A077937 -id:A077937 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A384646 Expansion of (1+x) / (1-x-5x^2-2x^3).

Original entry on oeis.org

1, 2, 7, 19, 58, 167, 495, 1446, 4255, 12475, 36642, 107527, 315687, 926606, 2720095, 7984499, 23438186, 68800871, 201960799, 592841526, 1740247263, 5108376491, 14995295858, 44017672839, 129210905111, 379289861022, 1113379732255, 3268250847587, 9593729230906

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 0 in the following graph:
    1---2
   /|\  |
  0 | \ |
   \|  \|
    4---3.
Also, by symmetry, the number of walks of length n starting at vertex 2 in the same graph.

			a(2)=7 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-4-0, 0-4-1, 0-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,5,2).

Cf. A384647 (vertex 1), A384648 (vertices 3 and 4), A077937 (missing edge {1,3}).

a:= n->  (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <1|1|0|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+x) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]

a(n) = A353964(n)+A353964(n-1). - R. J. Mathar, Jun 07 2025

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

A077981 Expansion of 1/(1+2x-2x^2-2*x^3).

Original entry on oeis.org

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

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

Cf. A077937.

a:=[1,-2,6];; 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, Jun 25 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1+2*x-2*x^2-2*x^3) )); // G. C. Greubel, Jun 25 2019

LinearRecurrence[{-2,2,2}, {1,-2,6}, 30] (* or *) CoefficientList[ Series[1/(1+2*x-2*x^2-2*x^3), {x,0,30}], x] (* G. C. Greubel, Jun 25 2019 *)

Vec(1/(1+2*x-2*x^2-2*x^3) + O(x^30)) \\ Michel Marcus, Jun 19 2015

(1/(1+2*x-2*x^2-2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 25 2019

a(n) = (-1)^n * A077937(n). - Ivan Neretin, Jun 19 2015

A091595 Triangle read by rows: T(n,m) := Sum_{k=0..floor((n-m)/2)} binomial(n-2k,m) * binomial(n-m-k,k) * 2^k.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 5, 5, 3, 1, 11, 12, 8, 4, 1, 21, 27, 22, 12, 5, 1, 43, 62, 55, 36, 17, 6, 1, 85, 137, 137, 99, 55, 23, 7, 1, 171, 304, 330, 264, 164, 80, 30, 8, 1, 341, 663, 784, 682, 466, 256, 112, 38, 9, 1, 683, 1442, 1833, 1720, 1278, 772, 382, 152, 47, 10, 1, 1365, 3109, 4235, 4257, 3402, 2234, 1218, 550, 201, 57, 11, 1

Offset: 0

Paul Barry, Jan 23 2004

A Jacobsthal related number number triangle.

			Rows begin:
   1,
   1,   1,
   3,   2,   1,
   5,   5,   3,  1,
  11,  12,   8,  4,  1,
  21,  27,  22, 12,  5,  1,
  43,  62,  55, 36, 17,  6, 1,
  85, 137, 137, 99, 55, 23, 7, 1,
  ...

Seiichi Manyama, Rows n = 0..139, flattened

Columns include A001045, A091596. Row sums are A077937.

k-th column has g.f. 1/(1-x-2x^2) * ( x*(1-2x^2)/(1-x-2x^2) )^k.

A107300 a(n) = 2a(n-1) + 2a(n-2) - 2*a(n-3) with a(0)=3, a(1)=2, a(3)=8.

Original entry on oeis.org

3, 2, 8, 14, 40, 92, 236, 576, 1440, 3560, 8848, 21936, 54448, 135072, 335168, 831584, 2063360, 5119552, 12702656, 31517696, 78201600, 194033280, 481434368, 1194532096, 2963866368, 7353928192

Offset: 0

Roger L. Bagula, May 20 2005

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

Cf. A077937.

