A005316 -id:A005316 - cp's OEIS frontend

A077460 Number of nonisomorphic ways a loop can cross a road (running East-West) 2n times.

1, 1, 1, 3, 12, 70, 464, 3482, 27779, 233556, 2038484, 18357672, 169599492, 1601270562, 15401735750, 150547249932, 1492451793728, 14980801247673, 152047178479946, 1558569469867824, 16119428039548246

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 03 2002

Nonisomorphic closed meanders, where two closed meanders are considered equivalent if one can be obtained from the other by reflections in an East-West or North-South mirror (a group of order 4).

Symmetries are possible by reflection in a North-South mirror, or by rotation through 180 degrees when n is odd.(see illustration). - Andrew Howroyd, Nov 24 2015

			A meander can be specified by marking 2n equally spaced points along a line and recording the order in which the meander visits the points.
For n = 2, 4, 6, 8 the solutions are as follows:
n=2: 1 2
n=4: 1 2 3 4
n=6: 1 2 3 4 5 6, 1 2 3 6 5 4, 1 2 5 4 3 6
n=8: 1 2 3 4 5 6 7 8, 1 2 3 4 5 8 7 6, 1 2 3 4 7 6 5 8, 1 2 7 6 3 4 5 8, 1 2 3 6 7 8 5 4, 1 2 3 6 5 4 7 8, 1 2 7 6 5 4 3 8, 1 2 3 8 5 6 7 4, 1 2 3 8 7 4 5 6, 1 2 5 6 7 4 3 8, 1 2 7 4 5 6 3 8, 1 4 3 2 7 6 5 8

Andrew Howroyd, Illustration of Closed Meander Symmetries

The total number of closed meanders with 2n crossings is given in A005315. Cf. A000682, A005316, A060206, A077055, A078104, A078105, A078591.

A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[0] = a[1] = 1;
a[n_] := If[OddQ[n], (A005316[[n + 1]] + A005316[[2n]] + A000682[[n]])/4, (A005316[[2n]] + 2 A005316[[n + 1]])/4];
a /@ Range[0, 20] (* Jean-François Alcover, Sep 06 2019, after Andrew Howroyd *)

From Andrew Howroyd, Nov 24 2015: (Start)
a(2n+1) = (A005315(2n+1) + A005316(2n+1) + A060206(n)) / 4.
a(2n) = (A005315(2n) + 2 * A005316(2n)) / 4. (End)

a(10)-a(20) from Andrew Howroyd, Nov 24 2015

A006659 Number of closed meander systems of order n+1 with n components.

Original entry on oeis.org

2, 12, 56, 240, 990, 4004, 16016, 63648, 251940, 994840, 3922512, 15452320, 60843510, 239519700, 942871200, 3711935040, 14615744220, 57562286760, 226760523600, 893550621600, 3522078700140, 13887053160552

Offset: 1

D. Ivanov, S. K. Lando and A. K. Zvonkin (zvonkin(AT)labri.u-bordeaux.fr)

a(n) is the total number of long interior inclines in all Dyck (n+2)-paths. An incline is a maximal subpath of like steps (all Us or all Ds); interior means it does not start or end the path; long means of length >= 2. Example: for n=1, the 5 Dyck 3-paths are shown with long interior inclines in uppercase: uuuddd, uududd, udUUdd, ududud, uuDDud and so a(1)=2. - David Callan, Jul 03 2006

a(n) is the number of corners in all parallelogram polyominoes of semiperimeter n+3. - Emeric Deutsch, Oct 09 2008

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
M. Delest, J. P. Dubernard and I. Dutour, Parallelogram polyominoes and corners, J. Symbolic Computation, 20(1995),503-515.
M. P. Delest, D. Gouyou-Beauchamps and B. Vauquelin, Enumeration of parallelogram polyominoes with given bond and site parameter, Graphs and Combinatorics, 3 (1987), 325-339.
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
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.
Simon Plouffe, Approximations of generating functions and a few conjectures, Master's Thesis UQAM 1992, arXiv:0911.4975 [math.NT], 2009.

