A052528 -id:A052528 - 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

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

A330039 Number of essential lattice congruences of the weak order on the symmetric group S_n.

Original entry on oeis.org

1, 1, 4, 47, 3322, 11396000

Offset: 1

Torsten Muetze, Nov 28 2019

			For n=3, the weak order on S_3 has the cover relations 123<132, 123<213, 132<312, 213<231, 312<321, 231<321, and there are a(3)=4 essential lattice congruences, namely {}, {132=312}, {213=231}, {132=312,213=231}.

Hung Phuc Hoang, Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
V. Pilaud and F. Santos, Quotientopes, arXiv:1711.05353 [math.CO], 2017-2019; Bull. Lond. Math. Soc., 51 (2019), no. 3, 406-420.

Cf. A091687, A001246, A052528, A024786, A123663, A330040, A330042.

A330040 Number of non-isomorphic cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_n.

Original entry on oeis.org

1, 1, 3, 19, 748, 2027309

Offset: 1

Torsten Muetze, Nov 28 2019

			For n=3, the weak order on S_3 has the cover relations 123<132, 123<213, 132<312, 213<231, 312<321, 231<321, and there are four essential lattice congruences, namely {}, {132=312}, {213=231}, {132=312,213=231}. The cover graph of the first one is a 6-cycle, the cover graph of the middle two is a 5-cycle, and the cover graph of the last one is a 4-cycle. These are 3 non-isomorphic graphs, showing that a(3)=3.

Hung Phuc Hoang, Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
V. Pilaud and F. Santos, Quotientopes, arXiv:1711.05353 [math.CO], 2017-2019; Bull. Lond. Math. Soc., 51 (2019), no. 3, 406-420.

Cf. A091687, A001246, A052528, A024786, A123663, A330039, A330042.

A330042 Number of non-isomorphic regular cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_n.

Original entry on oeis.org

1, 1, 3, 10, 51, 335, 2909

Offset: 1

Torsten Muetze, Nov 28 2019

			For n=3, the weak order on S_3 has the cover relations 123<132, 123<213, 132<312, 213<231, 312<321, 231<321, and there are four essential lattice congruences, namely {}, {132=312}, {213=231}, {132=312,213=231}. The cover graph of the first one is a 6-cycle, the cover graph of the middle two is a 5-cycle, and the cover graph of the last one is a 4-cycle. These are 3 non-isomorphic regular graphs, showing that a(3)=3.

Hung Phuc Hoang, Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
V. Pilaud and F. Santos, Quotientopes, arXiv:1711.05353 [math.CO], 2017-2019; Bull. Lond. Math. Soc., 51 (2019), no. 3, 406-420.

Cf. A091687, A001246, A052528, A024786, A123663, A330039, A330040.

A052575 Expansion of e.g.f. (1-x)/(1-2x-2x^2+2*x^3).

Original entry on oeis.org

1, 1, 8, 48, 528, 6240, 95040, 1632960, 32578560, 725760000, 18027878400, 491774976000, 14645952921600, 472356889804800, 16409046682828800, 610694391250944000, 24244324628299776000, 1022626965270822912000

Offset: 0

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

Robert Israel, Table of n, a(n) for n = 0..390
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 518.

Cf. A052528.

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

m = 17; Range[0, m]! * CoefficientList[Series[(1 - x)/(1 - 2*x - 2*x^2 + 2*x^3), {x, 0, m}], x] (* Amiram Eldar, Mar 07 2022 *)

my(x='x+O('x^25)); Vec(serlaplace((1-x)/(1-2*x-2*x^2+2*x^3))) \\ Michel Marcus, Mar 07 2022

E.g.f.: (1-x)/(1-2*x-2*x^2+2*x^3).
(12+2*n^3+12*n^2+22*n)*a(n) + (-2*n^2-10*n-12)*a(n+1) + (-2*n-6)*a(n+2) + a(n+3) = 0, with a(1)=1, a(0)=1, a(2)=8.
Sum_(-1/37*(-5+9*_alpha^2-12*_alpha)*_alpha^(-1-n), _alpha=RootOf(2*_Z^3-2*_Z^2-2*_Z+1))*n!.
a(n) = n!*A052528(n). - R. J. Mathar, Nov 27 2011

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A077937 Expansion of 1/(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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A330039 Number of essential lattice congruences of the weak order on the symmetric group S_n.

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A330040 Number of non-isomorphic cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_n.

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A330042 Number of non-isomorphic regular cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_n.

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A052575 Expansion of e.g.f. (1-x)/(1-2*x-2*x^2+2*x^3).

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Mathematica

PARI

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

A052575 Expansion of e.g.f. (1-x)/(1-2x-2x^2+2*x^3).