A078591 - OEIS

%I #19 Aug 10 2022 10:07:18

%S 1,1,1,4,21,131,914,6910,55477,466729,4076430,36712325,339195058,

%T 3202515525,30803440806,301094270964,2984903334517,29961600364523,

%U 304094354787062,3117138919265903,32238856059792302,336132907436386486,3530470987229030696,37330864330583904876,397168915877285183906

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

%C 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).

%H Jean-François Alcover, <a href="/A078591/b078591.txt">Table of n, a(n) for n = 0..28</a>

%F a(n) = A005315(n) / 2 for n >= 2. - _Andrew Howroyd_, Nov 23 2015

%e 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.

%e For n = 2, 4, 6, 8 the solutions are as follows:

%e n=2: 1 2

%e n=4: 1 2 3 4

%e 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

%e 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,

%e 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,

%e 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,

%t A005315 = Cases[Import["https://oeis.org/A005315/b005315.txt", "Table"], {_, _}][[All, 2]];

%t a[n_] := If[n < 3, 1, A005315[[n+1]]/2];

%t Table[a[n], {n, 0, 28}] (* _Jean-François Alcover_, Aug 10 2022, after _Andrew Howroyd_ *)

%Y 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).

%K nonn,nice

%O 0,4

%A _N. J. A. Sloane_ and _Jon Wild_, Dec 07 2002

%E a(10)-a(20) added by _Andrew Howroyd_, Nov 23 2015

%E a(21)-a(28) computed from A005315 added by _Jean-François Alcover_, Aug 10 2022