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

1, 1, 1, 3, 12, 70, 464, 3482, 27779, 233556, 2038484, 18357672, 169599492, 1601270562, 15401735750, 150547249932, 1492451793728, 14980801247673, 152047178479946, 1558569469867824, 16119428039548246

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 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

Links

Table of n, a(n) for n=0..20.

Andrew Howroyd, Illustration of Closed Meander Symmetries

Formula

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)

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
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

Mathematica

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 *)

Crossrefs

The total number of closed meanders with 2n crossings is given in A005315. Cf. A000682, A005316, A060206, A077055, A078104, A078105, A078591.

Sequence in context: A102078 A113341 A125862 * A001205 A346888 A330493

Adjacent sequences: A077457 A077458 A077459 * A077461 A077462 A077463

Keyword

nonn,nice

Author

N. J. A. Sloane and Jon Wild, Dec 03 2002

Extensions

a(10)-a(20) from Andrew Howroyd, Nov 24 2015

Status

approved