A214315 Floor of the real part of the zeros of the complex Fibonacci function on the right half-plane.

0, 1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 42, 43, 45, 47, 49, 51, 53, 54, 56, 58, 60, 62, 63, 65, 67, 69, 71, 73, 74, 76, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 96, 98, 100, 102, 104, 106, 107, 109, 111, 113, 115, 117, 118

Offset: 0

Author

Wolfdieter Lang, Jul 24 2012

Keywords

• nonn

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.

Examples

			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
		

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. A052952 (Fibonacci related formula), A214656.

Programs

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

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 *)

SageMath

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

Formula

a(n) = floor(alpha*n), n>=0, with alpha = x_0(1) given in the comment section.