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

1, 1, 2, 8, 42, 262, 1828, 13820, 110954, 933458, 8152860, 73424650, 678390116, 6405031050, 61606881612, 602188541928, 5969806669034, 59923200729046, 608188709574124, 6234277838531806, 64477712119584604, 672265814872772972, 7060941974458061392

Offset: 0

Author

N. J. A. Sloane, J. A. Reeds (reeds(AT)idaccr.org)

Keywords

• nonn
• hard
• nice
• changed

Comments

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

References

• 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.
• S. K. Lando and A. K. Zvonkin, Meanders, Selecta Mathematica Sovietica, Vol. 11, Number 2, pp. 117-144, 1992.
• 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.
• V. R. Pratt, personal communication.
• J. A. Reeds and L. A. Shepp, An upper bound on the meander constant, preprint, May 25, 1999. [Obtains upper bound of 13.01]
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• For additional references, see A005316.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..28 (first 24 terms from Iwan Jensen)
• Oswin Aichholzer, Carlos Alegría Galicia, Irene Parada, Alexander Pilz, Javier Tejel, Csaba D. Tóth, Jorge Urrutia, and Birgit Vogtenhuber, Hamiltonian meander paths and cycles on bichromatic point sets, XVIII Spanish Meeting on Computational Geometry (Girona, 2019).
• 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.
• Roland Bacher, Meander algebras
• David Bevan, Random Closed Meanders - _David Bevan_, Jun 25 2010
• Alexander E. Black, Kevin Liu, Alex Mcdonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, Sampling planar tanglegrams and pairs of disjoint triangulations, arXiv:2304.05318 [math.CO], 2023.
• B. Bobier and J. Sawada, A fast algorithm to generate open meandric systems and meanders, Transactions on Algorithms, Vol. 6 No. 2 (2010) article #42, 12 pages.
• P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 525.
• Reinhard O. W. Franz, and Berton A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), no. 1, 7-17.
• Erich Friedman, Illustration of initial terms
• Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of components, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
• Iwan Jensen, Home page
• Iwan Jensen, Closed meanders, a(n) for n = 1..24
• Iwan Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
• Iwan Jensen, A transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33, 5953-5963 (2000).
• Iwan Jensen and A. J. Guttmann, Critical exponents of plane meanders J. Phys. A 33, L187-L192 (2000).
• Michael La Croix, Approaches to the Enumerative Theory of Meanders, 2003.
• 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. (Annotated scanned copy)
• S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
• A. Panayotopoulos, On Meandric Colliers, Mathematics in Computer Science, (2018).
• A. Panayotopoulos and P. Tsikouras, Meanders and Motzkin Words, J. Integer Seqs., Vol. 7, 2004.
• A. Phillips, Mazes
• A. Phillips, Simple, Alternating, Transit Mazes
• J. A. Reeds, D. E. Knuth, & N. J. A. Sloane, Email Correspondence
• J. Reeds, L. Shepp, & D. McIlroy, Numerical bounds for the Arnol'd "meander" constant, Preprint.
• M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Crossrefs

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

A column of triangle A008828.

Programs

Mathematica

A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 0, 1, A005316[[2n]]];
a /@ Range[0, 28] (* Jean-François Alcover, Sep 25 2019 *)

Formula

a(n) = A005316(2n-1) for n>0.