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%I #23 Mar 09 2024 08:16:58
%S 0,2,4,6,8,10,11,13,15,17,19,21,22,24,26,28,30,31,33,35,37,39,41,42,
%T 44,46,48,50,52,53,55,57,59,61,63,64,66,68,70,72,74,75,77,79,81,83,85,
%U 86,88,90,92,94,95,97,99,101,103,105,106,108,110,112,114,116,117,119
%N Floor of the real parts of the zeros of the complex Lucas function on the right half-plane.
%C For the complex Lucas function and its zeros see the Koshy reference.
%C This function is L: C -> C, z -> L(z), with L(z) = exp(log(phi)*z) + exp(i*Pi*z)*exp(-log(phi)*z), with the complex unit i and the golden section phi = (1+sqrt(5))/2. The complex zeros are z_0(k) = x_0(k) + y_0(k)*i, with x_0(k) = (k+1/2)*alpha and y_0(k) = (k+1/2)*b, where alpha and b appear in the Fibonacci case as alpha = 2*(Pi^2)/(Pi^2 + (2*log(phi))^2) and b = 4*Pi*log(phi)/(Pi^2 + (2*log(phi))^2). The x_0 and y_0 values are shifted compared to the zeros of the Fibonacci case by alpha/2 = 0.9142023918..., respectively b/2 = 0.2800649542....
%D Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%H G. C. Greubel, <a href="/A214671/b214671.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = floor((n+1/2)*alpha), with alpha/2 = x_0(0) = Pi^2/(Pi^2 + (2*log(phi))^2).
%t Table[Floor[(2*n+1)*(Pi^2)/(Pi^2+(2*Log[GoldenRatio])^2)], {n,0,100}] (* _G. C. Greubel_, Mar 09 2024 *)
%o (Magma) R:= RealField(100); [Floor((2*n+1)*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+1)*pi^2/(pi^2 +4*(log(golden_ratio))^2)) for n in range(101)] # _G. C. Greubel_, Mar 09 2024
%Y Cf. A214315 (Fibonacci case), A214672 (floor of imaginary parts), A214673 (moduli).
%K nonn
%O 0,2
%A _Wolfdieter Lang_, Jul 25 2012
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