This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A214657 #20 Mar 09 2024 08:15:15
%S A214657 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,
%T A214657 47,49,51,53,55,57,59,61,63,65,66,68,70,72,74,76,78,80,82,84,86,87,89,
%U A214657 91,93,95,97,99,101,103,105,107,108,110,112,114,116,118,120,122,124
%N A214657 Floor of the moduli of the zeros of the complex Fibonacci function.
%C A214657 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.
%C A214657 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.
%C A214657 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.
%C A214657 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.
%D A214657 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%H A214657 G. C. Greubel, <a href="/A214657/b214657.txt">Table of n, a(n) for n = 0..10000</a>
%F A214657 a(n) = floor(n*tau), n>=0, with tau = |z_0(1)| = 2*Pi/sqrt(Pi^2 + (2*log(phi))^2).
%e A214657 The complementary Beatty sequences a(n) and b(n) start:
%e A214657 n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
%e A214657 a(n): 0 1 3 5 7 9 11 13 15 17 19 21 22 24 26 28 30 32 34 ...
%e A214657 b(n): (0)2 4 6 8 10 12 14 16 18 20 23 25 27 29 31 33 35 37 ...
%t A214657 Table[Floor[2*n*Pi/Sqrt[Pi^2 + (2*Log[GoldenRatio])^2]], {n,0,100}] (* _G. C. Greubel_, Mar 09 2024 *)
%o A214657 (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
%o A214657 (SageMath) [floor(2*n*pi/sqrt(pi^2 +4*(log(golden_ratio))^2)) for n in range(101)] # _G. C. Greubel_, Mar 09 2024
%Y A214657 Cf. A214315, A214656.
%K A214657 nonn
%O A214657 0,3
%A A214657 _Wolfdieter Lang_, Jul 25 2012