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 A069959 #26 Aug 17 2022 16:19:45
%S A069959 1,5,8,25,61,164,425,1117,2920,7649,20021,52420,137233,359285,940616,
%T A069959 2462569,6447085,16878692,44188985,115688269,302875816,792939185,
%U A069959 2075941733,5434886020,14228716321,37251262949,97525072520,255323954617,668446791325,1750016419364
%N A069959 Define C(n) by the recursion C(0) = 2*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 2*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of the complex number z.
%C A069959 If we define C(n) with C(0) = i in the recurrence C(n+1) = 1/(1 + C(n)) then Im(C(n)) = 1/Fibonacci(2*n+1).
%C A069959 Here, C(n) is defined with C(0) = 2*i in C(n) = (F(n) + C(0)*F(n-1))/(F(n+1) + C(0)*F(n)) = (F(n)*(F(n+1) + 4*F(n-1)) + (-1)^n*2*i)/(F(n+1)^2 + 4*F(n)^2), where F(n) = Fibonacci(n), for which Im(C(n)) = 2*(-1)^n/(F(n+1)^2 + 4*F(n)^2).
%H A069959 Colin Barker, <a href="/A069959/b069959.txt">Table of n, a(n) for n = 0..1000</a>
%H A069959 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A069959 a(n) = 4*F(n)^2 + F(n+1)^2, where F(n) = A000045(n) is the n-th Fibonacci number.
%F A069959 From _Colin Barker_, Jun 14 2013: (Start)
%F A069959 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A069959 G.f.: (1-x)*(1+4*x) / ((1+x)*(1-3*x+x^2)). (End)
%F A069959 a(n) = (2^(-1-n)*(-3*(-1)^n*2^(2+n) - (3-sqrt(5))^n*(-11+sqrt(5)) + (3+sqrt(5))^n*(11+sqrt(5))))/5. - _Colin Barker_, Sep 28 2016
%t A069959 a[n_]:= 4Fibonacci[n]^2+Fibonacci[n+1]^2;
%t A069959 4#[[1]]^2+#[[2]]^2&/@Partition[Fibonacci[Range[0,30]],2,1] (* or *) LinearRecurrence[{2,2,-1},{1,5,8},30] (* _Harvey P. Dale_, Aug 25 2017 *)
%o A069959 (PARI) a(n) = round((2^(-1-n)*(-3*(-1)^n*2^(2+n)-(3-sqrt(5))^n*(-11+sqrt(5))+(3+sqrt(5))^n*(11+sqrt(5))))/5) \\ _Colin Barker_, Sep 28 2016
%o A069959 (PARI) Vec(-(x-1)*(4*x+1)/((x+1)*(x^2-3*x+1)) + O(x^40)) \\ _Colin Barker_, Sep 28 2016
%o A069959 (Magma) F:=Fibonacci; [F(n+1)^2 +4*F(n)^2: n in [0..40]]; // _G. C. Greubel_, Aug 17 2022
%o A069959 (SageMath) f=fibonacci; [f(n+1)^2+4*f(n)^2 for n in (0..40)] # _G. C. Greubel_, Aug 17 2022
%Y A069959 Cf. A000045, A069921, A069960, A069961, A069962, A069963.
%K A069959 easy,nonn
%O A069959 0,2
%A A069959 _Benoit Cloitre_, Apr 28 2002
%E A069959 Edited by _Dean Hickerson_, May 08 2002