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.

1, 17, 20, 73, 169, 464, 1193, 3145, 8212, 21521, 56321, 147472, 386065, 1010753, 2646164, 6927769, 18137113, 47483600, 124313657, 325457401, 852058516, 2230718177, 5840095985, 15289569808, 40028613409, 104796270449, 274360197908, 718284323305, 1880492771977

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) + 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).

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) = 16*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+16*x) / ((1+x)*(1-3*x+x^2)). (End)

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

Mathematica

a[n_]:= 16Fibonacci[n]^2+Fibonacci[n+1]^2; Array[a, 30, 0]
16First[#]^2+Last[#]^2&/@Partition[Fibonacci[Range[0, 30]], 2, 1] (* Harvey P. Dale, Nov 08 2011 *)

Prog

(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
(PARI) Vec(-(x-1)*(16*x+1)/((x+1)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Oct 01 2016
(Magma) F:=Fibonacci; [F(n+1)^2 + 16*F(n)^2: n in [0..40]]; // G. C. Greubel, Aug 17 2022
(SageMath) f=fibonacci; [f(n+1)^2 +16*f(n)^2 for n in (0..40)] # G. C. Greubel, Aug 17 2022

Crossrefs

Cf. A000045, A069921, A069959, A069960, A069962, A069963.

Sequence in context: A146169 A218864 A045020 * A140146 A039502 A039505

Adjacent sequences: A069958 A069959 A069960 * A069962 A069963 A069964

Keyword

easy,nonn

Author

Benoit Cloitre, Apr 28 2002

Extensions

Edited by Dean Hickerson, May 08 2002

Status

approved