A077460 - OEIS

%I #28 Aug 22 2024 09:16:49

%S 1,1,1,3,12,70,464,3482,27779,233556,2038484,18357672,169599492,

%T 1601270562,15401735750,150547249932,1492451793728,14980801247673,

%U 152047178479946,1558569469867824,16119428039548246

%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 or North-South mirror (a group of order 4).

%C 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

%H Andrew Howroyd, <a href="/A077460/a077460.pdf">Illustration of Closed Meander Symmetries</a>

%F From _Andrew Howroyd_, Nov 24 2015: (Start)

%F a(2n+1) = (A005315(2n+1) + A005316(2n+1) + A060206(n)) / 4.

%F a(2n) = (A005315(2n) + 2 * A005316(2n)) / 4. (End)

%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

%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, 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

%t A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];

%t A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {_, _}][[All, 2]];

%t a[0] = a[1] = 1;

%t a[n_] := If[OddQ[n], (A005316[[n + 1]] + A005316[[2n]] + A000682[[n]])/4, (A005316[[2n]] + 2 A005316[[n + 1]])/4];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Sep 06 2019, after _Andrew Howroyd_ *)

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

%K nonn,nice

%O 0,4

%A _N. J. A. Sloane_ and _Jon Wild_, Dec 03 2002

%E a(10)-a(20) from _Andrew Howroyd_, Nov 24 2015