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

0,5

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

Formula

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

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 *)

Prog

(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
(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
(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

Crossrefs

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

Sequence in context: A276796 A309093 A085268 * A194979 A366502 A098294

Adjacent sequences: A214653 A214654 A214655 * A214657 A214658 A214659

Keyword

nonn

Author

Wolfdieter Lang, Jul 24 2012

Status

approved