

A052528


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


7



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 vertextransitive 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^32*Z^22*Z+1)} (1/37)*(5  9*alpha^2 + 12*alpha)* alpha^(1  n).
a(n) = 2*a(n2) + Sum_{i=0..n1} 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((1x)/(12*x2*x^2+2*x^3)) \\ G. C. Greubel, May 12 2019
(MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1x)/(1 2*x2*x^2+2*x^3) )); // G. C. Greubel, May 12 2019
(Sage) ((1x)/(12*x2*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[n1]+2*a[n2]2*a[n3]; od; a; # G. C. Greubel, May 12 2019


CROSSREFS

Cf. A077937, A052987.
Sequence in context: A175655 A000639 A190795 * A058855 A297339 A290138
Adjacent sequences: A052525 A052526 A052527 * A052529 A052530 A052531


KEYWORD

nonn,easy


AUTHOR

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


EXTENSIONS

More terms from James A. Sellers, Jun 06 2000


STATUS

approved



