs/hep-th/9506030">Meander, folding and arch statistics</a>, arXiv:hep-th/9506030, 1995.
%H A005315 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 525.
%H A005315 Reinhard O. W. Franz, and Berton A. Earnshaw, <a href="http://dx.doi.org/10.1007/s00026-002-8026-z">A constructive enumeration of meanders</a>, Ann. Comb. 6 (2002), no. 1, 7-17.
%H A005315 Erich Friedman, <a href="/A005315/a005315.gif">Illustration of initial terms</a>
%H A005315 Motohisa Fukuda, Ion Nechita, <a href="https://arxiv.org/abs/1609.02756">Enumerating meandric systems with large number of components</a>, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
%H A005315 Iwan Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/">Home page</a>
%H A005315 Iwan Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/meanders/series/closed.meanders.ser">Closed meanders, a(n) for n = 1..24</a>
%H A005315 Iwan Jensen, <a href="http://arxiv.org/abs/cond-mat/9910313">Enumeration of plane meanders</a>, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
%H A005315 Iwan Jensen, <a href="http://dx.doi.org/10.1088/0305-4470/33/34/301">A transfer matrix approach to the enumeration of plane meanders</a>, J. Phys. A 33, 5953-5963 (2000).
%H A005315 Iwan Jensen and A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/33/21/101">Critical exponents of plane meanders</a> J. Phys. A 33, L187-L192 (2000).
%H A005315 Michael La Croix, <a href="http://www.math.uwaterloo.ca/~malacroi/Latex/Meanders.pdf">Approaches to the Enumerative Theory of Meanders</a>, 2003.
%H A005315 S. K. Lando and A. K. Zvonkin, <a href="/A005316/a005316_1.pdf">Plane and projective meanders</a>, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
%H A005315 S. K. Lando and A. K. Zvonkin, <a href="http://dx.doi.org/10.1016/0304-3975(93)90316-L">Plane and projective meanders</a>, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
%H A005315 A. Panayotopoulos, <a href="https://doi.org/10.1007/s11786-018-0389-6">On Meandric Colliers</a>, Mathematics in Computer Science, (2018).
%H A005315 A. Panayotopoulos and P. Tsikouras, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Panayotopoulos/panayo4.html">Meanders and Motzkin Words</a>, J. Integer Seqs., Vol. 7, 2004.
%H A005315 A. Phillips, <a href="http://www.math.sunysb.edu/~tony/mazes/index.html">Mazes</a>
%H A005315 A. Phillips, <a href="http://www.math.sunysb.edu/~tony/mazes/satmaze.html">Simple, Alternating, Transit Mazes</a>
%H A005315 J. A. Reeds, D. E. Knuth, & N. J. A. Sloane, <a href="/A005315/a005315.pdf">Email Correspondence</a>
%H A005315 J. Reeds, L. Shepp, & D. McIlroy, <a href="/A005315/a005315_1.pdf">Numerical bounds for the Arnol'd "meander" constant</a>, Preprint.
%H A005315 M. A. Sainte-Laguë, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k34029008.texteImage">Les Réseaux (ou Graphes)</a>, Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.
%H A005315 N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%F A005315 a(n) = A005316(2n-1) for n>0.
%t A005315 A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {_, _}][[All, 2]];
%t A005315 a[n_] := If[n == 0, 1, A005316[[2n]]];
%t A005315 a /@ Range[0, 28] (* _Jean-François Alcover_, Sep 25 2019 *)
%Y A005315 These are the odd-numbered terms of A005316. Cf. A077054. For nonisomorphic solutions, see A077460.
%Y A005315 A column of triangle A008828.
%K A005315 nonn,hard,nice,changed
%O A005315 0,3
%A A005315 _N. J. A. Sloane_, J. A. Reeds (reeds(AT)idaccr.org)