cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A214671 Floor of the real parts of the zeros of the complex Lucas function on the right half-plane.

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%I A214671 #23 Mar 09 2024 08:16:58
%S A214671 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 A214671 44,46,48,50,52,53,55,57,59,61,63,64,66,68,70,72,74,75,77,79,81,83,85,
%U A214671 86,88,90,92,94,95,97,99,101,103,105,106,108,110,112,114,116,117,119
%N A214671 Floor of the real parts of the zeros of the complex Lucas function on the right half-plane.
%C A214671 For the complex Lucas function and its zeros see the Koshy reference.
%C A214671 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 A214671 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%H A214671 G. C. Greubel, <a href="/A214671/b214671.txt">Table of n, a(n) for n = 0..10000</a>
%F A214671 a(n) = floor((n+1/2)*alpha), with alpha/2 = x_0(0) = Pi^2/(Pi^2 + (2*log(phi))^2).
%t A214671 Table[Floor[(2*n+1)*(Pi^2)/(Pi^2+(2*Log[GoldenRatio])^2)], {n,0,100}] (* _G. C. Greubel_, Mar 09 2024 *)
%o A214671 (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 A214671 (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 A214671 Cf. A214315 (Fibonacci case), A214672 (floor of imaginary parts), A214673 (moduli).
%K A214671 nonn
%O A214671 0,2
%A A214671 _Wolfdieter Lang_, Jul 25 2012