A052987 Expansion of (1-2x^2)/(1-2x-2x^2+2x^3).

1, 2, 4, 10, 24, 60, 148, 368, 912, 2264, 5616, 13936, 34576, 85792, 212864, 528160, 1310464, 3251520, 8067648, 20017408, 49667072, 123233664, 305766656, 758666496, 1882398976, 4670597632, 11588660224, 28753717760, 71343560704

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Form the graph with matrix A=[1,1,1,1;1,0,0,0;1,0,0,0;1,0,0,1]. Then the sequence 1,1,2,4,... with g.f. (1-x-2x^2)/(1-2x-2x^2+2x^3) counts closed walks of length n at the degree 3 vertex. - Paul Barry, Oct 02 2004

Equals INVERT transform of the Jacobsthal sequence A001045 prefaced with a 1:

[1, 1, 1, 3, 5, 11, 21, 43, ...]. - Gary W. Adamson, May 27 2009

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1061
• Index entries for linear recurrences with constant coefficients, signature (2,2,-2).

Crossrefs

Cf. A077847, A052528, A077937, A001045.

Programs

Maple

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

Mathematica

InvertTransform[ser_, n_] := CoefficientList[ Series[1/(1 - x ser), {x,0,n}],x];
Jacobsthal := (2x^2-1)/((x + 1)(2x - 1));
PadLeft[InvertTransform[Jacobsthal, 29],29,1] (* Peter Luschny, Jan 10 2019 *)

Formula

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

Recurrence: {a(0)=1, a(2)=4, a(1)=2, 2*a(n)-2*a(n+1)-2*a(n+2)+a(n+3)=0}

Sum(1/37*(6+7*_alpha+4*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(2*_Z^3-2*_Z^2-2*_Z+1)).

Extensions

More terms from James Sellers, Jun 05 2000