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A005315 Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.
(Formerly M1862)
30

%I M1862

%S 1,1,2,8,42,262,1828,13820,110954,933458,8152860,73424650,678390116,

%T 6405031050,61606881612,602188541928,5969806669034,59923200729046,

%U 608188709574124,6234277838531806,64477712119584604,672265814872772972,7060941974458061392

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

%C There is a 1-to-1 correspondence between loops crossing a road 2n times and lines crossing a road 2n-1 times.

%D V. I. Arnol'd, A branched covering of CP^2->S^4, hyperbolicity and projective topology [ Russian ], Sibir. Mat. Zhurn., 29 (No. 2, 1988), 36-47 = Siberian Math. J., 29 (1988), 717-725.

%D B. Bobier and J. Sawada, A fast algorithm to generate open meandric systems and meanders, ACM Transactions on Algorithms, Vol. 6, No. 2, 2010, article #42.

%D Franz, Reinhard O. W. and Earnshaw, Berton A. A constructive enumeration of meanders. Ann. Comb. 6 (2002), no. 1, 7-17.

%D Iwan Jensen, A transfer matrix approach to the enumeration of plane meanders. J. Phys. A 33, 5953-5963 (2000).

%D Iwan Jensen and A. J. Guttmann, Critical exponents of plane meanders. J. Phys. A 33, L187-L192 (2000).

%D 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 S. K. Lando and A. K. Zvonkin, Meanders, Selecta Mathematica Sovietica, Vol. 11, Number 2, pp. 117-144, 1992.

%D S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, p227-241, 1993.

%D 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 V. R. Pratt, personal communication.

%D 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 M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.

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

%D For additional references see A005316.

%H Iwan Jensen and Andrew Howroyd, <a href="/A005315/b005315.txt">Table of n, a(n) for n = 0..28</a> (first 24 terms from I. Jensen)

%H R. Bacher, <a href="http://www-fourier.ujf-grenoble.fr/sites/default/files/ref_478.pdf">Meander algebras</a>

%H David Bevan, <a href="http://demonstrations.wolfram.com/RandomClosedMeanders">Random Closed Meanders</a> - _David Bevan_, Jun 25 2010

%H 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 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 525

%H Erich Friedman, <a href="/A005315/a005315.gif">Illustration of initial terms</a>

%H 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 Iwan Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/">Home page</a>

%H 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 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 Michael La Croix, <a href="http://www.math.uwaterloo.ca/~malacroi/Latex/Meanders.pdf">Approaches to the Enumerative Theory of Meanders</a>, 2003.

%H 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 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 A. Phillips, <a href="http://www.math.sunysb.edu/~tony/mazes/index.html">Mazes</a>

%H A. Phillips, <a href="http://www.math.sunysb.edu/~tony/mazes/satmaze.html">Simple, Alternating, Transit Mazes</a>

%H J. A. Reeds, D. E. Knuth, & N. J. A. Sloane, <a href="/A005315/a005315.pdf">Email Correspondence</a>

%H J. Reeds, L. Shepp, & D. McIlroy, <a href="/A005315/a005315_1.pdf">Numerical bounds for the Arnol'd "meander" constant</a>, Preprint.

%H 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 a(n) = A005316(2n-1).

%Y These are the odd-numbered terms of A005316. Cf. A077054. For nonisomorphic solutions see A077460.

%Y A column of triangle A008828.

%K nonn,hard,nice

%O 0,3

%A _N. J. A. Sloane_, J. A. Reeds (reeds(AT)idaccr.org)

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Last modified August 20 20:36 EDT 2018. Contains 313927 sequences. (Running on oeis4.)