cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A214656 Floor of the imaginary part of the zeros of the complex Fibonacci function on the left half-plane.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39
Offset: 0

Views

Author

Wolfdieter Lang, Jul 24 2012

Keywords

Comments

See the comment on the Fibonacci Function F(z) and its zeros in A214315, where also the T. Koshy reference is given.
The imaginary part of the zeros, corresponding to the real part x_0(k) given in A214315, is y_0(k) = -b*k, with b = 4*Pi*log(phi)/(Pi^2 + (2*log(phi))^2) and phi = (1+sqrt(5))/2. Note that b is approximately 0.5601299084.

References

  • Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Links

Crossrefs

Cf. A052952 (Fibonacci related formula), A214315 (real part).

Programs

  • Magma
    R:= RealField(100); [Floor(4*n*Pi(R)*Log((1+Sqrt(5))/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[4*n*Pi*Log[GoldenRatio]/(Pi^2 + 4*Log[GoldenRatio]^2)];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Jul 03 2013 *)
  • PARI
    A214656(n,phi=(sqrt(5)+1)/2)=n*4*Pi*log(phi)\(Pi^2+(2*log(phi))^2)  \\ M. F. Hasler, Jul 24 2012
  • SageMath
    [floor(4*n*pi*log(golden_ratio)/(pi^2 +4*(log(golden_ratio))^2)) for n in range(101)] # G. C. Greubel, Mar 09 2024

Formula

a(n) = floor(b*n), n>=0, with b = -y_0(1) = 4*Pi*log(phi)/(Pi^2 + (2*log(phi))^2).

A214671 Floor of the real parts of the zeros of the complex Lucas function on the right half-plane.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 11, 13, 15, 17, 19, 21, 22, 24, 26, 28, 30, 31, 33, 35, 37, 39, 41, 42, 44, 46, 48, 50, 52, 53, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 74, 75, 77, 79, 81, 83, 85, 86, 88, 90, 92, 94, 95, 97, 99, 101, 103, 105, 106, 108, 110, 112, 114, 116, 117, 119
Offset: 0

Views

Author

Wolfdieter Lang, Jul 25 2012

Keywords

Comments

For the complex Lucas function and its zeros see the Koshy reference.
This function is L: C -> C, z -> L(z), with L(z) = exp(log(phi)*z) + exp(i*Pi*z)*exp(-log(phi)*z), with the complex unit i and the golden section phi = (1+sqrt(5))/2. The complex zeros are z_0(k) = x_0(k) + y_0(k)*i, with x_0(k) = (k+1/2)*alpha and y_0(k) = (k+1/2)*b, where alpha and b appear in the Fibonacci case as alpha = 2*(Pi^2)/(Pi^2 + (2*log(phi))^2) and b = 4*Pi*log(phi)/(Pi^2 + (2*log(phi))^2). The x_0 and y_0 values are shifted compared to the zeros of the Fibonacci case by alpha/2 = 0.9142023918..., respectively b/2 = 0.2800649542....

References

  • Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Links

Crossrefs

Cf. A214315 (Fibonacci case), A214672 (floor of imaginary parts), A214673 (moduli).

Programs

  • Magma
    R:= RealField(100); [Floor((2*n+1)*Pi(R)^2/(Pi(R)^2 + (2*Log((1+Sqrt(5))/2))^2)) : n in [0..100]]; // G. C. Greubel, Mar 09 2024
  • Mathematica
    Table[Floor[(2*n+1)*(Pi^2)/(Pi^2+(2*Log[GoldenRatio])^2)], {n,0,100}] (* G. C. Greubel, Mar 09 2024 *)
  • SageMath
    [floor((2*n+1)*pi^2/(pi^2 +4*(log(golden_ratio))^2)) for n in range(101)] # G. C. Greubel, Mar 09 2024

Formula

a(n) = floor((n+1/2)*alpha), with alpha/2 = x_0(0) = Pi^2/(Pi^2 + (2*log(phi))^2).

A214657 Floor of the moduli of the zeros of the complex Fibonacci function.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Jul 25 2012

Keywords

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.

Examples

			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 ...

References

  • Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Links

Crossrefs

Cf. A214315, A214656.

Programs

  • 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
  • Mathematica
    Table[Floor[2*n*Pi/Sqrt[Pi^2 + (2*Log[GoldenRatio])^2]], {n,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

Formula

a(n) = floor(n*tau), n>=0, with tau = |z_0(1)| = 2*Pi/sqrt(Pi^2 + (2*log(phi))^2).
Showing 1-3 of 3 results.

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