A214672 Floor of the imaginary parts of the zeros of the complex Lucas function on the left half-plane.

0, 0, 1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 31, 31, 32, 32, 33, 33, 34, 35, 35, 36

Offset: 0

Author

Wolfdieter Lang, Jul 25 2012

Keywords

• nonn

Comments

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

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. A214656 (Fibonacci case), A214671 (floor of real parts), A214673 (moduli).

Programs

Magma

R:= RealField(100); [Floor((2*n+1)*(2*Pi(R)*Log((1 + Sqrt(5))/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)*(2*Pi*Log[GoldenRatio])/(Pi^2 + (2*Log[GoldenRatio])^2)], {n, 0, 100}] (* G. C. Greubel, Mar 09 2024 *)

SageMath

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

Formula

a(n) = floor((2*n+1)*b/2), n >= 0, with b/2 = -y_0(0) = 2*Pi*log(phi) / (Pi^2 + (2*log(phi))^2), with phi = (1+sqrt(5))/2. Note that b/2 is approximately 0.2800649542... . The constant b appears in the corresponding Fibonacci case A214656.