Cf. A002057, A001622.
Equals 2*A002694(n+1).
Cf. A005315, A005316.
A diagonal of triangle A008828.

a006659 n = 2 * a007318' (2 * n + 2) (n - 1)
-- Reinhard Zumkeller, Jun 18 2012

seq(2*binomial(2*n+2,n-1),n=1..22); # Emeric Deutsch, Oct 09 2008

f[x_] := 32/((1 + Sqrt[1 - 4x])^4*Sqrt[1 - 4x]); CoefficientList[ Series[ f[x], {x, 0, 21}], x] (* Jean-François Alcover, Dec 07 2011 *)
CoefficientList[Series[4*Exp[2x](BesselI[1,2*x]+ x(x-1)(BesselI[0,2x]+BesselI[1,2x]))/x^2,{x,0,22}],x]Table[n!,{n,0,22}] (* Stefano Spezia, May 10 2022 *)

a(n)=2*binomial(2*n+2,n-1) \\ Charles R Greathouse IV, Dec 07 2011

x='x+O('x^100); Vec(32/(sqrt(1-4*x)*(1+sqrt(1-4*x))^4)) \\ Altug Alkan, Oct 14 2015

G.f.: 32/(sqrt(1-4x)*(1+sqrt(1-4x))^4).
a(n) = (n+1) * A002057(n). - Ralf Stephan, Aug 31 2003
a(n) = 2*binomial(2n+2, n-1). - Emeric Deutsch, Oct 09 2008
a(n) = {(-56 - 30*n - 4*n^2)*a(n+1) + (8*n+12+n^2)*a(n+2), a(0)=2, a(1)=12}. - Simon Plouffe (master's thesis, 1992)
a(n) ~ 2^(2*n+3)/sqrt(n*Pi). - Charles R Greathouse IV, Dec 07 2011
E.g.f.: 4*exp(2*x)*(I_1(2*x) + x*(x - 1)*(I_0(2*x) + I_1(2*x)))/x^2, where I_n(x) is the modified Bessel function of the first kind. - Stefano Spezia, May 09 2022
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 23/12 - 13*Pi/(18*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 53*log(phi)/(5*sqrt(5)) - 37/20, where phi is the golden ratio (A001622). (End)

A259974 Irregular triangle read by rows: T(n,k) = number of meanders with n bridges in which the first bridge is bridge k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 8, 3, 3, 14, 7, 7, 14, 42, 14, 11, 14, 81, 36, 28, 36, 81, 262, 81, 57, 57, 81, 538, 221, 155, 155, 221, 538, 1828, 538, 353, 316, 353, 538, 3926, 1530, 1003, 902, 1003, 1530, 3926, 13820, 3926, 2458, 2053, 2053, 2458, 3926, 30694, 11510, 7214, 6059, 6059, 7214, 11510, 30694, 110954, 30694, 18575, 14810, 13827, 14810, 18575, 30694

Offset: 1

N. J. A. Sloane, Jul 12 2015

			Triangle begins:
1,
1,
1,1,
2,1,
3,2,3,
8,3,3,
14,7,7,14,
42,14,11,14,
81,36,28,36,81,
262,81,57,57,81,
538,221,155,155,221,538,
...

Lando, S. K. and Zvonkin, A. K. Plane and projective meanders. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991).
Lando, S. K. and Zvonkin, A. K. Meanders. In Selected translations. Selecta Math. Soviet. 11 (1992), no. 2, 117-144.

Andrew Howroyd, Table of n, a(n) for n = 1..420
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 (1993), no. 1-2, 227-241.

Diagonals are A005316, A006660, A006661, A006662. Cf. A008828.

T(12,k)-T(40,k) from Andrew Howroyd, Dec 15 2015

A006657 Number of closed meanders with 2 components and 2n bridges.

Original entry on oeis.org

2, 12, 84, 640, 5236, 45164, 406012, 3772008, 35994184, 351173328, 3490681428, 35253449296, 360946635312, 3739935635756, 39159200588780, 413836299216608, 4409705753032648, 47337525317450816, 511563350415103008

Offset: 2

N. J. A. Sloane, Simon Plouffe

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

P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
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)

Cf. A005315.

