cp's OEIS Frontend

A069961 Define C(n) by the recursion C(0) = 4*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 4*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of z.

%I A069961 #29 Aug 18 2022 01:42:33
%S A069961 1,17,20,73,169,464,1193,3145,8212,21521,56321,147472,386065,1010753,
%T A069961 2646164,6927769,18137113,47483600,124313657,325457401,852058516,
%U A069961 2230718177,5840095985,15289569808,40028613409,104796270449,274360197908,718284323305,1880492771977
%N A069961 Define C(n) by the recursion C(0) = 4*i where i^2 = -1, C(n+1) = 1/(1 + C(n)), then a(n) = 4*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of z.
%C A069961 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 A069961 Here, C(n) = (F(n) + C(0)*F(n-1))/(F(n+1) + C(0)*F(n)) = (F(n)*(F(n+1) + 16*F(n-1)) + (-1)^n*4*i)/(F(n+1)^2 + 16*F(n)^2), where F(n) = Fibonacci(n), for which Im(C(n)) = 4*(-1)^n/(F(n+1)^2 + 16*F(n)^2).
%H A069961 Colin Barker, <a href="/A069961/b069961.txt">Table of n, a(n) for n = 0..1000</a>
%H A069961 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A069961 a(n) = 16*F(n)^2 + F(n+1)^2, where F(n) = A000045(n) is the n-th Fibonacci number.
%F A069961 From _Colin Barker_, Jun 14 2013: (Start)
%F A069961 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A069961 G.f.: (1-x)*(1+16*x) / ((1+x)*(1-3*x+x^2)). (End)
%F A069961 a(n) = (2^(-1-n)*(-15*(-1)^n*2^(2+n) - (3-sqrt(5))^n*(-35+sqrt(5)) + (3+sqrt(5))^n*(35+sqrt(5))))/5. - _Colin Barker_, Oct 01 2016
%t A069961 a[n_]:= 16Fibonacci[n]^2+Fibonacci[n+1]^2; Array[a,30,0]
%t A069961 16First[#]^2+Last[#]^2&/@Partition[Fibonacci[Range[0,30]],2,1] (* _Harvey P. Dale_, Nov 08 2011 *)
%o A069961 (PARI) a(n) = round((2^(-1-n)*(-15*(-1)^n*2^(2+n)-(3-sqrt(5))^n*(-35+sqrt(5))+(3+sqrt(5))^n*(35+sqrt(5))))/5) \\ _Colin Barker_, Oct 01 2016
%o A069961 (PARI) Vec(-(x-1)*(16*x+1)/((x+1)*(x^2-3*x+1)) + O(x^30)) \\ _Colin Barker_, Oct 01 2016
%o A069961 (Magma) F:=Fibonacci; [F(n+1)^2 + 16*F(n)^2: n in [0..40]]; // _G. C. Greubel_, Aug 17 2022
%o A069961 (SageMath) f=fibonacci; [f(n+1)^2 +16*f(n)^2 for n in (0..40)] # _G. C. Greubel_, Aug 17 2022
%Y A069961 Cf. A000045, A069921, A069959, A069960, A069962, A069963.
%K A069961 easy,nonn
%O A069961 0,2
%A A069961 _Benoit Cloitre_, Apr 28 2002
%E A069961 Edited by _Dean Hickerson_, May 08 2002