

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
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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*phi1), 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/(tau1), 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



