|
|
A077460
|
|
Number of nonisomorphic ways a loop can cross a road (running East-West) 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 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
|
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] (* Jean-François 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
|
|
|
|