A005315 -id:A005315 - cp's OEIS frontend

A060149 Number of homogeneous generators of degree n for graded algebra associated with meanders.

1, 3, 2, 13, 16, 106, 166, 1073, 1934, 12142, 24076, 147090, 312906, 1865772, 4191822, 24463905, 57433950, 328887346, 800740450, 4508608610, 11319707546, 62781858592, 161841539812, 885513974674, 2335765140994, 12624162072740, 33979681977530, 181611275997040

Offset: 1

N. J. A. Sloane, Apr 10 2001

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Roland Bacher, Meander algebras

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

Cf. A018224.

seq(n) = Vec(1 - 1/sum(k=0, n, binomial(k, k\2)^2*x^k, O(x*x^n))) \\ Andrew Howroyd, Feb 07 2025

G.f.: 1 - 1/B(x) where B(x) is the g.f. of A018224. - Andrew Howroyd, Feb 07 2025

a(11) onwards from Andrew Howroyd, Feb 07 2025

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

Original entry on oeis.org

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)

A078105 Number of nonisomorphic ways a loop can cross three roads meeting in a Y n times (orbits under symmetry group of order 6).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 8, 8, 48, 54, 331, 439, 2558, 3734, 21057, 33384, 182293, 307719, 1638465, 2913775, 15181584, 28194412, 144206012, 277887666, 1398566992

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 05 2002

There is no constraint on touching any particular sector.

The Mercedes-Benz problem: closed meanders crossing a Y.

			With three crossings the loop must cut each road exactly once, so a(3) = 1.
With 4 crossings the loop can cut one road 4 times (one possibility), or two roads twice each (one possibility), so a(4) = 2.

Anonymous, Illustration for a(3) = 1

Cf. A078104 (total number of solutions), A077460 and A005315 (loop crossing one road).

A078104 Number of ways a loop can cross three roads meeting in a Y n times. The loop must touch the southwest sector.

Original entry on oeis.org

1, 0, 2, 1, 7, 6, 37, 42, 237, 320, 1715, 2610, 13478, 22404, 112480, 200158, 982561, 1846314, 8897089, 17481864

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 05 2002

nonn

The Mercedes-Benz problem: closed meanders crossing a Y.

			With three crossings the loop must cut each road exactly once, so a(3) = 1.
With 4 crossings the loop can cut one road 4 times (giving A005315(2)*2 = 4 possibilities), or two roads twice each (3 ways), so a(4) = 7.

Anonymous, Illustration for a(3) = 1

See A085919 for another version. Cf. A078105 (nonisomorphic solutions), A077460 and A005315 (loop crossing one road).

Cf. also A077550.

More terms added Aug 25 2003

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

A078591 Number of nonisomorphic ways a loop can cross a road (running East-West) 2n times.

Original entry on oeis.org

1, 1, 1, 4, 21, 131, 914, 6910, 55477, 466729, 4076430, 36712325, 339195058, 3202515525, 30803440806, 301094270964, 2984903334517, 29961600364523, 304094354787062, 3117138919265903, 32238856059792302, 336132907436386486, 3530470987229030696, 37330864330583904876, 397168915877285183906

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 07 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 mirror (a group of order 2).

			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, 1 4 3 2 5 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,
n=8 (cont.): 1 2 5 4 3 6 7 8, 1 2 3 8 7 6 5 4, 1 2 5 4 3 8 7 6, 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,
n=8 (cont.): 1 2 7 4 5 6 3 8, 1 4 3 2 5 6 7 8, 1 4 5 6 3 2 7 8, 1 4 3 2 5 8 7 6, 1 4 3 2 7 6 5 8, 1 6 5 4 3 2 7 8, 1 6 5 2 3 4 7 8, 1 6 3 4 5 2 7 8,

Jean-François Alcover, Table of n, a(n) for n = 0..28

The total number of closed meanders with 2n crossings is given in A005315. Cf. A077055, A078104, A078105, A077460 (same but with group of order 4).

