A259974 -id:A259974 - cp's OEIS frontend

A006660 Number of meanders in which first bridge is 3.

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

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).

A006661 Number of meanders in which first bridge is 5.

Original entry on oeis.org

3, 3, 7, 11, 28, 57, 155, 353, 1003, 2458, 7214, 18575, 55880, 149183, 457639, 1255933, 3914103, 10978240, 34663182, 98953078, 315884786, 915008430, 2948378068, 8645874055, 28084475514, 83222134020

Offset: 5

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.

Offset corrected terms a(19)-a(30) added, Andrew Howroyd, Dec 15 2015

A006662 Number of meanders in which first bridge is 7.

Original entry on oeis.org

14, 14, 36, 57, 155, 316, 902, 2053, 6059, 14810, 44842, 115009, 355293, 943860, 2963536, 8086913, 25733325, 71725012, 230811370, 654472364, 2126296860, 6115504594, 20032488714, 58309793101

Offset: 7

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.

Offset corrected, terms a(19)-a(30) added, Andrew Howroyd, Dec 15 2015

cp's OEIS Frontend

A006660 Number of meanders in which first bridge is 3.

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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.

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A006661 Number of meanders in which first bridge is 5.

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A006662 Number of meanders in which first bridge is 7.

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