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.

A189961 Decimal expansion of (5+7*sqrt(5))/10.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 02 2011

Keywords

Comments

The constant at A189961 is the shape of a rectangle whose continued fraction partition consists of 3 golden rectangles. For a general discussion, see A188635.

Links

Crossrefs

Cf. A000032, A000045, A001622, A188635, A189962, A189963.

Programs

  • Magma
    (5+7*Sqrt(5))/10 // G. C. Greubel, Jan 13 2018
  • Mathematica
    r=(1+5^(1/2))/2;
FromContinuedFraction[{r,r,r}]
FullSimplify[%]
N[%,130]
RealDigits[%]  (* A189961 *)
ContinuedFraction[%%]
RealDigits[(5+7*Sqrt[5])/10,10,150][[1]] (* Harvey P. Dale, Mar 30 2024 *)
  • PARI
    (5+7*sqrt(5))/10 \\ G. C. Greubel, Jan 13 2018

Formula

Continued fraction (as explained at A188635): [r,r,r], where r = (1 + sqrt(5))/2. Ordinary continued fraction, as given by Mathematica program shown below:
[2,15,3,15,3,15,3,15,3,...]
From Amiram Eldar, Feb 06 2022: (Start)
Equals phi^4/sqrt(5) - 1, where phi is the golden ratio (A001622).
Equals lim_{k->oo} Fibonacci(k+4)/Lucas(k) - 1. (End)

A189962 Decimal expansion of 3*(1 + 3*sqrt(5))/11.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 02 2011

Keywords

Comments

The constant at A189961 is the shape of a rectangle whose continued fraction partition consists of 4 golden rectangles. For a general discussion, see A188635.

Examples

			2.10223743613619156978932391078013510172414229422761...

Links

Crossrefs

Cf. A188635, A189961, A189963.

Programs

  • Magma
    3*(1+3*Sqrt(5))/11 // G. C. Greubel, Jan 13 2018
  • Mathematica
    r=(1+5^(1/2))/2;
FromContinuedFraction[{r,r,r,r}]
FullSimplify[%]
N[%,130]
RealDigits[%]  (*A189962*)
ContinuedFraction[%%]
RealDigits[3 (1+3*Sqrt[5])/11,10,150][[1]] (* Harvey P. Dale, Sep 11 2023 *)
  • PARI
    3*(1+3*sqrt(5))/11 \\ G. C. Greubel, Jan 13 2018

Formula

Continued fraction (as explained at A188635): [r,r,r,r], where r = (1 + sqrt(5))/2. Ordinary continued fraction, as given by Mathematica program shown below:
[2,9,1,3,1,1,3,9,1,3,1,1,3,9,1,3,1,1,3,...]

Extensions

Definition corrected by G. C. Greubel, Jan 13 2018

A386753 Decimal expansion of the surface area of a bilunabirotunda with unit edges.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Aug 01 2025

Keywords

Comments

The bilunabirotunda is Johnson solid J_91.

Examples

			12.346011217493622278090940592566183131515107...

Links

Crossrefs

Cf. A189963 (volume - 1).
Cf. A010469, A010476.

Programs

  • Mathematica
    First[RealDigits[2 + Sqrt[12] + Sqrt[5*(5 + Sqrt[20])], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J91", "SurfaceArea"], 10, 100]]

Formula

Equals 2 + 2*sqrt(3) + sqrt(5*(5 + 2*sqrt(5))) = 2 + A010469 + sqrt(5*(5 + A010476)).
Equals the largest root of x^8 - 16*x^7 - 36*x^6 + 1328*x^5 - 2946*x^4 - 16624*x^3 + 34796*x^2 + 61392*x - 10679.
Showing 1-3 of 3 results.

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