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

Author

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

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A077937, A052987.

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Sage

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

Formula

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

Extensions

More terms from James Sellers, Jun 06 2000