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%I #33 Nov 03 2020 13:28:06
%S 1,0,3,1,10,7,35,36,127,165,474,715,1807,3004,6995,12393,27370,50559,
%T 107883,204820,427351,826045,1698458,3321891,6765175,13333932,
%U 26985675,53457121,107746282,214146295,430470899,857417220,1720537327
%N Expansion of (1-x)/((1-2x)(1+x-x^2)).
%C Number of walks of length n+1 between two adjacent vertices in the cycle graph C_5. Example: a(2)=3 because in the cycle ABCDE we have three walks of length 3 between A and B: ABAB, ABCB and AEAB. - _Emeric Deutsch_, Apr 01 2004
%C In general a(n,m)=2^n/m*Sum(k,0,m-1,Cos(2Pi*k/m)^(n+1)) gives number of walks of length n between two adjacent vertices in the cycle graph C_m. Here we have m=5. - _Herbert Kociemba_, May 31 2004
%C Counts walks of length n at the vertex of degree 3 of the graph with adjacency matrix A=[0,1,1,1;1,0,0,0;1,0,0,0;1,0,0,1]. Binomial transform is (L(n-2)+2*3^n)/5, or A099159. - _Paul Barry_, Oct 01 2004
%H Harvey P. Dale, <a href="/A052964/b052964.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1035">Encyclopedia of Combinatorial Structures 1035</a>
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2).
%F G.f.: -(-1+x)/(1-x-3*x^2+2*x^3)
%F Recurrence: {a(1)=0, a(0)=1, a(2)=3, 2*a(n)-3*a(n+1)-a(n+2)+a(n+3)=0}
%F Sum(-1/25*(-1-11*_alpha+6*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-_Z-3*_Z^2+2*_Z^3))
%F a(n-1)=2^n/5*Sum(k, 0, 4, Cos(2Pi*k/5)^(n+1)), n>=1 - _Herbert Kociemba_, May 31 2004
%F a(n)=((sqrt(5)-1)/2)^n(3/10-sqrt(5)/10)+((-sqrt(5)-1)/2)^n(3/10+sqrt(5)/10)+2^(n+1)/5 - _Paul Barry_, Oct 01 2004
%F a(n) = (2^(n+1) + A000032(n+2)*(-1)^n)/5 - _Ross La Haye_, May 31 2006
%F a(n) = |A084179(n+1)|-|A084179(n)|. - _R. J. Mathar_, Feb 27 2019
%p spec := [S,{S=Sequence(Prod(Union(Prod(Sequence(Z),Z),Z,Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%t CoefficientList[Series[(1-x)/((1-2x)(1+x-x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{1,3,-2},{1,0,3},40] (* _Harvey P. Dale_, Jun 03 2019 *)
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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