A052987 -id:A052987 - cp's OEIS frontend

A077937 Expansion of 1/(1-2x-2x^2+2*x^3).

1, 2, 6, 14, 36, 88, 220, 544, 1352, 3352, 8320, 20640, 51216, 127072, 315296, 782304, 1941056, 4816128, 11949760, 29649664, 73566592, 182532992, 452899840, 1123732480, 2788198656, 6918062592, 17165057536, 42589842944, 105673675776, 262196922368

Offset: 0

N. J. A. Sloane, Nov 17 2002

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 0, 1, 2, 6, ... counts walks of length n between the degree 5 vertex and the degree 3 vertex. - Paul Barry, Oct 02 2004
From Sean A. Irvine, Jun 05 2025: (Start)
Also, the number of walks of length n starting at vertex 0 in the graph:
    1-2
   /| |
  0 | |
   \| |
    4-3. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (2,2,-2).

Cf. A052528, A052987, A107300.

[n le 3 select Factorial(n) else 2*(Self(n-1) +Self(n-2) -Self(n-3)): n in [1..51]]; // G. C. Greubel, May 02 2022

LinearRecurrence[{2,2,-2}, {1,2,6}, 50] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
CoefficientList[Series[1/(1-2*x-2*x^2+2*x^3),{x,0,40}],x] (* Harvey P. Dale, Dec 05 2018 *)

Vec(1/(1-2*x-2*x^2+2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

def A077937_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2+2*x^3) ).list()
A077937_list(50) # G. C. Greubel, May 02 2022

a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) with a(0) = 1, a(1) = 2, and a(3) = 8. - G. C. Greubel, May 02 2022

A052528 Expansion of (1 - x)/(1 - 2x - 2x^2 + 2*x^3).

Original entry on oeis.org

1, 1, 4, 8, 22, 52, 132, 324, 808, 2000, 4968, 12320, 30576, 75856, 188224, 467008, 1158752, 2875072, 7133632, 17699904, 43916928, 108966400, 270366848, 670832640, 1664466176, 4129863936, 10246994944, 25424785408, 63083832832

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 a(n) counts closed walks of length n at the degree 5 vertex. - Paul Barry, Oct 02 2004
Equals the INVERT transform of (1, 3, 1, 1, 1, ...). - Gary W. Adamson, Apr 27 2009
a(n) is also the number of vertex-transitive cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_{n+1}. See Table 1 in the Hoang/Mütze reference in the Links section. - Torsten Muetze, Nov 28 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Hung Phuc Hoang, Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 455
Index entries for linear recurrences with constant coefficients, signature (2,2,-2).

Cf. A077937, A052987.

a:=[1,1,4];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, May 12 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1 -2*x-2*x^2+2*x^3) )); // G. C. Greubel, May 12 2019

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

LinearRecurrence[{2,2,-2}, {1,1,4}, 30] (* G. C. Greubel, May 12 2019 *)

my(x='x+O('x^30)); Vec((1-x)/(1-2*x-2*x^2+2*x^3)) \\ G. C. Greubel, May 12 2019

((1-x)/(1-2*x-2*x^2+2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

G.f.: (1 - x)/(1 - 2*x - 2*x^2 + 2*x^3).
Recurrence: a(1) = 1, a(0) = 1, a(2) = 4, 2*a(n) - 2*a(n+1) - 2*a(n+2) + a(n+3) = 0.
a(n) = Sum_{alpha=RootOf(2*Z^3-2*Z^2-2*Z+1)} (1/37)*(5 - 9*alpha^2 + 12*alpha)* alpha^(-1 - n).
a(n) = 2*a(n-2) + Sum_{i=0..n-1} a(i). - Yuchun Ji, Dec 29 2018

More terms from James Sellers, Jun 06 2000

A077847 Expansion of (1-x)^(-1)/(1-2x-2x^2+2*x^3).

Original entry on oeis.org

1, 3, 9, 23, 59, 147, 367, 911, 2263, 5615, 13935, 34575, 85791, 212863, 528159, 1310463, 3251519, 8067647, 20017407, 49667071, 123233663, 305766655, 758666495, 1882398975, 4670597631, 11588660223, 28753717759, 71343560703, 177017236479, 439214158847

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3, 0, -4, 2).

Cf. A052987.

