OFFSET
0,3
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).
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
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 *)
PROG
(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
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 06 2000
STATUS
approved