A214657 Floor of the moduli of the zeros of the complex Fibonacci function.

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

Offset: 0

Author

Wolfdieter Lang, Jul 25 2012

Keywords

• nonn

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.

Examples

			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 ...
		

References

• Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A214315, A214656.

Programs

Magma

R:= RealField(100); [Floor(2*n*Pi(R)/Sqrt(Pi(R)^2 + (2*Log((1+Sqrt(5))/2))^2)) : n in [0..100]]; // G. C. Greubel, Mar 09 2024

Mathematica

Table[Floor[2*n*Pi/Sqrt[Pi^2 + (2*Log[GoldenRatio])^2]], {n,0,100}] (* G. C. Greubel, Mar 09 2024 *)

SageMath

[floor(2*n*pi/sqrt(pi^2 +4*(log(golden_ratio))^2)) for n in range(101)] # G. C. Greubel, Mar 09 2024

Formula

a(n) = floor(n*tau), n>=0, with tau = |z_0(1)| = 2*Pi/sqrt(Pi^2 + (2*log(phi))^2).