A052964 Expansion of (1-x)/((1-2x)(1+x-x^2)).

1, 0, 3, 1, 10, 7, 35, 36, 127, 165, 474, 715, 1807, 3004, 6995, 12393, 27370, 50559, 107883, 204820, 427351, 826045, 1698458, 3321891, 6765175, 13333932, 26985675, 53457121, 107746282, 214146295, 430470899, 857417220, 1720537327

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

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

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

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

Also, the cogrowth sequence for the 10-element dihedral group D5; that is a(n) is the number of words of length n+2 that reduce to the identity using the presentation . - Sean A. Irvine, Nov 04 2024

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1035
• Index entries for linear recurrences with constant coefficients, signature (1,3,-2).

Programs

Maple

spec := [S,{S=Sequence(Prod(Union(Prod(Sequence(Z),Z),Z,Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

Mathematica

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 *)

Formula

G.f.: -(-1+x)/(1-x-3*x^2+2*x^3)

Recurrence: {a(1)=0, a(0)=1, a(2)=3, 2*a(n)-3*a(n+1)-a(n+2)+a(n+3)=0}

Sum(-1/25*(-1-11*_alpha+6*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-_Z-3*_Z^2+2*_Z^3))

a(n-1)=2^n/5*Sum(k, 0, 4, Cos(2Pi*k/5)^(n+1)), n>=1 - Herbert Kociemba, May 31 2004

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

a(n) = (2^(n+1) + A000032(n+2)*(-1)^n)/5 - Ross La Haye, May 31 2006

a(n) = |A084179(n+1)|-|A084179(n)|. - R. J. Mathar, Feb 27 2019