A column of triangle A008828.

A008828 = Import["https://oeis.org/A008828/b008828.txt", "Table"][[All, 2]];
a[n_] := A008828[[(n^2 - n + 4)/2]];
a /@ Range[2, 20] (* Jean-François Alcover, Sep 25 2019 *)

a(13)-a(20) from Andrew Howroyd, Nov 22 2015

A006664 Number of irreducible systems of meanders.

Original entry on oeis.org

1, 1, 2, 8, 46, 322, 2546, 21870, 199494, 1904624, 18846714, 191955370, 2002141126, 21303422480, 230553207346, 2531848587534, 28159614749270, 316713536035464, 3597509926531778, 41225699113145888, 476180721050626814, 5539597373695447322, 64863295574835126394, 763984568163192551672, 9047263176444565467566

Offset: 0

N. J. A. Sloane, Simon Plouffe

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

Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of components, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
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.

terms = 25;
S[x_] = Sum[CatalanNumber[k]^2 x^k, {k, 0, 2 terms}];
inv = InverseSeries[x S[x^2] + O[x]^(2 terms), x] // Normal;
(S[x^2] /. x -> inv) + O[x]^(2 terms) // CoefficientList[#, x]& // DeleteCases[#, 0]& (* Jean-François Alcover, Sep 04 2018 *)
terms = 25;
S = Sum[CatalanNumber[k]^2 x^k, {k, 0, terms}] + O[x]^terms;
ComposeSeries[S,InverseSeries[x S^2]] // CoefficientList[#, x] &
(* Mamuka Jibladze, Jun 04 2025 *)

A(x^2) = S(x^2)#inv(x*S(x^2)) where # is functional composition, S(x) is g.f. of A001246 (squares of Catalan numbers) and inv(.) is functional inverse. A(x) consists of even-numbered terms of A(x^2), odd terms of which are 0. - Doug McIlroy (doug(AT)cs.dartmouth.edu), Mar 22 2006
A(x) = S(x)#inv(x*S(x)^2) where # is functional composition, S(x) is g.f. of A001246 (squares of Catalan numbers) and inv(.) is functional inverse. - Mamuka Jibladze, Jun 04 2025
A(x) satisfies ODE A^2*(A^2 - A - 4*x) - x*A*(3*A^2 - 6*A + 8*x)*A' + 4*x^2*(A^2 - 3*A + 4*x)*(A')^2 - 2*x^3*(A - 4)*(A')^3 + x^2*A*(A^2 - 16*x)*A'' = 0 (derived from the functional equation using S(x^2) = (2F_1(-1/2,-1/2,1,16x^2)-1) / (4x^2)). - _Mamuka Jibladze, Aug 25 2025

More terms from Doug McIlroy (doug(AT)cs.dartmouth.edu), Mar 22 2006

A077014 Number of ways that a directed line (or river) that starts in the south can cross an east-west road n times.

Original entry on oeis.org

2, 1, 2, 2, 6, 8, 28, 42, 162, 262, 1076, 1828, 7852, 13820, 61388, 110954, 505878, 933458, 4345660, 8152860, 38608380, 73424650, 352686780, 678390116, 3298016912, 6405031050, 31461151108, 61606881612, 305327366988, 602188541928, 3007925909860, 5969806669034

Offset: 0

N. J. A. Sloane and Jon Wild, Nov 29 2002

nonn

The difference between this and A005316 is that here the river can also end up in the southwest. In A005316 it was required to end up in the northeast or southeast.

Or, number of ways that an undirected line that starts and ends anywhere can cross an east-west road n times.

S. Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
S. Legendre, Foldings and Meanders, Australasian Journal of Combinatorics 58(2) (2014), 275-291.

A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[n_] := If[OddQ[n], A005316[[n+1]], 2*A005316[[n+1]] ];
a /@ Range[0, 31] (* Jean-François Alcover, Sep 07 2019 *)

Equals A005316(n) if n odd, or 2*A005316(n) if n even (because then each solution to A005316 can be reflected in a vertical mirror to give another solution).

