OFFSET
0,2
COMMENTS
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.
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).
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
FORMULA
a(n) = 9*F(n)^2 + F(n+1)^2, where F(n) = A000045(n) is the n-th Fibonacci number.
From Colin Barker, Jun 14 2013: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: (1-x)*(1+9*x) / ((1+x)*(1-3*x+x^2)). (End)
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
MATHEMATICA
a[n_] := 9Fibonacci[n]^2+Fibonacci[n+1]^2
9*First[#]+Last[#]&/@(Partition[Fibonacci[Range[0, 30]], 2, 1]^2) (* Harvey P. Dale, Mar 06 2012 *)
PROG
(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
(PARI) Vec(-(x-1)*(9*x+1)/((x+1)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Sep 30 2016
(Magma) F:=Fibonacci; [F(n+1)^2 +9*F(n)^2: n in [0..40]]; // G. C. Greubel, Aug 17 2022
(SageMath) f=fibonacci; [f(n+1)^2 +9*f(n)^2 for n in (0..40)] # G. C. Greubel, Aug 17 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, Apr 28 2002
EXTENSIONS
Edited by Dean Hickerson, May 08 2002
STATUS
approved