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 A373566 #18 Jun 12 2024 11:42:20
%S A373566 1,3,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,
%T A373566 10946,17711,28657,46368,75025,121393,196418,317811,514229,832040,
%U A373566 1346269,2178309,3524578,5702887,9227465,14930352,24157817,39088169,63245986,102334155
%N A373566 Expansion of x - 1/(x - 1/(x + 1)).
%H A373566 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A373566 a(n) = [x^n] (x^3 + x^2 - 2*x - 1) / (x^2 + x - 1).
%F A373566 From _Stefano Spezia_, Jun 10 2024: (Start)
%F A373566 a(n) = 2^(-n-1)*((1 - sqrt(5))^n*(sqrt(5) - 3) + (1 + sqrt(5))^n*(sqrt(5) + 3))/sqrt(5) for n <> 1.
%F A373566 E.g.f.: x + exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5. (End)
%t A373566 CoefficientList[Series[x - 1/(x - 1/(x + 1)), {x, 0, 38}], x] (* _Michael De Vlieger_, Jun 10 2024 *)
%Y A373566 Essentially the same as A071679 and A020695.
%K A373566 nonn,easy
%O A373566 0,2
%A A373566 _Peter Luschny_, Jun 10 2024