This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A176981 #39 Sep 01 2025 11:22:42
%S A176981 1,0,-1,-2,-5,-12,-29,-70,-169,-408,-985,-2378,-5741,-13860,-33461,
%T A176981 -80782,-195025,-470832,-1136689,-2744210,-6625109,-15994428,
%U A176981 -38613965,-93222358,-225058681,-543339720,-1311738121,-3166815962,-7645370045,-18457556052,-44560482149
%N A176981 Expansion of 2+(1-2*x)/(-1+2*x+x^2).
%C A176981 It is essentially A000129, A077985 and A069306 except for signs and offsets.
%H A176981 Vincenzo Librandi, <a href="/A176981/b176981.txt">Table of n, a(n) for n = 0..1000</a>
%H A176981 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).
%F A176981 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.
%F A176981 So lim_{n->infinity} a(n+1)/a(n) = 1+sqrt(2).
%F A176981 Matches the A000045 formula: Fibonacci(n) = Fibonacci(n-1)*(1 + sqrt(5+4*(-1)^(n-1)/Fibonacci(n-1)^2))/2.
%F A176981 a(0)=1, a(1)=0, a(2)=-1, a(n) = 2*a(n-1)+a(n-2). - _Harvey P. Dale_, Dec 24 2011
%F A176981 G.f.: 1 / (1 + x^2 / (1 - 2*x / (1 - x / (1 + x)))). - _Michael Somos_, Jan 03 2013
%F A176981 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
%F A176981 For n>0, a(n) = ((1 - sqrt(2))^(n-1) - (1 + sqrt(2))^(n-1)) / 2^(3/2). - _Vaclav Kotesovec_, Sep 01 2025
%e A176981 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 + ...
%t A176981 a[0] = 1; a[n_] := a[n] = a[n - 1] - Sqrt[2*a[n - 1]^2 + (-1)^n]; Table[a[n], {n, 0, 30}]
%t A176981 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 *)
%t A176981 FullSimplify[Join[{1}, Table[((1 - Sqrt[2])^(n-1) - (1 + Sqrt[2])^(n-1)) / 2^(3/2), {n, 1, 30}]]] (* _Vaclav Kotesovec_, Sep 01 2025 *)
%Y A176981 Cf. A000129.
%K A176981 sign,easy,changed
%O A176981 0,4
%A A176981 _Roger L. Bagula_, Apr 30 2010
%E A176981 Name corrected by _Jason Yuen_, Sep 01 2025