A214673 Floor of the moduli of the zeros of the complex Lucas function.

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset

0,2

Comments

For the complex Lucas function L(z) and its zeros see the commens in A214671 and the Koshy reference.

The modulus rho(k) of the zeros is sqrt(x_0(k)^2 + y_0(k)^2), with x_0(k) = (2*k+1)*(alpha/2) and y_0(k) = (2*k+1)*(b/2), where alpha = 2*(Pi^2)/(Pi^2 + (2*log(phi))^2) and b = 4*Pi*log(phi)/(Pi^2 + (2*log(phi))^2) (see the Fibonacci case A214657) and phi =(1+sqrt(5))/2. This leads to rho(k) = (k+1/2)*tau, with tau = 2*Pi/sqrt(Pi^2 + (2*log(phi))^2), known from the Fibonacci case. tau is approximately 1.912278633.

The zeros lie in the complex plane on a straight line with angle Phi = -arctan(2*log(phi)/Pi). They are equally spaced with distance tau given above. Phi is approximately -.2972713044, corresponding to about -17.03 degrees. This is the same line like in the Fibonacci case A214657, and the zeros of the Lucas function are just shifted on this line by tau/2, approximately 0.9561393165.

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((2*n+1)*tau/2), n>=0, with tau/2 = rho(0) = 2*Pi / sqrt(Pi^2 + (2*log(phi))^2).

Mathematica

Table[Floor[(2*n+1)*Pi/Sqrt[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)/Sqrt(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/sqrt(pi^2 +4*(log(golden_ratio))^2)) for n in range(101)] # G. C. Greubel, Mar 09 2024

Crossrefs

Cf. A214657 (Fibonacci case), A214671, A214672.

Sequence in context: A007928 A257219 A092451 * A055962 A337718 A246410

Adjacent sequences: A214670 A214671 A214672 * A214674 A214675 A214676

Keyword

nonn

Author

Wolfdieter Lang, Jul 25 2012

Status

approved