OFFSET
0,3
COMMENTS
For the complex Fibonacci function and its complex zeros see the Koshy reference, pp. 523-524. See also the formula for F(z) given in the formula section of A052952. The real parts of the zeros of F are x_0(k) = alpha*k, with alpha = 2*(Pi^2)/(Pi^2 + (2*log(phi))^2), where phi = (1+sqrt(5))/2, and integer k. The corresponding imaginary parts are y_0(k) = - 4*Pi*log(phi)*k/(Pi^2 + (2*log(phi))^2). alpha is approximately 1.828404783. The zeros lie in the lower right and the upper left half-planes, and there is a zero at the origin.
a(n) = floor(alpha*n), n>=0, is a Beatty sequence with the complementary sequence b(n) = floor(beta*n), with beta = alpha/(alpha-1), approximately 2.207139336.
For the floor of the negative imaginary part see A214656.
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(alpha*n), n>=0, with alpha = x_0(1) given in the comment section.
EXAMPLE
The complementary Beatty sequences start with:
n: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
a(n): 0 1 3 5 7 9 10 12 14 16 18 20 21 23 25 27
b(n): (0) 2 4 6 8 11 13 15 17 19 22 24 26 28 30 33
MATHEMATICA
a[n_]:= Floor[2*n*Pi^2/(Pi^2 + 4*Log[GoldenRatio]^2)]; Table[a[n], {n, 0, 65}] (* Jean-François Alcover, Jul 03 2013 *)
PROG
(Magma) R:= RealField(100); [Floor(2*n*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*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 24 2012
STATUS
approved