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

0,5

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

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.

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A361234 A065603 A225215 * A268060 A084242 A182281

Adjacent sequences: A214669 A214670 A214671 * A214673 A214674 A214675

Keyword

nonn

Author

Wolfdieter Lang, Jul 25 2012

Status

approved