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A214657
Floor of the moduli of the zeros of the complex Fibonacci function.
2
0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 108, 110, 112, 114, 116, 118, 120, 122, 124
OFFSET
0,3
COMMENTS
For the complex Fibonacci function F(z) and its zeros see the T. Koshy reference given in A214315. There the formula for the real and imaginary parts of the zeros is also given.
F: C -> C, z -> F(z) with F(z) := (exp(log(phi)*z) - exp(i*Pi*z)*exp(-log(phi)*z))/(2*phi-1), with phi = (1+sqrt(5))/2 and the imaginary unit i.
The zeros in the complex plane lie on a straight line with angle Phi = -arctan(2*log(phi)/Pi). They are equally spaced with distance tau defined below. Phi is approximately -0.2972713044, corresponding to about -17.03 degrees. The moduli are |z_0(k)| = tau*k, with tau = 2*Pi/sqrt(Pi^2 + (2*log(phi))^2), which is approximately 1.912278633.
a(n) = floor(tau*n) is a Beatty sequence with the complementary sequence b(n) = floor(sigma*n), with sigma:= tau/(tau-1), approximately 2.096156332.
REFERENCES
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
LINKS
FORMULA
a(n) = floor(n*tau), n>=0, with tau = |z_0(1)| = 2*Pi/sqrt(Pi^2 + (2*log(phi))^2).
EXAMPLE
The complementary Beatty sequences a(n) and b(n) start:
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
a(n): 0 1 3 5 7 9 11 13 15 17 19 21 22 24 26 28 30 32 34 ...
b(n): (0)2 4 6 8 10 12 14 16 18 20 23 25 27 29 31 33 35 37 ...
MATHEMATICA
Table[Floor[2*n*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*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*pi/sqrt(pi^2 +4*(log(golden_ratio))^2)) for n in range(101)] # G. C. Greubel, Mar 09 2024
CROSSREFS
Sequence in context: A251238 A059547 A064719 * A137803 A059533 A189397
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Jul 25 2012
STATUS
approved