A052923 Expansion of (1-x)/(1 - x - 4*x^2).

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

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

First differences of A006131.

This sequence {a(n)} appears in the formula for powers of c = (1 + sqrt(17))/2 = A222132, the fundamental (integer) algebraic number of Q(sqrt(17)): c^n = a(n) + A006131(n-1)*c. This is also valid for positive powers of 1/c = (-1 + sqrt(17)) /8. See the formula below and in A006131 in terms of Chebyshev or Fibonacci polynomials. - Wolfdieter Lang, Nov 27 2023

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).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A006131, A026581, A049310, A222132.

Programs

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
a := n -> -(2*I)^n*ChebyshevU(n-2, -I/4):
seq(simplify(a(n)), n = 0..28);  # Peter Luschny, Dec 03 2023

Magma

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

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 *)

PARI

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

Sage

def A052923_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1 -x -4*x^2)).list()
A052923_list(30) # G. C. Greubel, Oct 16 2019

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

From Wolfdieter Lang, Nov 27 2023: (Start)

a(n) = 4*A006131(n-2), with A006131(-2) = 1/4 and A006131(-1) = 0.

a(n) = -(-2*i)^n*S(n-2, i/2), with i = sqrt(-1), and the S-Chebyshev polynomials (see A049310). S(-n, x) = -S(n-2, x). The Fibonacci polynomials are F(n, x) = (-i)^(n-1)*S(n-1, i*x). (End)

Extensions

More terms from James Sellers, Jun 06 2000