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 1-5 of 5 results.

A094886 Decimal expansion of phi*Pi, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jun 15 2004

Keywords

Comments

The area of a golden ellipse with a semi-major axis phi and a minor semi-axis 1. - Amiram Eldar, Jul 05 2020
phi*Pi = area of the region having boundaries y = 0, x = Pi/2, and y = (tan x)^(4/5). - Clark Kimberling, Oct 25 2020

Examples

			5.0832036923152598158...

Links

Crossrefs

Cf. A000796, A001622, A094881, A094885.

Programs

  • Mathematica
    First@ RealDigits[N[GoldenRatio Pi, 120]] (* Michael De Vlieger, May 24 2016 *)
  • PARI
    { default(realprecision, 20080); phi=(1+sqrt(5))/2; x=phi*Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b094886.txt", n, " ", d)); } \\ Harry J. Smith, Apr 27 2009
  • PARI
    Pi*(1+sqrt(5))/2 \\ Michel Marcus, May 25 2016

Formula

Equals the nested radical sqrt(Pi^2+sqrt(Pi^4+sqrt(Pi^8+...))). For a proof, see A094885. - Stanislav Sykora, May 24 2016
Equals Integral_{x=0..Pi/2} tan(x)^(4/5) dx. - Clark Kimberling, Nov 18 2020

A277112 a(n) = floor(n*(1+sqrt(5))/Pi).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70
Offset: 0

Views

Author

Paulo Romero Zanconato Pinto, Sep 30 2016

Keywords

Examples

			For n = 1000 we have that floor(1000*(1+sqrt(5))/Pi) = floor(1000*1.0300724296...) = floor(1030.0724296...) so a(1000) = 1030.

Links

Crossrefs

Complement of A277113.
Cf. A000796, A001622, A094881, A094882.

Programs

  • Maple
    A277112:=n->floor(n*(1+sqrt(5))/Pi): seq(A277112(n), n=0..100); # Wesley Ivan Hurt, Oct 31 2016
  • Mathematica
    f[n_] := Floor[n*(1+Sqrt[5])/Pi]; Array[f, 100, 0]

Formula

a(n) = floor(n*(1+sqrt(5))/Pi).

A277113 a(n) = floor(n/(1-Pi/(sqrt(5)+1))).

Original entry on oeis.org

34, 68, 102, 137, 171, 205, 239, 274, 308, 342, 376, 411, 445, 479, 513, 548, 582, 616, 650, 685, 719, 753, 787, 822, 856, 890, 924, 959, 993, 1027, 1061, 1096, 1130, 1164, 1198, 1233, 1267, 1301, 1335, 1370, 1404, 1438, 1472, 1507, 1541, 1575, 1609, 1644, 1678
Offset: 1

Views

Author

Paulo Romero Zanconato Pinto, Sep 30 2016

Keywords

Comments

The goal is to generate a ratio near 1 from two well-known constants.

Examples

			For n = 10 we have that floor(10/(1-Pi/(sqrt(5)+1))) = floor(10/0.02919448...) = floor(342.5304983...) so a(10) = 342.

Links

Crossrefs

Complement of A277112.
Cf. A000796, A001622, A094881, A094882.

Programs

  • Maple
    A277113:=n->floor(n/(1-Pi/(sqrt(5)+1))): seq(A277113(n), n=1..100);
  • Mathematica
    f[n_] := Floor[n/(1-Pi/(Sqrt[5]+1))]; Array[f, 100, 1]
  • PARI
    a(n) = n\(1-Pi/(sqrt(5)+1)) \\ Michel Marcus, Oct 29 2016

Formula

a(n) = floor(n/(1-Pi/(sqrt(5)+1))).

Extensions

More terms from Michel Marcus, Oct 29 2016

A309282 Decimal expansion of the circumference of a golden ellipse with a unit semi-major axis.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 05 2020

Keywords

Comments

A golden ellipse is an ellipse inscribed in a golden rectangle. The concept of a golden ellipse was introduced by H. E. Huntley in 1970.
The aesthetic preferences of rectangles and ellipses with relation to the golden ratio were studied by Gustav Fechner in 1876. His results for ellipses were published by Witmer in 1893.
A golden ellipse with a semi-major axis 1 has a minor semi-axis 1/phi and an eccentricity 1/sqrt(phi), where phi is the golden ratio (A001622).

Examples

			5.154273178025879962492835539113341955287972235708661...

Links

Crossrefs

Cf. A001622 (phi), A094881 (area of the golden ellipse), A197762 (eccentricity of the golden ellipse).
Similar sequences: A138500, A274014.

Programs

  • Mathematica
    RealDigits[4 * EllipticE[1/GoldenRatio], 10, 100][[1]]

Formula

Equals 4*E(1/phi), where E(x) is the complete elliptic integral of the second kind.

A341332 Decimal expansion of Pi/(2*phi).

Original entry on oeis.org

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

Views

Author

Bernard Schott, Feb 09 2021

Keywords

Comments

This is the middle angle (in radians) of the unique right triangle whose angles are in geometric progression; common ratio is phi and the angles are (Pi/(2*phi^2), Pi/(2*phi), Pi/2) in radians, corresponding to approximately (34.377, 55.623, 90) in degrees.

Examples

			0.970805519362733288673432814981347978817334946923024149753694108...

Links

Crossrefs

Cf. A000796 (Pi), A001622 (phi), A019669 (Pi/2), A180014 (Pi/(2*phi^2)).

Programs

  • Maple
    evalf(Pi/(1+sqrt(5)),150);
  • Mathematica
    RealDigits[Pi/(2*GoldenRatio), 10, 100][[1]] (* Amiram Eldar, Feb 09 2021 *)
  • PARI
    Pi/(1+sqrt(5)) \\ Michel Marcus, Feb 09 2021

Formula

Equals A019669/A001622 = A094881/2 = Pi/(1+sqrt(5)) = (Pi/4) * (sqrt(5)-1).
Showing 1-5 of 5 results.

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

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