CoefficientList[Series[(1-x)^(-1)/(1-2x-2x^2+2x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,0,-4,2},{1,3,9,23},40] (* Harvey P. Dale, Apr 02 2013 *)

Vec((1-x)^(-1)/(1-2*x-2*x^2+2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = -1+2*A077937(n)-2*A077937(n-2). [From R. J. Mathar, Nov 10 2009]

a(0)=1, a(1)=3, a(2)=9, a(3)=23, a(n)=3*a(n-1)-4*a(n-3)+2*a(n-4). - Harvey P. Dale, Apr 02 2013

A106666 Expansion of g.f. (1-4x^2+2x^3)/((1-x)(1-2x-2x^2+2x^3)).

Original entry on oeis.org

1, 3, 5, 13, 29, 73, 177, 441, 1089, 2705, 6705, 16641, 41281, 102433, 254145, 630593, 1564609, 3882113, 9632257, 23899521, 59299329, 147133185, 365065985, 905799681, 2247464961, 5576397313, 13836125185, 34330115073, 85179685889

Offset: 0

Creighton Dement, May 13 2005

Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[I*J*cyc(I)] with I = + .5'ii' + .5'kk' + .5'ik' + .5'jk' + .5'ki' + .5'kj' and J = + .5'i + .5i' - .5'ii' + .5'jj' + .5'kk' + .5e

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-4,2).

Cf. A052970, A052987, A077937.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(  (1-4*x^2+2*x^3)/((1-x)*(1-2*x-2*x^2+2*x^3)) )); // G. C. Greubel, Sep 08 2021

LinearRecurrence[{3,0,-4,2},{1,3,5,13},30] (* Harvey P. Dale, Jul 28 2015 *)

def A106666_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1-4*x^2+2*x^3)/((1-x)*(1-2*x-2*x^2+2*x^3)) ).list()
A106666_list(50) # G. C. Greubel, Sep 08 2021

Superseeker results: a(n+1) - a(n) = A052970(n+2); a(n+2) - a(n) = A052987(n+2).

a(0)=1, a(n) = 2*A077937(n-1) + 1.

A109545 a(n) = 2*a(n-1) + a(n-2) + a(n-3).

Original entry on oeis.org

1, 1, 2, 6, 15, 38, 97, 247, 629, 1602, 4080, 10391, 26464, 67399, 171653, 437169, 1113390, 2835602, 7221763, 18392518, 46842401, 119299083, 303833085, 773807654, 1970747476, 5019135691, 12782826512, 32555536191, 82913034585

Offset: 0

Roger L. Bagula, Jun 20 2005

David Garth and Kevin G. Hare, Comments on the spectra of Pisot numbers, J. Number Theory 121 (2006), 187-203.
Index entries for linear recurrences with constant coefficients, signature (2, 1, 1).

a = 2; b = -1; M = {{0, 1, 0, 0, 0}, { a - 2, a - 2, a - 2 - b, a - 2 - b, 0}, {1, 1, 1, 1, 0}, {0, 1, 1, 0, 0}, {0, 0, 0, 1, 1}} v[1] = {1, 1, 1, 1, 1} v[n_] := v[n] = M.v[n - 1] a0 = Table[Abs[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{2,1,1},{1,1,2},30] (* Harvey P. Dale, Aug 05 2015 *)
Lucas := 1 + x (1 + 2 x)/(1 - x - x^2); (* InvertTransform defined in A052987 *)
InvertTransform[Lucas, 28] (* Peter Luschny, Jan 10 2019 *)

lim_{n-> infinity} a(n)/a(n-1)= 2.54682...
G.f.: (1-x-x^2)/(1-2*x-x^2-x^3). [Sep 28 2009]
a(n) = A077939(n)-A116413(n-1).
G.f.: (-1+x+x^2)/(-1+2*x+x^2+x^3). a(n) = A077997(n)-A077939(n-2). [From R. J. Mathar, Sep 27 2009]

Definition replaced by recurrence by the Associate Editors of the OEIS, Sep 28 2009

cp's OEIS Frontend

A077937 Expansion of 1/(1-2*x-2*x^2+2*x^3).

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A052528 Expansion of (1 - x)/(1 - 2*x - 2*x^2 + 2*x^3).

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A077847 Expansion of (1-x)^(-1)/(1-2*x-2*x^2+2*x^3).

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A106666 Expansion of g.f. (1-4*x^2+2*x^3)/((1-x)*(1-2*x-2*x^2+2*x^3)).

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A109545 a(n) = 2*a(n-1) + a(n-2) + a(n-3).

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A077937 Expansion of 1/(1-2x-2x^2+2*x^3).

A052528 Expansion of (1 - x)/(1 - 2x - 2x^2 + 2*x^3).

A077847 Expansion of (1-x)^(-1)/(1-2x-2x^2+2*x^3).

A106666 Expansion of g.f. (1-4x^2+2x^3)/((1-x)(1-2x-2x^2+2x^3)).