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A069963 Define C(n) by the recursion C(0) = 6*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 6*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of z.

%I A069963 #25 Aug 18 2022 01:45:55
%S A069963 1,37,40,153,349,964,2473,6525,17032,44641,116821,305892,800785,
%T A069963 2096533,5488744,14369769,37620493,98491780,257854777,675072621,
%U A069963 1767363016,4627016497,12113686405,31714042788,83028441889,217371282949,569085406888,1489884937785
%N A069963 Define C(n) by the recursion C(0) = 6*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 6*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of z.
%C A069963 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 A069963 Here, C(n) is defined with C(0) = 6*i in C(n) = (F(n) + C(0)*F(n-1))/(F(n+1) + C(0)*F(n)) = (F(n)*(F(n+1) + 36*F(n-1)) + (-1)^n*6*i)/(F(n+1)^2 + 36*F(n)^2), where F(n) = Fibonacci(n), for which Im(C(n)) = 6*(-1)^n/(F(n+1)^2 + 36*F(n)^2).
%H A069963 Colin Barker, <a href="/A069963/b069963.txt">Table of n, a(n) for n = 0..1000</a>
%H A069963 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A069963 a(n) = 36*F(n)^2 + F(n+1)^2, where F(n) = A000045(n) is the n-th Fibonacci number.
%F A069963 From _Colin Barker_, Jun 14 2013: (Start)
%F A069963 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A069963 G.f.: (1-x) *(1+36*x) / ((1+x)*(1-3*x+x^2)). (End)
%F A069963 a(n) = (2^(-1-n)*(-35*(-1)^n*2^(2+n) - (3-sqrt(5))^n*(-75+sqrt(5)) + (3+sqrt(5))^n*(75+sqrt(5))))/5. - _Colin Barker_, Sep 28 2016
%t A069963 a[n_]:= 36*Fibonacci[n]^2 +Fibonacci[n+1]^2; Table[a[n], {n,0,30}]
%o A069963 (PARI) a(n)=36*fibonacci(n)^2+fibonacci(n+1)^2 \\ _Charles R Greathouse IV_, Jun 14 2013
%o A069963 (PARI) a(n) = round((2^(-1-n)*(-35*(-1)^n*2^(2+n)-(3-sqrt(5))^n*(-75+sqrt(5))+(3+sqrt(5))^n*(75+sqrt(5))))/5) \\ _Colin Barker_, Sep 28 2016
%o A069963 (PARI) Vec(-(x-1)*(36*x+1)/((x+1)*(x^2-3*x+1)) + O(x^30)) \\ _Colin Barker_, Sep 28 2016
%o A069963 (Magma) F:=Fibonacci; [F(n+1)^2 + 36*F(n)^2: n in [0..40]]; // _G. C. Greubel_, Aug 18 2022
%o A069963 (SageMath) f=fibonacci; [f(n+1)^2 +36*f(n)^2 for n in (0..40)] # _G. C. Greubel_, Aug 18 2022
%Y A069963 Cf. A000045, A069921, A069959, A069960, A069961, A069962.
%K A069963 easy,nonn
%O A069963 0,2
%A A069963 _Benoit Cloitre_, Apr 28 2002
%E A069963 Edited by _Dean Hickerson_, May 08 2002