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

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

0,4

Comments

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).

Links

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

Formula

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

Example

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,

Mathematica

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 *)

Crossrefs

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).

Sequence in context: A111177 A141052 A058308 * A090366 A273956 A131965

Adjacent sequences: A078588 A078589 A078590 * A078592 A078593 A078594

Keyword

nonn,nice

Author

N. J. A. Sloane and Jon Wild, Dec 07 2002

Extensions

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

Status

approved