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 1-4 of 4 results.

A188720 Decimal expansion of (e+sqrt(4+e^2))/2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 09 2011

Keywords

Comments

Decimal expansion of shape of an e-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles having shape r.
An e-extension rectangle matches the continued fraction A188721 of the shape L/W = (1/2) *(e+sqrt(4+e^2)). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for an e-extension rectangle, 3 squares are removed first, then 21 squares, then 2 squares, then 40 squares, then 1 square,..., so that the original rectangle is partitioned into an infinite collection of squares.
(e+sqrt(4+e^2))/2 = [e,e,e,... ] (continued fraction). - Clark Kimberling, Sep 23 2013

Examples

			3.046524695333472471811401766587155243274607058879794774577422496312...

Links

Crossrefs

Cf. A188640, A188721, A188726, A188722, A188725.

Programs

  • Magma
    SetDefaultRealField(RealField(100)); (Exp(1) +Sqrt(4+Exp(2)))/2; // G. C. Greubel, Oct 31 2018
  • Maple
    evalf((exp(1)+sqrt(4+exp(2)))/2,140); # Muniru A Asiru, Nov 01 2018
  • Mathematica
    r=E; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(E+Sqrt[4+E^2])/2,10,150][[1]] (* Harvey P. Dale, Jan 07 2015 *)
  • PARI
    default(realprecision, 100); (exp(1) + sqrt(4 + exp(2)))/2 \\ G. C. Greubel, Oct 31 2018

A086773 Decimal expansion of the continued fraction 1/(Pi+1/(Pi+1/(Pi+1/(Pi+...)))).

Original entry on oeis.org

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

Views

Author

Cino Hilliard, Aug 03 2003

Keywords

Comments

Repeat s = s + Pi; s=1/s. The initial value of s is irrelevant.
Solves log(x+Pi) = -log(x). This equation represents the first (by absolute value) self-intersection of the spiral defined by the polar equation r=log(theta), and this constant is the smaller value of theta in the self-intersection. - Jeremy Tan, Sep 03 2016

Examples

			1
------
Pi + 1
     ------
     Pi + 1
          --------
           Pi + ...

Links

Crossrefs

Cf. A188722.

Programs

  • Mathematica
    RealDigits[N[(Sqrt[Pi^2 + 4] - Pi)/2, 120]] // First (* Michael De Vlieger, Mar 31 2015 *)
  • PARI
    default(realprecision, 2000); f(n) = s=0; for(x=1,n,s=s+Pi; s=1/s); print(s)

Formula

Equals (sqrt(Pi^2+4)-Pi)/2 = 0.2912995... . - R. J. Mathar, Sep 15 2012

A188723 Continued fraction of (Pi + sqrt(4 + Pi^2))/2.

Original entry on oeis.org

3, 2, 3, 4, 2, 3, 1, 1, 105, 1, 2, 1, 13, 5, 16, 1, 44, 1, 1, 4, 2, 1, 2, 3, 100, 4, 1, 1, 18, 4, 2, 2, 2, 8, 2, 5, 2, 2, 3, 7, 184, 1, 8, 6, 2, 6, 2, 1, 5, 1, 38, 1, 2, 1, 1, 1, 4, 2, 6, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 2, 3, 8, 1, 1, 2, 1, 3, 1, 2, 1, 10, 1, 6, 1, 3, 1, 1, 1, 1, 2, 2, 1, 7, 1, 11, 1, 6, 1, 2, 13, 35, 1, 5, 2, 2, 1, 1, 2, 8, 2, 6, 2, 3, 1, 1, 2, 5
Offset: 0

Views

Author

Clark Kimberling, Apr 09 2011

Keywords

Comments

Continued fraction of the constant in A188722.

Crossrefs

Cf. A000796, A188640, A188722 (decimal expansion).

Programs

  • Maple
    Digits := 100 ;
(Pi+sqrt(4+Pi^2))/2 ;
evalf(%) ;
numtheory[cfrac](%,40,'quotients') ; # R. J. Mathar, Apr 11 2011
  • Mathematica
    r = Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

Extensions

Offset changed by Andrew Howroyd, Jul 07 2024

A188724 Decimal expansion of shape of a (Pi/2)-extension rectangle; shape = (1/4)*(Pi + sqrt(16 + Pi^2)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 09 2011

Keywords

Comments

See A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r.
A (Pi/2)-extension rectangle matches the continued fraction [2,17,1,1,3,1,3,2,2,1637,1,210,7,...] of the shape L/W = (1/4)*(Pi + sqrt(16 + Pi^2)). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (Pi/2)-extension rectangle, 2 squares are removed first, then 17 squares, then 1 square, then 1 square, then 3 squares, ..., so that the original rectangle is partitioned into an infinite collection of squares.

Examples

			2.0569524387109659093967879243788072585880991...

Links

Crossrefs

Cf. A188640, A188722.

Programs

  • Mathematica
    r = Pi/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

Extensions

a(130) corrected by Georg Fischer, Jul 16 2021
Showing 1-4 of 4 results.

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