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A214656
Floor of the imaginary part of the zeros of the complex Fibonacci function on the left half-plane.
5
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
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
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Jul 24 2012
STATUS
approved