A005315 = Cases[Import["https://oeis.org/A005315/b005315.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n < 3, 1, A005315[[n+1]]/2];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Aug 10 2022, after Andrew Howroyd *)

a(n) = A005315(n) / 2 for n >= 2. - Andrew Howroyd, Nov 23 2015

a(10)-a(20) added by Andrew Howroyd, Nov 23 2015

a(21)-a(28) computed from A005315 added by Jean-François Alcover, Aug 10 2022

A217310 The number of meandering curves of order n, with only one extremity covered by its arcs.

Original entry on oeis.org

0, 0, 4, 4, 32, 38, 264, 342, 2288, 3134, 20740, 29526, 194916, 285458, 1885840, 2822310, 18682016, 28440970, 188717116, 291294678, 1937706144, 3025232480, 20173268632, 31797822936, 212530874156, 337731551446, 2262235585956, 3620119437762, 24297593488468

Offset: 1

Panayotis Vlamos, Antonios Panayotopoulos, Georgia Theocharopoulou, Mar 17 2013

nonn

A meandering curve of order n is a continuous curve which does not intersect itself yet intersects a horizontal line n times.

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.

Panayotis Vlamos, Table of n, a(n) for n = 1..42
A. Panayotopoulos, P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 1-10.

Cf. A005315.

a(n) = A223093(n) * A000034(n). - Andrew Howroyd, Dec 06 2015

A217318 The number of meandering curves of order n, with both extremity covered by its arcs.

Original entry on oeis.org

0, 0, 0, 1, 10, 20, 156, 273, 1986, 3358, 23742, 39736, 277178, 462794, 3205896, 5355743, 36963722, 61856394, 426075994, 714515312, 4916833424, 8263479072, 56840484232, 95733461792, 658460090994, 1111253958664, 7644360501390, 12925362323004, 88938175307354

Offset: 1

Panayotis Vlamos, Antonios Panayotopoulos, and Georgia Theocharopoulou, Mar 18 2013

nonn

A meandering curve of order n is a continuous curve which does not intersect itself yet intersects a horizontal line n times.

A. Panayotopoulos and P. Tsikouras, Properties of meanders, JCMCC 46 (2003), 181-190.

Panayotis Vlamos, Table of n, a(n) for n = 1..42
A. Panayotopoulos, P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 1-10.

Cf. A005315.

a(n) = A223095(n) * A000034(n) / 2. - Andrew Howroyd, Dec 06 2015

A281442 Triangle read by rows: T(n,r), 0 <= r <= n, is the number of idempotents of rank r in the Kauffman monoid K_n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 4, 0, 1, 0, 8, 0, 6, 0, 1, 0, 0, 22, 0, 8, 0, 1, 0, 42, 0, 40, 0, 10, 0, 1, 0, 0, 140, 0, 62, 0, 12, 0, 1, 0, 262, 0, 288, 0, 88, 0, 14, 0, 1, 0, 0, 992, 0, 492, 0, 118, 0, 16, 0, 1

Offset: 0

James East, Oct 05 2017

Values were computed using the Semigroups package for GAP.

T(n,r) is also the number of idempotent basis elements of rank r in the Temperley-Lieb algebra of degree n in the generic case (when the twisting parameter is not an m-th root of unity for any m <= n).

Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015-2016.

Cf. A281438 (row sums), A281441, A289620.

T(2n-1,1) = A005315(n). Empirical: T(2n,2) = A077056(n); T(n+2,n-2) = 2*A028875(n) for n>2. - Andrey Zabolotskiy, Oct 19 2017

cp's OEIS Frontend

A060149 Number of homogeneous generators of degree n for graded algebra associated with meanders.

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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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A078105 Number of nonisomorphic ways a loop can cross three roads meeting in a Y n times (orbits under symmetry group of order 6).

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A078104 Number of ways a loop can cross three roads meeting in a Y n times. The loop must touch the southwest sector.

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

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A078591 Number of nonisomorphic ways a loop can cross a road (running East-West) 2n times.

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A217310 The number of meandering curves of order n, with only one extremity covered by its arcs.

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