A005316 -id:A005316 - cp's OEIS frontend

A077055 Call two meanders from A005316 equivalent if they differ by a reflection in the Y axis (if n even) or by reflections in the X or Y axes (if n odd). Sequence gives number of inequivalent meanders with n crossings.

1, 1, 1, 2, 3, 8, 13, 42, 72, 273, 475, 1970, 3506, 15368, 27888, 126510, 233809, 1086546, 2039564, 9652364, 18360296, 88172609, 169610371, 824506191, 1601297937, 7865294687, 15401847339, 76331857094, 150547538649, 751981532942, 1492452957398

Offset: 0

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

Meander shapes. [Stéphane Legendre, Apr 09 2013]

			For n=7 the A005316(7) = 42 meanders with 7 crossings fall into 5 equivalence classes of size 2 and 8 of size 4, so a(7) = 5+8 = 13.

N. J. A. Sloane, Table of n, a(n) for n = 0..42 [from Legendre, 2013]
CombOS - Combinatorial Object Server, Generate meanders and stamp foldings
Stéphane Legendre, Illustration of initial terms
Stéphane Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
J. Sawada and R. Li, Stamp foldings, semi-meanders, and open meanders: fast generation algorithms, Electronic Journal of Combinatorics, Volume 19 No. 2 (2012), P#43 (16 pages).

Cf. A005316, A077460, A078592, A223096.

For n even a(n) = (A005316(n)+A005316(n/2))/2 (this is A078592).

For n odd a(n) = (A005316(n)+2*A223096(floor(n/2)))/4. [Stéphane Legendre, Apr 09 2013]

More terms from the Sawada-Li paper from Daniel Recoskie, Jul 11 2012

A077056 Total number of "humps" in all A005316(2n) open meanders with 2n crossings.

Original entry on oeis.org

0, 1, 4, 22, 140, 992, 7584, 61420, 520320, 4570292, 41354416, 383609434, 3634231140, 35059333218, 343580813492, 3413862553800, 34336784362232, 349131316634164, 3584622273967324, 37128374058146138, 387630493693010928, 4076329834240413392, 43150916852592230992

Offset: 0

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

nonn

A "hump" is a local maxima of the upper envelope of the part of the meander that extends above the East-West road.

A "hump" can also be described as an exterior top arch. - Andrew Howroyd, Feb 01 2025

			The three open meanders with 4 crossings have respectively 2, 1 and 1 humps, so a(2) = 4.

Cf. A005316 (open meanders), A077054, A380369.

a(n) = Sum_{k=1..n} k*A380369(n,k). - Andrew Howroyd, Feb 01 2025

a(9)-a(20) from Andrew Howroyd, Nov 22 2015
a(21)-a(22) from Andrew Howroyd, Feb 01 2025
a(0) corrected by Andrew Howroyd, Feb 02 2025

A078592 Call two meanders from A005316 with 2n crossings equivalent if they differ by reflections in the X or Y axes. Sequence gives number of inequivalent meanders.

Original entry on oeis.org

1, 1, 2, 8, 42, 273, 1970, 15368, 126510, 1086546, 9652364, 88172609, 824506191, 7865294687, 76331857094, 751981532942, 7506432993145, 75811326673326, 773682540353704, 7969986193751019, 82798726340037900, 866804540900696571

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 08 2002

nonn

Symmetry group has order 4.

Jean-François Alcover, Table of n, a(n) for n = 0..27

A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[n_] := (A005316[[2 n + 1]] + A005316[[n + 1]])/2;
a /@ Range[0, 27] (* Jean-François Alcover, Sep 20 2019 *)

a(n) = (A005316(2n)+A005316(n))/2.

A107321 a(n)=Sum(i+j+k+l+...+r+s=n) A005316(i)A005316(j)...*A005316(s) and the ordered partition of n runs over all odd i, all even j,k,.., r, all i,j,...,r,s>=1.

