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.

A180434 Decimal expansion of constant (2 - Pi/2).

Original entry on oeis.org

4, 2, 9, 2, 0, 3, 6, 7, 3, 2, 0, 5, 1, 0, 3, 3, 8, 0, 7, 6, 8, 6, 7, 8, 3, 0, 8, 3, 6, 0, 2, 4, 8, 5, 5, 7, 9, 0, 1, 4, 1, 5, 3, 0, 0, 3, 1, 2, 4, 4, 7, 0, 8, 9, 5, 1, 2, 5, 2, 7, 7, 0, 3, 8, 4, 6, 0, 9, 1, 7, 9, 6, 8, 5, 6, 8, 9, 5, 5, 0, 0, 6, 8, 5, 9, 8, 2, 5, 8, 7, 3, 2, 8, 9, 4, 1, 4, 6, 6
Offset: 0

Views

Author

Jonathan Vos Post, Sep 05 2010

Keywords

Comments

(2-Pi/2)*a^2 is the area of the loop of the right strophoid (also called the Newton strophoid) whose polar equation is r = a*cos(2*t)/cos(t) and whose Cartesian equation is x*(x^2+y^2) = a*(x^2-y^2) or y = +- x*sqrt((a-x)/(a+x)). See the curve with its loop at the Mathcurve link; the loop appears for -Pi/4 <= t <= Pi/4. - Bernard Schott, Jan 28 2020

Examples

			0.42920367320510338076867830836024855790141530...

Links

Crossrefs

Cf. A004601, A153799, A180433, A222362.

Programs

  • Mathematica
    RealDigits[2-Pi/2,10,120][[1]] (* Harvey P. Dale, Oct 12 2013 *)

Formula

Equals Integral_{t=0..Pi/4} ((cos(2*t))/cos(t))^2 dt. - Bernard Schott, Jan 28 2020
From Amiram Eldar, May 30 2021: (Start)
Equals Sum_{k>=1} 2^k/(binomial(2*k,k)*k*(2*k + 1)).
Equals Integral_{x=0..1} arcsin(x)*arccos(x) dx. (End)
Equals Integral_{x=0..1} sqrt(x)/(1+x) dx. - Andy Nicol, Mar 23 2024
Equals A153799/2. - Hugo Pfoertner, Mar 23 2024

Extensions

Corrected by Carl R. White, Sep 09 2010
More terms from N. J. A. Sloane, Sep 23 2010

A030315 Write n in base 2, complement each digit (d -> 1-d) and append the complemented digits to the sequence.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			To get the initial terms:
  0 -> 1
  1 -> 0
  2 = 10 -> 0, 1
  3 = 11 -> 0, 0
  4 = 100 -> 0, 0, 1
  5 = 101 -> 0, 1, 0,
  ...
and concatenating 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0,  ... gives the start of the sequence. - _N. J. A. Sloane_, Jan 30 2023

Links

Crossrefs

Cf. A030190, A180433. [Jonathan Vos Post, Sep 05 2010]

Programs

  • Mathematica
    Table[IntegerDigits[n,2]/.{1->0,0->1},{n,0,30}]//Flatten (* Harvey P. Dale, Jan 30 2023 *)

Formula

a(n) = 1 - A030190(n). - Amiram Eldar, May 22 2023

Extensions

a(0)=1 added by Jonathan Vos Post, Sep 05 2010
Edited by N. J. A. Sloane, Jan 30 2023

A180443 Decimal expansion of constant defined in A030315.

Original entry on oeis.org

5, 6, 8, 8, 7, 9, 9, 3, 7, 0, 6, 5, 9, 7, 2, 7, 1, 4, 2, 2, 1, 1, 0, 4, 8, 5, 8, 3, 7, 5, 3, 0, 2, 7, 1, 0, 7, 1, 7, 1, 1, 7, 6, 2, 8, 6, 1, 5, 8, 5, 0, 4, 5, 2, 7, 4, 1, 9, 6, 4, 3, 9, 2, 7, 2, 1, 3, 4, 6, 6, 2, 9, 7, 0, 4, 7, 4, 1, 7, 7, 0, 9, 7, 8, 9, 8, 0
Offset: 0

Views

Author

Jonathan Vos Post, Sep 05 2010

Keywords

Comments

This constant is 2-normal. - Charles R Greathouse IV, Feb 06 2015

Examples

			0.56887993706597271422110485837530271071711762861585045274...

Links

Crossrefs

Cf. A030190, A030315, A066716, A180433.

Programs

  • Mathematica
    First[RealDigits[1 - ChampernowneNumber[2]/2, 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

Formula

Equals 1 - A066716 / 2. - Amiram Eldar, May 22 2023

Extensions

a(30) onward corrected by Sean A. Irvine, Jun 09 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.