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A052923 Expansion of (1-x)/(1 - x - 4*x^2). 3
1, 0, 4, 4, 20, 36, 116, 260, 724, 1764, 4660, 11716, 30356, 77220, 198644, 507524, 1302100, 3332196, 8540596, 21869380, 56031764, 143509284, 367636340, 941673476, 2412218836, 6178912740, 15827788084, 40543439044, 103854591380 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

First differences of A006131.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 908

Index entries for linear recurrences with constant coefficients, signature (1,4).

FORMULA

G.f.: (1-x)/(1 - x - 4*x^2).

a(n) = a(n-1) + 4*a(n-2), with a(0)=1, a(1)=0.

a(n) = Sum_{alpha=RootOf(-1+z+4*z^2)} (1/17)*(-1+9*alpha)*alpha^(-1-n).

If p[1]=0, and p[i]=4, ( i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010

MAPLE

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

seq(coeff(series((1-x)/(1 -x -4*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 16 2019

MATHEMATICA

LinearRecurrence[{1, 4}, {1, 0}, 30] (* G. C. Greubel, Oct 16 2019 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((1-x)/(1 -x -4*x^2)) \\ G. C. Greubel, Oct 16 2019

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1 -x -4*x^2) )); // G. C. Greubel, Oct 16 2019

(Sage)

def A052923_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P((1-x)/(1 -x -4*x^2)).list()

A052923_list(30) # G. C. Greubel, Oct 16 2019

(GAP) a:=[1, 0];; for n in [3..30] do a[n]:=a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Oct 16 2019

CROSSREFS

Cf. A006131, A026581.

Sequence in context: A165559 A180967 A231884 * A321691 A014433 A191366

Adjacent sequences:  A052920 A052921 A052922 * A052924 A052925 A052926

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

More terms from James A. Sellers, Jun 06 2000

STATUS

approved

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Last modified November 14 12:21 EST 2019. Contains 329114 sequences. (Running on oeis4.)