Original entry on oeis.org

1, 1, 2, 3, 8, 14, 42, 79, 254, 506, 1702, 3548, 12320, 26666, 94794, 211751, 766362, 1758352, 6453812, 15150922, 56238710, 134659120

Offset: 0

R. J. Mathar, May 06 2006

A lower bound to A076876. a(n) counts the cases where all crossings k are East of the crossings i, all crossings l East of the crossings j etc. Some backwinding interlaced crossings are counted in A076876(n) but not here.

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

Original entry on oeis.org

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

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

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

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.

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).

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

A column of triangle A008828.

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 *)

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

A076875 Meandric numbers for a curve crossing two perpendicular lines at n points.

Original entry on oeis.org

1, 2, 4, 10, 22, 62, 176, 436

Offset: 0

N. J. A. Sloane and Jon Wild, Nov 24 2002

a(n) = number of ways that a curve can start in the (-,-) quadrant, cross the x and y axes at exactly n points and end in any quadrant. Line is undirected.

			See illustration for a(4)=22: each of the 12 solutions shown crosses the x-axis first and ten of them are related by mirror symmetry to a corresponding curve that crosses the y-axis first, making the total a(4)=22.

Jon Wild, Illustration of a(4) = 22 (ignore the arrowheads)

Cf. A005316, A076876, A076906, A076907 (directed case).

For n odd a(n) = 2*A076906(n).

a(6) and a(7) corrected Aug 23 2003

a(7) corrected by Robert Price, Jul 29 2012

A076876 Meandric numbers for a river crossing two parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 3, 8, 14, 43, 81, 272, 538, 1920, 3926, 14649, 30694, 118489, 252939, 1002994, 2172830, 8805410, 19304190, 79648888, 176343390, 738665040, 1649008456, 6996865599, 15730575554, 67491558466, 152663683494, 661370687363, 1503962954930, 6571177867129

Offset: 0

N. J. A. Sloane and Jon Wild, Nov 26 2002

a(n) = number of ways that a curve can start in the (-,-) quadrant, cross two parallel lines and end up in the (+,+) or (+,-) quadrant if n is even or head East between the two roads if n is odd.
A107321 is a lower bound. - R. J. Mathar, May 06 2006
It appears that for odd n, A076876(n) = A005316(n+1). And for even n, A076876(n) >= A005316(n+1). - Robert Price, Jul 27 2013.

			Let b(n) = A005316(n). Then a(0) = b(0), a(1) = b(1), a(2) = b(1) + b(2), a(3) = b(3) + b(2), a(4) = b(4) + 2*b(3) + 1, a(5) = b(5) + b(3)*b(2) + b(4) + 1.
Consider n=5: if we do not cross the second road there are b(5) = 8 solutions. If we cross the first road 3 times and then the second road twice there are b(3)*b(2) = 2 solutions. If we cross the first road once and the second road 4 times there are b(4) = 3 solutions. The only other possibility is to cross road 1, road 2 twice, road 1 twice and exit to the right.
For larger n it is convenient to give the vector of the number of times the same road is crossed. For example for n=6 the vectors and the numbers of possibilities are as follows:
[6] ...... 14
[5 1] ..... 8
[3 3] ..... 4
[3 2 1] ... 2
[1 5] ..... 8
[1 4 1] ... 3
[1 2 3] ... 2
[1 2 2 1] . 2
Total .... 43

Andrew Howroyd, Table of n, a(n) for n = 0..40
R. J. Mathar, ASCII representations

Column k=2 of A380367.

Cf. A005316, A206432, A204352, A076875, A076906, A076907.

More terms from R. J. Mathar, Mar 04 2007
a(12)-a(20) from Robert Price, Apr 15 2012
a(21)-a(40) from Andrew Howroyd, Dec 07 2015

A076907 Meandric numbers for a river crossing two perpendicular roads at n points, beginning in the (-,-) quadrant and ending in any quadrant.

