login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A069960 Define C(n) by the recursion C(0)=3*I where I^2=-1, C(n+1)=1/(1+C(n)); then a(n)=3*(-1)^n/Im(C(n)) where Im(z) denotes the imaginary part of the complex number z. 4
1, 10, 13, 45, 106, 289, 745, 1962, 5125, 13429, 35146, 92025, 240913, 630730, 1651261, 4323069, 11317930, 29630737, 77574265, 203092074, 531701941, 1392013765, 3644339338, 9541004265, 24978673441, 65395016074, 171206374765, 448224108237, 1173465949930 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

If we define C(n) with C(0)=I then Im(C(n))=1/F(2n+1) where F(k) are the Fibonacci numbers.

C(n) = (F(n)+F(n-1)*C(0))/(F(n+1)+F(n)*C(0)) = (F(n)*(F(n+1)+9*F(n-1))+(-1)^n*3*I)/(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.

a(n) = 2*a(n-1)+2*a(n-2)-a(n-3). G.f.: -(x-1)*(9*x+1) / ((x+1)*(x^2-3*x+1)). - Colin Barker, Jun 14 2013

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

CROSSREFS

Cf. A069921, A069959, A069961-A069963.

Sequence in context: A195313 A219829 A062370 * A219715 A154142 A219804

Adjacent sequences:  A069957 A069958 A069959 * A069961 A069962 A069963

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, Apr 28 2002

EXTENSIONS

Edited by Dean Hickerson, May 08 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 6 20:37 EST 2021. Contains 341850 sequences. (Running on oeis4.)