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A095930
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Number of walks of length 2n between two nodes at distance 2 in the cycle graph C_10.
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1
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1, 4, 15, 57, 220, 859, 3381, 13380, 53143, 211585, 843756, 3368259, 13455325, 53774932, 214978335, 859595529, 3437550076, 13748021995, 54986385093, 219930610020, 879683351911, 3518631073489, 14074256379660
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OFFSET
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1,2
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COMMENTS
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In general 2^n/m*Sum_{r=0..m-1} Cos(2Pi*k*r/m)Cos(2Pi*r/m)^n is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=10 and k=2.
Equals INVERT transform of A014138: (1, 3, 8, 22, 64, 196,...). [From Gary W. Adamson, May 15 2009]
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LINKS
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Table of n, a(n) for n=1..23.
Index entries for linear recurrences with constant coefficients, signature (7,-13,4)
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FORMULA
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a(n)= 4^n/10*Sum_{r=0..9} Cos(2Pi*r/5)Cos(Pi*r/5)^(2n).
a(n)= 7a(n-1)-13a(n-2)+4a(n-3).
G.f.: (-x+3x^2)/((-1+4x)(1-3x+x^2))
a(n) = (4^n + Lucas(2n-1))/5. With a(0) = 0, binomial transform of A098703. - Ross La Haye, May 31 2006
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MATHEMATICA
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f[n_]:=FullSimplify[TrigToExp[(4^n/10)Sum[Cos[2Pi*k/5]Cos[Pi*k/5]^(2n), {k, 0, 9}]]]; Table[f[n], {n, 1, 35}]
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CROSSREFS
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A014138 [From Gary W. Adamson, May 15 2009]
Sequence in context: A047108 A125145 A242781 * A026850 A109642 A164589
Adjacent sequences: A095927 A095928 A095929 * A095931 A095932 A095933
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KEYWORD
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nonn
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AUTHOR
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Herbert Kociemba, Jul 12 2004
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STATUS
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approved
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