A005316 -id:A005316 - cp's OEIS frontend

A209657 Meandric numbers for a river crossing up to 13 parallel roads at n points.

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240369, 665127, 2123408, 5964627, 19301713, 54897139, 179687084, 516448412, 1707004865, 4950415081, 16501638058, 48228801029, 161963084065, 476588705579, 1611037922998, 4769064680579, 16215807145689

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 13 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=13 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

A209660 Meandric numbers for a river crossing up to 14 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665128, 2123437, 5964658, 19302248, 54897742, 179695133, 516457890, 1707112980, 4950547189, 16502992754, 48230509790, 161979310981, 476609746441, 1611226513378, 4769315213007, 16217954185533

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 14 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=14 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

A209707 Meandric numbers for a river crossing up to 15 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123438, 5964689, 19302281, 54898345, 179695808, 516467363, 1707124038, 4950679082, 16503152306, 48232213024, 161981435856, 476630676669, 1611253336964, 4769563838975, 16218280461398

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 15 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=15 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

A210344 Meandric numbers for a river crossing up to 16 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964690, 19302314, 54898380, 179696483, 516468114, 1707135091, 4950691884, 16503311633, 48232404028, 161983554724, 476633293902, 1611280036392, 4769597694865, 16218604538134

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 16 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=16 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

A210478 Meandric numbers for a river crossing up to 17 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964691, 19302315, 54898415, 179696520, 516468865, 1707135922, 4950704681, 16503326351, 48232594797, 161983781556, 476635904628, 1611283232139, 4769631412013, 16218646880868

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 17 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=17 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

A210567 Meandric numbers for a river crossing up to 18 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964691, 19302316, 54898416, 179696557, 516468904, 1707136753, 4950705596, 16503341064, 48232611611, 161984008143, 476636172048, 1611286420859, 4769635283158, 16218689069176

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 18 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=18 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

A210592 Meandric numbers for a river crossing up to 19 parallel roads at n points.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 63, 155, 448, 1152, 3452, 9158, 28178, 76539, 240370, 665129, 2123439, 5964691, 19302316, 54898417, 179696558, 516468943, 1707136794, 4950706511, 16503342067, 48232628420, 161984027241, 476636439213, 1611286734027, 4769639146736, 16218693724179

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 19 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=19 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

A380367 Array read by antidiagonals: meandric numbers for a river crossing up to k parallel roads at n points, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 4, 8, 8, 1, 1, 2, 4, 9, 14, 14, 1, 1, 2, 4, 10, 21, 43, 42, 1, 1, 2, 4, 10, 22, 52, 81, 81, 1, 1, 2, 4, 10, 23, 61, 131, 272, 262, 1, 1, 2, 4, 10, 23, 62, 142, 345, 538, 538, 1, 1, 2, 4, 10, 23, 63, 153, 420, 915, 1920, 1828

Offset: 0

Andrew Howroyd, Jan 31 2025

Illustrations of the initial terms for the case of two parallel roads can be found in A076876.

			Array begins:
===================================================
n\k |   1   2   3    4    5    6    7    8    9 ...
----+----------------------------------------------
  0 |   1   1   1    1    1    1    1    1    1 ...
  1 |   1   1   1    1    1    1    1    1    1 ...
  2 |   1   2   2    2    2    2    2    2    2 ...
  3 |   2   3   4    4    4    4    4    4    4 ...
  4 |   3   8   9   10   10   10   10   10   10 ...
  5 |   8  14  21   22   23   23   23   23   23 ...
  6 |  14  43  52   61   62   63   63   63   63 ...
  7 |  42  81 131  142  153  154  155  155  155 ...
  8 |  81 272 345  420  433  446  447  448  448 ...
  9 | 262 538 915 1017 1120 1135 1150 1151 1152 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..945 (first 43 antidiagonals)

Columns 1..19 are A005316, A076876, A204352, A208062, A208126, A208452, A208453, A209383, A209621, A209622, A209626, A209656, A209657, A209660, A209707, A210344, A210478, A210567, A210592.
Main diagonal is A206432.
Cf. A076875 (perpendicular roads).

T(n,k) = T(n,n) for k > n.

A008828 Triangle read by rows: T(n,k) = number of closed meander systems of order n with k<=n components.

Original entry on oeis.org

1, 2, 2, 8, 12, 5, 42, 84, 56, 14, 262, 640, 580, 240, 42, 1828, 5236, 5894, 3344, 990, 132, 13820, 45164, 60312, 42840, 17472, 4004, 429, 110954, 406012, 624240, 529104, 271240, 85904, 16016, 1430, 933458, 3772008, 6540510, 6413784, 3935238, 1569984, 405552, 63648, 4862

Offset: 1

D. Ivanov, S. K. Lando, A. K. Zvonkin ( LabRI, Bordeaux, France)

A meander of order n has 2n bridges. For many more references, see A005315 and A005316.

			Triangle starts:
   1;
   2  2;
   8 12  5;
  42 84 56 14;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..210
Bertrand Duplantier and Emmanuel Guitter, Liouville Quantum Duality and Random Planar Maps, arXiv:2507.12203 [math-ph], 2025. See p. 50.
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of components, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
Iwan Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
Michael La Croix, Approaches to the Enumerative Theory of Meanders [_Gerald McGarvey_, Oct 26 2008]
Sergei K. Lando and Alexander 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)
Sergei K. Lando and Alexander K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117 (1993) p. 232.

Columns include A005315, A006657, A006658. Diagonals include A000108 (Catalan numbers), A006659, A007746. Row sums are in A001246.

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 10 2004
Edited by Ralf Stephan, Dec 29 2004
T(10,k)-T(20,k) from Andrew Howroyd, Nov 22 2015

A077054 Number of ways a river can cross a road 2n times.

Original entry on oeis.org

1, 1, 3, 14, 81, 538, 3926, 30694, 252939, 2172830, 19304190, 176343390, 1649008456, 15730575554, 152663683494, 1503962954930, 15012865733351, 151622652413194, 1547365078534578, 15939972379349178, 165597452660771610, 1733609081727968492

Offset: 0

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

nonn

More precisely, number of ways that a river (or directed line) that starts in the southwest and flows east can cross an east-west road 2n times (bisection of A005316).

Also number of ways a loop can cross two parallel roads 2n times. Some portion of loop must lie below lower road.

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

Bisection of A005316. Cf. A005315, A085873, A086031.

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

a(n) = A005316(2*n).

cp's OEIS Frontend

A209657 Meandric numbers for a river crossing up to 13 parallel roads at n points.

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A209660 Meandric numbers for a river crossing up to 14 parallel roads at n points.

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A209707 Meandric numbers for a river crossing up to 15 parallel roads at n points.

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A210344 Meandric numbers for a river crossing up to 16 parallel roads at n points.

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A210478 Meandric numbers for a river crossing up to 17 parallel roads at n points.

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A210567 Meandric numbers for a river crossing up to 18 parallel roads at n points.

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A210592 Meandric numbers for a river crossing up to 19 parallel roads at n points.

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A380367 Array read by antidiagonals: meandric numbers for a river crossing up to k parallel roads at n points, n >= 0, k >= 1.

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Formula

A008828 Triangle read by rows: T(n,k) = number of closed meander systems of order n with k<=n components.

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A077054 Number of ways a river can cross a road 2n times.

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