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A051437
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Number of undirected walks of length n+1 on an oriented triangle, visiting n+2 vertices, with n "corners"; the symmetry group is C3. Walks are not self-avoiding.
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2
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1, 3, 4, 10, 16, 36, 64, 136, 256, 528, 1024, 2080, 4096, 8256, 16384, 32896, 65536, 131328, 262144, 524800, 1048576, 2098176, 4194304, 8390656, 16777216, 33558528, 67108864, 134225920, 268435456, 536887296, 1073741824, 2147516416, 4294967296, 8590000128
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OFFSET
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0,2
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COMMENTS
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For a way to obtain this sequence from symmetry in quilts, see the Tom Young web page.
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LINKS
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Table of n, a(n) for n=0..33.
Tom Young, Math Research Quilt Pattern Symmetry
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FORMULA
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n=2m: a(n)=2^(n-1)+2^((n-2)/2); n=2m+1: a(n)=2^(n-1).
Binomial transform is 3^n+Pell(n) (A000244(n)+A000129(n)). G.f. : (1+x-4x^2)/((1-2x)(1-2x^2)); a(n)=2^n+2^(n/2)(1-(-1)^n)/(2sqrt(2)). - Paul Barry, Apr 28 2004
a(0)=1, a(1)=3, a(2)=4, a(n)=2*a(n-1)+2*a(n-2)-4*a(n-3) [From Harvey P. Dale, June 06 2011]
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EXAMPLE
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For n=3 the walks visit vertices 1212, 1213, 1232, 1231.
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MATHEMATICA
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LinearRecurrence[{2, 2, -4}, {1, 3, 4}, 50] (* or *) CoefficientList[ Series[ (1+x-4x^2)/((1-2x)(1-2x^2)), {x, 0, 50}], x] (* From Harvey P. Dale, June 06 2011 *)
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CROSSREFS
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Cf. A005418.
Sequence in context: A037952 A093512 A081160 * A224073 A034774 A172416
Adjacent sequences: A051434 A051435 A051436 * A051438 A051439 A051440
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Colin Mallows colinm(AT)research.avayalabs.com
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EXTENSIONS
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More terms from Harvey P. Dale, June 06 2011.
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STATUS
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approved
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