LINKS
G. C. Greubel, <a href="/A214657/b214657_1.txt">Table of n, a(n) for n = 0..10000</a>
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Revision History for A214657
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
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REFERENCES
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
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COMMENTS
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.
LINKS
G. C. Greubel, <a href="/A214657/b214657_1.txt">Table of n, a(n) for n = 0..10000</a>
FORMULA
a(n) = floor(n*tau*n), n>=0, with tau = |z_0(1)| given in the comment section= 2*Pi/sqrt(Pi^2 + (2*log(phi))^2).
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
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