A176981 Expansion of 1+(1-2*x)/(-1+2*x+x^2).

1, 0, -1, -2, -5, -12, -29, -70, -169, -408, -985, -2378, -5741, -13860, -33461, -80782, -195025, -470832, -1136689, -2744210, -6625109, -15994428, -38613965, -93222358, -225058681, -543339720, -1311738121, -3166815962, -7645370045

Offset

0,4

Comments

It is essentially A000129, A077985 and A069306 except for signs and offsets.

a(n)=a(n - 1) - sqrt(2*a(n - 1)^2 + (-1)^n) = a(n-1)*(1-sqrt(2-(-1)^n/a(n-1)^2)) for n>0.

So in the limit of n->infinity, a(n+1)/a(n)= 1-Sqrt(2).

This matches the following formula in A000045:

Fibonacci(n) = Fibonacci(n - 1)*(1 + sqrt( 5+4*(-1)^(n-1)/Fibonacci(n-1)^2) )/2.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,1).

Formula

a(0)=1, a(1)=0, a(2)=-1, a(n)=2*a(n-1)+a(n-2) [From Harvey P. Dale, Dec 24 2011]

G.f.: 1 / (1 + x^2 / (1 - 2*x / (1 - x / (1 + x)))). - Michael Somos, Jan 03 2013

G.f.: 1 - Q(0)*x^2/2 , where Q(k) = 1 + 1/(1 - x*(4*k+2 + x)/( x*(4*k+4 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013

Example

1 - x^2 - 2*x^3 - 5*x^4 - 12*x^5 - 29*x^6 - 70*x^7 - 169*x^8 - 408*x^9 - 985*x^10 + ...

Mathematica

a[0] = 1; a[n_] := a[n] = a[n - 1] - Sqrt[2*a[n - 1]^2 + (-1)^n]; Table[a[n], {n, 0, 30}]
Join[{1}, LinearRecurrence[{2, 1}, {0, -1}, 30]] (* or *) Join[{1}, Rest[ CoefficientList[Series[1 + (1 - 2 x)/(-1 + 2 x + x^2), {x, 0, 30}] , x]]] (* Harvey P. Dale, Dec 24 2011 *)

Crossrefs

Cf. A000129.

Sequence in context: A130009 A324979 A048624 * A215936 A000129 A077985

Adjacent sequences: A176978 A176979 A176980 * A176982 A176983 A176984

Keyword

sign,easy

Author

Roger L. Bagula, Apr 30 2010

Status

approved