Original entry on oeis.org

2, 2, 6, 10, 32, 62, 210, 436, 1540, 3346, 12192, 27344, 102054, 234388, 891574, 2085940, 8057844, 19134786, 74864648, 179968564, 711708544

Offset: 0

N. J. A. Sloane and Jon Wild, Nov 26 2002

a(n) = number of ways that a directed curve (or arrow) can start in the (-,-) quadrant, cross the x and y axes at exactly n points and end in any quadrant.

Cf. A005316, A076876, A076906, A076875 (undirected case).

Cf. A077551 (cross x axis first).

a(2n+1) = 2*A076906(2n+1).

a(6) and a(7) corrected Aug 25 2003
a(7) corrected by Robert Price, Jul 29 2012
a(8)-a(20) from Robert Price, Aug 01 2012

A076906 Meandric numbers for a river (or directed line) crossing two perpendicular roads at n points, beginning in the (-,-) quadrant and ending in the (+,+) quadrant if n even, in the (+,-) quadrant if n odd.

Original entry on oeis.org

0, 1, 2, 5, 12, 31, 82, 218, 612, 1673, 4892, 13672, 41192, 117194, 361302, 1042970, 3274712, 9567393, 30490688, 89984282, 290353456

Offset: 0

N. J. A. Sloane and Jon Wild, Nov 26 2002

Cf. A005316, A076875 (undirected line), A076876, A076907 (end anywhere).

a(7) corrected Aug 25 2003

a(7) corrected and a(8)-a(20) added by Robert Price, Jul 29 2012

A209656 Meandric numbers for a river crossing up to 12 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76538, 240368, 665100, 2123379, 5964156, 19301178, 54890366, 179679030, 516360755, 1706896545, 4949350203, 16500278295, 48216373545, 161946759019, 476447428528, 1610847688579, 4767486352733, 16213635060406

Offset: 0

Robert Price, May 07 2012

nonn

Number of ways that a river (or directed line) that starts in the South and flows East can cross up to 12 parallel East-West roads n times.

Sequence derived from list of solutions described in A206432.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=12 of A380367.
Cf. A005316 (sequence for one road; extensive references and links).
Cf. A076876 (sequence for two parallel roads).
Cf. A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592 (sequences for 3 to 19 parallel roads).
Cf. A206432 (sequence for unlimited number of parallel roads).
Cf. A076875, A076906, A076907.

a(21) onwards from Andrew Howroyd, Jan 31 2025

cp's OEIS Frontend

A077055 Call two meanders from A005316 equivalent if they differ by a reflection in the Y axis (if n even) or by reflections in the X or Y axes (if n odd). Sequence gives number of inequivalent meanders with n crossings.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A077056 Total number of "humps" in all A005316(2n) open meanders with 2n crossings.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A078592 Call two meanders from A005316 with 2n crossings equivalent if they differ by reflections in the X or Y axes. Sequence gives number of inequivalent meanders.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A107321 a(n)=Sum(i+j+k+l+...+r+s=n) A005316(i)*A005316(j)*...*A005316(s) and the ordered partition of n runs over all odd i, all even j,k,.., r, all i,j,...,r,s>=1.

Views

Author

Keywords

Comments

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A076875 Meandric numbers for a curve crossing two perpendicular lines at n points.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A076876 Meandric numbers for a river crossing two parallel roads at n points.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A076907 Meandric numbers for a river crossing two perpendicular roads at n points, beginning in the (-,-) quadrant and ending in any quadrant.

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

A076906 Meandric numbers for a river (or directed line) crossing two perpendicular roads at n points, beginning in the (-,-) quadrant and ending in the (+,+) quadrant if n even, in the (+,-) quadrant if n odd.

Views

Author

Keywords

Crossrefs

Extensions

A209656 Meandric numbers for a river crossing up to 12 parallel roads at n points.

Views

A107321 a(n)=Sum(i+j+k+l+...+r+s=n) A005316(i)A005316(j)...*A005316(s) and the ordered partition of n runs over all odd i, all even j,k,.., r, all i,j,...,r,s>=1.