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.

1, 37, 40, 153, 349, 964, 2473, 6525, 17032, 44641, 116821, 305892, 800785, 2096533, 5488744, 14369769, 37620493, 98491780, 257854777, 675072621, 1767363016, 4627016497, 12113686405, 31714042788, 83028441889, 217371282949, 569085406888, 1489884937785

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) 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).

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) = 36*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+36*x) / ((1+x)*(1-3*x+x^2)). (End)

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

Mathematica

a[n_]:= 36*Fibonacci[n]^2 +Fibonacci[n+1]^2; Table[a[n], {n, 0, 30}]

Prog

(PARI) a(n)=36*fibonacci(n)^2+fibonacci(n+1)^2 \\ Charles R Greathouse IV, Jun 14 2013
(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
(PARI) Vec(-(x-1)*(36*x+1)/((x+1)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Sep 28 2016
(Magma) F:=Fibonacci; [F(n+1)^2 + 36*F(n)^2: n in [0..40]]; // G. C. Greubel, Aug 18 2022
(SageMath) f=fibonacci; [f(n+1)^2 +36*f(n)^2 for n in (0..40)] # G. C. Greubel, Aug 18 2022

Crossrefs

Cf. A000045, A069921, A069959, A069960, A069961, A069962.

Sequence in context: A295801 A111198 A081645 * A071855 A137675 A161725

Adjacent sequences: A069960 A069961 A069962 * A069964 A069965 A069966

Keyword

easy,nonn

Author

Benoit Cloitre, Apr 28 2002

Extensions

Edited by Dean Hickerson, May 08 2002

Status

approved