A077847 -id:A077847 - cp's OEIS frontend

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

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

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

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

Cf. A077847, A052528, A077937, A001045.

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

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 *)

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)).

More terms from James Sellers, Jun 05 2000

A110158 Expansion of x^4 / ((x+1)(2x^3-2x^2-2x+1)*(x-1)^2).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 10, 26, 69, 173, 436, 1084, 2699, 6699, 16634, 41274, 102425, 254137, 630584, 1564600, 3882103, 9632247, 23899510, 59299318, 147133173, 365065973, 905799668, 2247464948, 5576397299, 13836125171

Offset: 0

Creighton Dement, Sep 05 2005

Floretion Algebra Multiplication Program, FAMP Code: 1jbasejcycsumseq[ + .5'k + .5k' + 'ij'], sumtype: (Y[15], *, vesy)

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-7,2,4,-2).

Cf. A077847.

CoefficientList[Series[x^4/((x+1)(2x^3-2x^2-2x+1)(x-1)^2),{x,0,30}],x] (* Harvey P. Dale, Jan 23 2019 *)

concat([0,0,0,0], Vec(x^4 / ((1 - x)^2*(1 + x)*(1 - 2*x - 2*x^2 + 2*x^3)) + O(x^40))) \\ Colin Barker, May 16 2019

a(n) = -(-1)^n/4 + 7/4 - A077937(n) + A077937(n-1) + 4*A077937(n-2) - (n+1)/2. - R. J. Mathar, Nov 10 2009

a(n) = 3*a(n-1) + a(n-2) - 7*a(n-3) + 2*a(n-4) + 4*a(n-5) - 2*a(n-6) for n > 5. - Colin Barker, May 16 2019

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

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A110158 Expansion of x^4 / ((x+1)(2x^3-2x^2-2x+1)*(x-1)^2).

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cp's OEIS Frontend

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

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A110158 Expansion of x^4 / ((x+1)*(2*x^3-2*x^2-2*x+1)*(x-1)^2).

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A110158 Expansion of x^4 / ((x+1)(2x^3-2x^2-2x+1)*(x-1)^2).