A214656 Floor of the imaginary part of the zeros of the complex Fibonacci function on the left half-plane.

0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39

Offset: 0

Author

Wolfdieter Lang, Jul 24 2012

Keywords

• nonn

Comments

See the comment on the Fibonacci Function F(z) and its zeros in A214315, where also the T. Koshy reference is given.

The imaginary part of the zeros, corresponding to the real part x_0(k) given in A214315, is y_0(k) = -b*k, with b = 4*Pi*log(phi)/(Pi^2 + (2*log(phi))^2) and phi = (1+sqrt(5))/2. Note that b is approximately 0.5601299084.

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. A052952 (Fibonacci related formula), A214315 (real part).

Programs

Magma

R:= RealField(100); [Floor(4*n*Pi(R)*Log((1+Sqrt(5))/2)/(Pi(R)^2 + (2*Log((1+Sqrt(5))/2))^2)) : n in [0..100]]; // G. C. Greubel, Mar 09 2024

Mathematica

a[n_]:= Floor[4*n*Pi*Log[GoldenRatio]/(Pi^2 + 4*Log[GoldenRatio]^2)];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Jul 03 2013 *)

PARI

A214656(n,phi=(sqrt(5)+1)/2)=n*4*Pi*log(phi)\(Pi^2+(2*log(phi))^2)  \\ M. F. Hasler, Jul 24 2012

SageMath

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

Formula

a(n) = floor(b*n), n>=0, with b = -y_0(1) = 4*Pi*log(phi)/(Pi^2 + (2*log(phi))^2).