A287220 Number of single-crossing partitions on n nodes.

Original entry on oeis.org

0, 0, 1, 4, 10, 36, 85, 312, 737, 2760, 6604, 25176

Offset: 0

Benedict W. J. Irwin, May 22 2017

The meandric numbers A005316 are the numbers of paths which cross themselves 0 times.

This sequence is the number of paths that must cross themselves exactly once.

			a(2) = 1, this is from the partition (2,1).
a(3) = 4, this is from the partitions (1,3,2),(2,1,3),(2,3,1),(3,1,2).

B. W. J. Irwin, On the number of k-crossing partitions

Cf. A005316, A287221.

A006660 Number of meanders in which first bridge is 3.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 36, 81, 221, 538, 1530, 3926, 11510, 30694, 92114, 252939, 773259, 2172830, 6743122, 19304190, 60658606, 176343390, 559952824, 1649008456, 5283239596, 15730575554, 50789908772, 152663683494

Offset: 3

Simon Plouffe

nonn

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

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)

Cf. A259974, A005316.

a(19)-a(30) from Andrew Howroyd, Dec 15 2015

A287221 Number of twice-crossing partitions on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 8, 42, 168, 760, 2418, 10490, 30842, 131676

Offset: 0

Benedict W. J. Irwin, May 22 2017

The meandric numbers A005316 are the numbers of paths which cross themselves 0 times.

This sequence is the number of paths that must cross themselves exactly twice.

			a(4) = 8; this is from the partitions
(2,1,4,3), (2,4,3,1),
(3,2,4,1), (3,4,2,1),
(4,1,3,2), (4,2,1,3),
(4,2,3,1), (4,3,1,2).

B. W. J. Irwin, On the number of k-crossing partitions

Cf. A005316, A287220.

A380369 Triangle read by rows: T(n,k) is the number of open meanders with 2n crossings and k exterior top arches, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 7, 6, 1, 0, 36, 32, 12, 1, 0, 221, 202, 94, 20, 1, 0, 1530, 1417, 728, 220, 30, 1, 0, 11510, 10752, 5854, 2090, 445, 42, 1, 0, 92114, 86554, 48942, 19300, 5160, 812, 56, 1, 0, 773259, 729716, 423778, 178478, 54758, 11396, 1372, 72, 1, 0, 6743122, 6384353, 3781926, 1669062, 561514, 138866, 23072, 2184, 90, 1

Offset: 0

Andrew Howroyd, Feb 01 2025

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,     7,     6,     1;
  0,    36,    32,    12,     1;
  0,   221,   202,    94,    20,    1;
  0,  1530,  1417,   728,   220,   30,   1;
  0, 11510, 10752,  5854,  2090,  445,  42,  1;
  0, 92114, 86554, 48942, 19300, 5160, 812, 56, 1;
  ...
The T(2,1) = 2 open meanders are:
         __           __
        /  \         /  \
   ... / /\ \..  .. / /\ \ ...
      / /  \/       \/  \ \
The T(2,2) = 1 open meander is:
   ... /\../\ ...
      /  \/  \

Andrew Howroyd, Table of n, a(n) for n = 0..230

Row sums are A077054.
Main diagonal is A000012.
Second diagonal is A002378.
Cf. A005316, A006660 (bisection gives column 1), A077056 (total number of exterior top arches), A259689 (for semi-meanders), A259974.

A077056(n) = Sum_{k=1..n} k*T(n,k).

T(n,1) = A006660(2*n + 1).

cp's OEIS Frontend

A077460 Number of nonisomorphic ways a loop can cross a road (running East-West) 2n times.

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A006659 Number of closed meander systems of order n+1 with n components.

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A259974 Irregular triangle read by rows: T(n,k) = number of meanders with n bridges in which the first bridge is bridge k.

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A006657 Number of closed meanders with 2 components and 2n bridges.

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A006664 Number of irreducible systems of meanders.

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A077014 Number of ways that a directed line (or river) that starts in the south can cross an east-west road n times.

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A287220 Number of single-crossing partitions on n nodes.

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A006660 Number of meanders in which first bridge is 3.

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