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A069960 Define C(n) by the recursion C(0) = 3*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 3*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of the complex number z.

%I A069960 #22 Aug 17 2022 22:31:05
%S A069960 1,10,13,45,106,289,745,1962,5125,13429,35146,92025,240913,630730,
%T A069960 1651261,4323069,11317930,29630737,77574265,203092074,531701941,
%U A069960 1392013765,3644339338,9541004265,24978673441,65395016074,171206374765,448224108237,1173465949930
%N A069960 Define C(n) by the recursion C(0) = 3*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 3*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of the complex number z.
%C A069960 If we define C(n) with C(0) = i then Im(C(n)) = 1/F(2*n+1) where F(k) are the Fibonacci numbers.
%C A069960 Here, C(n) = (F(n) + C(0)*F(n-1))/(F(n+1) + C(0)*F(n)) = (F(n)*(F(n+1) + 9*F(n-1)) + 3*i*(-1)^n)/(F(n+1)^2 + 9*F(n)^2), where F(n) = Fibonacci(n), for which Im(C(n)) = 3*(-1)^n/(F(n+1)^2 + 9*F(n)^2).
%H A069960 Colin Barker, <a href="/A069960/b069960.txt">Table of n, a(n) for n = 0..1000</a>
%H A069960 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A069960 a(n) = 9*F(n)^2 + F(n+1)^2, where F(n) = A000045(n) is the n-th Fibonacci number.
%F A069960 From _Colin Barker_, Jun 14 2013: (Start)
%F A069960 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A069960 G.f.: (1-x)*(1+9*x) / ((1+x)*(1-3*x+x^2)). (End)
%F A069960 a(n) = (2^(-1-n)*(-(-1)^n*2^(5+n) - (3-sqrt(5))^n*(-21+sqrt(5)) + (3+sqrt(5))^n*(21+sqrt(5))))/5. - _Colin Barker_, Sep 30 2016
%t A069960 a[n_] := 9Fibonacci[n]^2+Fibonacci[n+1]^2
%t A069960 9*First[#]+Last[#]&/@(Partition[Fibonacci[Range[0,30]],2,1]^2) (* _Harvey P. Dale_, Mar 06 2012 *)
%o A069960 (PARI) a(n) = round((2^(-1-n)*(-(-1)^n*2^(5+n)-(3-sqrt(5))^n*(-21+sqrt(5))+(3+sqrt(5))^n*(21+sqrt(5))))/5) \\ _Colin Barker_, Sep 30 2016
%o A069960 (PARI) Vec(-(x-1)*(9*x+1)/((x+1)*(x^2-3*x+1)) + O(x^30)) \\ _Colin Barker_, Sep 30 2016
%o A069960 (Magma) F:=Fibonacci; [F(n+1)^2 +9*F(n)^2: n in [0..40]]; // _G. C. Greubel_, Aug 17 2022
%o A069960 (SageMath) f=fibonacci; [f(n+1)^2 +9*f(n)^2 for n in (0..40)] # _G. C. Greubel_, Aug 17 2022
%Y A069960 Cf. A000045, A069921, A069959, A069961, A069962, A069963.
%K A069960 easy,nonn
%O A069960 0,2
%A A069960 _Benoit Cloitre_, Apr 28 2002
%E A069960 Edited by _Dean Hickerson_, May 08 2002