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

0, 2, 4, 6, 8, 10, 11, 13, 15, 17, 19, 21, 22, 24, 26, 28, 30, 31, 33, 35, 37, 39, 41, 42, 44, 46, 48, 50, 52, 53, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 74, 75, 77, 79, 81, 83, 85, 86, 88, 90, 92, 94, 95, 97, 99, 101, 103, 105, 106, 108, 110, 112, 114, 116, 117, 119

Offset: 0

Author

Wolfdieter Lang, Jul 25 2012

Keywords

• nonn

Comments

For the complex Lucas function and its zeros see the Koshy reference.

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....

References

• Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A214315 (Fibonacci case), A214672 (floor of imaginary parts), A214673 (moduli).

Programs

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

Mathematica

Table[Floor[(2*n+1)*(Pi^2)/(Pi^2+(2*Log[GoldenRatio])^2)], {n,0,100}] (* G. C. Greubel, Mar 09 2024 *)

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

Formula

a(n) = floor((n+1/2)*alpha), with alpha/2 = x_0(0) = Pi^2/(Pi^2 + (2*log(phi))^2).