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

%I A214656 #27 Mar 09 2024 10:29:09
%S A214656 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,
%T A214656 16,16,17,17,18,19,19,20,20,21,21,22,22,23,24,24,25,25,26,26,27,28,28,
%U A214656 29,29,30,30,31,31,32,33,33,34,34,35,35,36,36,37,38,38,39
%N A214656 Floor of the imaginary part of the zeros of the complex Fibonacci function on the left half-plane.
%C A214656 See the comment on the Fibonacci Function F(z) and its zeros in A214315, where also the T. Koshy reference is given.
%C A214656 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.
%D A214656 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%H A214656 G. C. Greubel, <a href="/A214656/b214656.txt">Table of n, a(n) for n = 0..10000</a>
%F A214656 a(n) = floor(b*n), n>=0, with b = -y_0(1) = 4*Pi*log(phi)/(Pi^2 + (2*log(phi))^2).
%t A214656 a[n_]:= Floor[4*n*Pi*Log[GoldenRatio]/(Pi^2 + 4*Log[GoldenRatio]^2)];
%t A214656 Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Jul 03 2013 *)
%o A214656 (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
%o A214656 (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
%o A214656 (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
%Y A214656 Cf. A052952 (Fibonacci related formula), A214315 (real part).
%K A214656 nonn
%O A214656 0,5
%A A214656 _Wolfdieter Lang_, Jul 24 2012