

A077460


Number of nonisomorphic ways a loop can cross a road (running EastWest) 2n times.


7



1, 1, 1, 3, 12, 70, 464, 3482, 27779, 233556, 2038484, 18357672, 169599492, 1601270562, 15401735750, 150547249932, 1492451793728, 14980801247673, 152047178479946, 1558569469867824, 16119428039548246
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 EastWest or NorthSouth mirror (a group of order 4).
Symmetries are possible by reflection in a NorthSouth 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

a(2n+1) = (A005315(2n+1) + A005316(2n+1) + A060206(n)) / 4.  Andrew Howroyd, Nov 24 2015
a(2n) = (A005315(2n) + 2 * A005316(2n)) / 4.  Andrew Howroyd, Nov 24 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
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] (* JeanFrançois Alcover, Sep 06 2019, after Andrew Howroyd *) *)


CROSSREFS

The total number of closed meanders with 2n crossings is given in A005315. Cf. A077055, A078104, A078105, A078591.
Sequence in context: A102078 A113341 A125862 * A001205 A330493 A112320
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



