A078104 -id:A078104 - cp's OEIS frontend

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

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

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

			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

Andrew Howroyd, Illustration of Closed Meander Symmetries

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

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 *)

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)

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

A078105 Number of nonisomorphic ways a loop can cross three roads meeting in a Y n times (orbits under symmetry group of order 6).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 8, 8, 48, 54, 331, 439, 2558, 3734, 21057, 33384, 182293, 307719, 1638465, 2913775, 15181584, 28194412, 144206012, 277887666, 1398566992

Offset: 0

N. J. A. Sloane and Jon Wild, Dec 05 2002

There is no constraint on touching any particular sector.

The Mercedes-Benz problem: closed meanders crossing a Y.

			With three crossings the loop must cut each road exactly once, so a(3) = 1.
With 4 crossings the loop can cut one road 4 times (one possibility), or two roads twice each (one possibility), so a(4) = 2.

Anonymous, Illustration for a(3) = 1

Cf. A078104 (total number of solutions), A077460 and A005315 (loop crossing one road).

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

Original entry on oeis.org

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

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

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).

			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,

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

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).

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 *)

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

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

A085919 Number of ways a loop can cross three roads meeting in a Y n times.

Original entry on oeis.org

3, 0, 3, 1, 9, 6, 45, 42, 279, 320, 1977, 2610, 15306, 22404, 126300, 200158, 1093515, 1846314, 9830547, 17481864

Offset: 0

N. J. A. Sloane and Jon Wild, Aug 25 2003

nonn

The Mercedes-Benz problem: closed meanders crossing a Y.

			With three crossings the loop must cut each road exactly once, so a(3) = 1.

Anonymous, Illustration for a(3) = 1

Similar to A078104, but without the constraint of touching the (-, -) quadrant.

cp's OEIS Frontend

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

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A078105 Number of nonisomorphic ways a loop can cross three roads meeting in a Y n times (orbits under symmetry group of order 6).

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

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A085919 Number of ways a loop can cross three roads meeting in a Y n times.

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