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A077460
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Number of nonisomorphic ways a loop can cross a road (running East-West) 2n times.
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7
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1, 1, 1, 3, 12, 70, 464, 3482, 27779, 233556, 2038484, 18357672, 169599492, 1601270562, 15401735750, 150547249932, 1492451793728, 14980801247673, 152047178479946, 1558569469867824, 16119428039548246
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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Table of n, a(n) for n=0..20.
Andrew Howroyd, Illustration of Closed Meander Symmetries
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *) *)
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CROSSREFS
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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
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane and Jon Wild, Dec 03 2002
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EXTENSIONS
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a(10)-a(20) from Andrew Howroyd, Nov 24 2015
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STATUS
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approved
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