login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A214657 Floor of the moduli of the zeros of the complex Fibonacci function. 1
0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 108, 110, 112, 114, 116, 118, 120, 122, 124 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For the complex Fibonacci function F(z) and its zeros see the T. Koshy reference given in A214315. There the formula for the real and imaginary parts of the zeros is also given.

  F: C -> C, z -> F(z) with F(z) := (exp(log(phi)*z) - exp(I*Pi*z)*exp(-log(phi)*z))/(2*phi-1), with phi := (1+sqrt(5))/2 and the imaginary unit I.

The zeros in the complex plane lie on a straight line with angle Phi = -arctan(2*log(phi)/Pi). They are equally spaced with distance tau defined below. Phi is approximately -0.2972713044, corresponding to about -17.03 degrees. The moduli are |z_0(k)| = tau*k, with tau:= 2*Pi/sqrt(Pi^2 + (2*log(phi))^2), which is approximately 1.912278633.

a(n) = floor(tau*n) is a Beatty sequence with the complementary sequence b(n) := floor(sigma*n), with sigma:= tau/(tau-1), approximately 2.096156332.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

a(n) = floor(tau*n), n>=0, with tau = |z_0(1)|given in the comment section.

EXAMPLE

The complementary Beatty sequences a(n) and  b(n) start:

n:     0 1 2 3 4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...

a(n):  0 1 3 5 7  9 11 13 15 17 19 21 22 24 26 28 30 32 34 ...

b(n): (0)2 4 6 8 10 12 14 16 18 20 23 25 27 29 31 33 35 37 ...

CROSSREFS

Cf. A214315, A214656.

Sequence in context: A251238 A059547 A064719 * A137803 A059533 A189397

Adjacent sequences:  A214654 A214655 A214656 * A214658 A214659 A214660

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Jul 25 2012

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 September 17 06:52 EDT 2019. Contains 327119 sequences. (Running on oeis4.)