OFFSET
0,2
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
FORMULA
a(n) = floor((n+1/2)*alpha), with alpha/2 = x_0(0) = Pi^2/(Pi^2 + (2*log(phi))^2).
MATHEMATICA
Table[Floor[(2*n+1)*(Pi^2)/(Pi^2+(2*Log[GoldenRatio])^2)], {n, 0, 100}] (* G. C. Greubel, Mar 09 2024 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Jul 25 2012
STATUS
approved