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

%I A052923 #33 Jul 02 2025 16:01:58
%S A052923 1,0,4,4,20,36,116,260,724,1764,4660,11716,30356,77220,198644,507524,
%T A052923 1302100,3332196,8540596,21869380,56031764,143509284,367636340,
%U A052923 941673476,2412218836,6178912740,15827788084,40543439044,103854591380
%N A052923 Expansion of (1-x)/(1 - x - 4*x^2).
%C A052923 First differences of A006131.
%C A052923 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
%H A052923 G. C. Greubel, <a href="/A052923/b052923.txt">Table of n, a(n) for n = 0..1000</a>
%H A052923 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=908">Encyclopedia of Combinatorial Structures 908</a>
%H A052923 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,4).
%H A052923 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A052923 G.f.: (1-x)/(1 - x - 4*x^2).
%F A052923 a(n) = a(n-1) + 4*a(n-2), with a(0)=1, a(1)=0.
%F A052923 a(n) = Sum_{alpha=RootOf(-1+z+4*z^2)} (1/17)*(-1+9*alpha)*alpha^(-1-n).
%F A052923 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
%F A052923 From _Wolfdieter Lang_, Nov 27 2023: (Start)
%F A052923 a(n) = 4*A006131(n-2), with A006131(-2) = 1/4 and A006131(-1) = 0.
%F A052923 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)
%p A052923 spec := [S,{S=Sequence(Prod(Sequence(Z),Z,Union(Z,Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052923 seq(coeff(series((1-x)/(1 -x -4*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 16 2019
%t A052923 LinearRecurrence[{1,4}, {1,0}, 30] (* _G. C. Greubel_, Oct 16 2019 *)
%o A052923 (PARI) my(x='x+O('x^30)); Vec((1-x)/(1 -x -4*x^2)) \\ _G. C. Greubel_, Oct 16 2019
%o A052923 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1 -x -4*x^2) )); // _G. C. Greubel_, Oct 16 2019
%o A052923 (Sage)
%o A052923 def A052923_list(prec):
%o A052923     P.<x> = PowerSeriesRing(ZZ, prec)
%o A052923     return P((1-x)/(1 -x -4*x^2)).list()
%o A052923 A052923_list(30) # _G. C. Greubel_, Oct 16 2019
%o A052923 (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
%o A052923 a := n -> -(2*I)^n*ChebyshevU(n-2, -I/4):
%o A052923 seq(simplify(a(n)), n = 0..28);  # _Peter Luschny_, Dec 03 2023
%Y A052923 Cf. A006131, A026581, A049310, A222132.
%K A052923 easy,nonn
%O A052923 0,3
%A A052923 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052923 More terms from _James Sellers_, Jun 06 2000