I:=[3,2,8]; [n le 3 select I[n] else 2*(Self(n-1) +Self(n-2) -Self(n-3)): n in [1..46]]; // G. C. Greubel, May 02 2022

Mathematica
```
LinearRecurrence[{2,2,-2}, {3,2,8}, 46]
```

def A107300_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (3-4*x-2*x^2)/(1-2*x-2*x^2+2*x^3) ).list()
A107300_list(45) # G. C. Greubel, May 02 2022

G.f.: (3-4*x-2*x^2)/(1-2*x-2*x^2+2*x^3). [Sep 28 2009]
a(n) = 3*A077937(n) - 4*A077937(n-1) - 2*A077937(n-2). [Sep 28 2009]
a(n) = 2*(b1^n + b2^n + b3^n)/(b1 + b2 + b3), where b1, b2, and b3 and the roots of x^3 = 2*x^2 + 2*x - 2.

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

A384647 Expansion of (1+3x+x^2) / (1-x-5x^2-2*x^3).

Original entry on oeis.org

1, 4, 10, 32, 90, 270, 784, 2314, 6774, 19912, 58410, 171518, 503392, 1477802, 4337798, 12733592, 37378186, 109721742, 322079856, 945444938, 2775287702, 8146672104, 23914000490, 70197936414, 206061283072, 604878966122, 1775581254310, 5212098651064

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 1 in the following graph:
    1---2
   /|\  |
  0 | \ |
   \|  \|
    4---3.

			a(2)=10 because we have the walks 1-0-1, 1-0-4, 1-2-1, 1-2-3, 1-3-1, 1-3-2, 1-3-4, 1-4-0, 1-4-1, 1-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,5,2).

Cf. A384646 (vertices 0, 2), A384648 (vertices 3 and 4), A077937 (missing edge {1,3}).

a:= n->  (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <1|1|0|1|0>>^n. <<1,1,1,1,1>>)[2,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+3*x+x^2) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]

A384648 Expansion of (1+2x+x^2) / (1-x-5x^2-2*x^3).

Original entry on oeis.org

1, 3, 9, 26, 77, 225, 662, 1941, 5701, 16730, 49117, 144169, 423214, 1242293, 3646701, 10704594, 31422685, 92239057, 270761670, 794802325, 2333088789, 6848623754, 20103672349, 59012968697, 173228577950, 508500766133, 1492669593277, 4381630579842

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 0 in the following graph:
    1---2
   /|\  |
  0 | \ |
   \|  \|
    4---3.
Also, by symmetry, the number of walks of length n starting at vertex 4 in the same graph.

			a(2)=9 because we have the walks 3-1-0, 3-1-2, 3-1-3, 3-1-4, 3-2-1, 3-2-3, 3-4-0, 3-4-1, 3-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,5,2).

Cf. A384646 (vertices 0 and 2), A384647 (vertex 1), A077937 (missing edge {1,3}).

a:= n->  (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <1|1|0|1|0>>^n. <<1,1,1,1,1>>)[4,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+2*x+x^2) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]

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.

cp's OEIS Frontend

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

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

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A384646 Expansion of (1+x) / (1-x-5*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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A077981 Expansion of 1/(1+2*x-2*x^2-2*x^3).

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A091595 Triangle read by rows: T(n,m) := Sum_{k=0..floor((n-m)/2)} binomial(n-2k,m) * binomial(n-m-k,k) * 2^k.

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A107300 a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) with a(0)=3, a(1)=2, a(3)=8.

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

A384646 Expansion of (1+x) / (1-x-5x^2-2x^3).

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

A077981 Expansion of 1/(1+2x-2x^2-2*x^3).

A107300 a(n) = 2a(n-1) + 2a(n-2) - 2*a(n-3) with a(0)=3, a(1)=2, a(3)=8.

A384647 Expansion of (1+3x+x^2) / (1-x-5x^2-2*x^3).

A384648 Expansion of (1+2x+x^2) / (1-x-5x^2-2*x^3).

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