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%I #18 Mar 09 2024 08:14:25
%S 0,1,3,5,7,9,10,12,14,16,18,20,21,23,25,27,29,31,32,34,36,38,40,42,43,
%T 45,47,49,51,53,54,56,58,60,62,63,65,67,69,71,73,74,76,78,80,82,84,85,
%U 87,89,91,93,95,96,98,100,102,104,106,107,109,111,113,115,117,118
%N Floor of the real part of the zeros of the complex Fibonacci function on the right half-plane.
%C For the complex Fibonacci function and its complex zeros see the Koshy reference, pp. 523-524. See also the formula for F(z) given in the formula section of A052952. The real parts of the zeros of F are x_0(k) = alpha*k, with alpha = 2*(Pi^2)/(Pi^2 + (2*log(phi))^2), where phi = (1+sqrt(5))/2, and integer k. The corresponding imaginary parts are y_0(k) = - 4*Pi*log(phi)*k/(Pi^2 + (2*log(phi))^2). alpha is approximately 1.828404783. The zeros lie in the lower right and the upper left half-planes, and there is a zero at the origin.
%C a(n) = floor(alpha*n), n>=0, is a Beatty sequence with the complementary sequence b(n) = floor(beta*n), with beta = alpha/(alpha-1), approximately 2.207139336.
%C For the floor of the negative imaginary part see A214656.
%D Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%H G. C. Greubel, <a href="/A214315/b214315.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = floor(alpha*n), n>=0, with alpha = x_0(1) given in the comment section.
%e The complementary Beatty sequences start with:
%e n: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
%e a(n): 0 1 3 5 7 9 10 12 14 16 18 20 21 23 25 27
%e b(n): (0) 2 4 6 8 11 13 15 17 19 22 24 26 28 30 33
%t a[n_]:= Floor[2*n*Pi^2/(Pi^2 + 4*Log[GoldenRatio]^2)]; Table[a[n], {n, 0, 65}] (* _Jean-François Alcover_, Jul 03 2013 *)
%o (Magma) R:= RealField(100); [Floor(2*n*Pi(R)^2/(Pi(R)^2 + (2*Log((1+Sqrt(5))/2))^2)) : n in [0..100]]; // _G. C. Greubel_, Mar 09 2024
%o (SageMath) [floor(2*n*pi^2/(pi^2 +4*(log(golden_ratio))^2)) for n in range(101)] # _G. C. Greubel_, Mar 09 2024
%Y Cf. A052952 (Fibonacci related formula), A214656.
%K nonn
%O 0,3
%A _Wolfdieter Lang_, Jul 24 2012
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