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A095929
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Number of closed walks of length 2n at a vertex of the cyclic graph on 10 nodes C_10.
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1
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1, 2, 6, 20, 70, 254, 948, 3614, 13990, 54740, 215766, 854702, 3396916, 13530350, 53971350, 215492564, 860941798, 3441074654, 13757249460, 55010542910, 219993856006, 879848932052, 3519064567926, 14075391282830
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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=0.
The number of round trips of odd length is zero. See the array w(N,L) and triangle a(K,N) given in A199571 for the general case. - Wolfdieter Lang, Nov 08 2011
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (7,-13,4).
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FORMULA
| a(n)= 4^n/10*Sum_{r=0..9} Cos(Pi*r/5)^(2n); a(n)=7a(n-1)-13a(n-2)+4a(n-3);
G.f.: (-1+5x-5x^2)/((-1+4x)(1-3x+x^2)).
a(n/2) = ( 2^n +2*phi^n +2*(phi-1)^n )*(1+(-1)^n)/10, with the golden section phi = A001622. - Wolfdieter Lang, Nov 08 2011
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EXAMPLE
| a(2)=6 from the six round trips from ,say, vertex no. 1: 12121, 1(10)1(10)1, 121(10)1, 1(10)121, 12321 and 1(10)9(10)1. - Wolfdieter Lang, Nov 08 2011
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MATHEMATICA
| f[n_]:=FullSimplify[TrigToExp[(4^n/10)Sum[Cos[Pi*k/5]^(2n), {k, 0, 9}]]]; Table[f[n], {n, 0, 35}]
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CROSSREFS
| Cf. A094233, A199571.
Sequence in context: A119373 A151284 A049138 * A078482 A049128 A192540
Adjacent sequences: A095926 A095927 A095928 * A095930 A095931 A095932
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KEYWORD
| nonn
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AUTHOR
| Herbert Kociemba (kociemba(AT)t-online.de), Jul 12 2004
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