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.

A188619 Decimal expansion of (diagonal)/(shortest side) of 2nd electrum rectangle.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 06 2011

Keywords

Comments

The 2nd electrum rectangle is introduced here as a rectangle whose length L and width W satisfy L/W=1+sqrt(3). The name of this shape refers to the alloy of gold and silver known as electrum, in view of the existing names "golden rectangle" and "silver rectangle" and these continued fractions:
golden ratio: L/W=[1,1,1,1,1,1,1,1,1,1,1,...]
silver ratio: L/W=[2,2,2,2,2,2,2,2,2,2,2,...]
1st electrum ratio: L/W=[1,2,1,2,1,2,1,2,...]
2nd electrum ratio: L/W=[2,1,2,1,2,1,2,1,...].
Recall that removal of 1 square from a golden rectangle leaves a golden rectangle, and that removal of 2 squares from a silver rectangle leaves a silver rectangle. Removal of a square from a 1st electrum rectangle leaves a silver rectangle; removal of 2 squares from a 2nd electrum rectangle leaves a golden rectangle.

Examples

			(diagonal/shortest side) = 2.9093129111764094646 approximately.

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.13 Steinitz Constants, p. 241.

Links

Crossrefs

Cf. A188593 (golden), A121601 (silver), A188618 (1st electrum).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); Sqrt(5+2*Sqrt(3)); // G. C. Greubel, Nov 02 2018
  • Mathematica
    h = 1 + 3^(1/2); r = (1 + h^2)^(1/2)
FullSimplify[r]
N[r, 130] (* ratio of diagonal h to shortest side; h=[1,2,1,2,1,2,...] *)
RealDigits[N[r, 130]][[1]]
RealDigits[Sqrt[5 + 2*Sqrt[3]], 10, 100][[1]] (* G. C. Greubel, Nov 02 2018 *)
  • PARI
    default(realprecision, 100); sqrt(5+2*sqrt(3)) \\ G. C. Greubel, Nov 02 2018

Formula

Equals sqrt(5+2*sqrt(3)).

A188614 Decimal expansion of (circumradius)/(inradius) of side-silver right triangle.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 05 2011

Keywords

Comments

This ratio is invariant of the size of the side-silver right triangle ABC. The shape of ABC is given by sidelengths a,b,c, where a=r*b, and c=sqrt(a^2+b^2), where r=(silver ratio)=(1+sqrt(2)). This is the unique right triangle matching the continued fraction [2,2,2,...] of r; i.e., under the side-partitioning procedure described in the 2007 reference, there are exactly 2 removable subtriangles at each stage. (This is analogous to the removal of 2 squares at each stage of the partitioning of the silver rectangle as a nest of squares.)

Examples

			ratio=3.26197262739566856105805510300327465221450 approx.

Links

Crossrefs

Cf. A188543, A152149, A188614, A188618.

Programs

  • Maple
    a179260 := sqrt(2+sqrt(2)) ; a014176 := 1+sqrt(2) ; 1/(a014176/a179260-1) ; evalf(%) ; # R. J. Mathar, Apr 05 2011
  • Mathematica
    r= 1+2^(1/2); b=1; a=r*b; c=(a^2+b^2)^(1/2); area=(1/4)((a+b+c)(b+c-a)(c+a-b)(a+b-c))^(1/2); RealDigits[N[a*b*c*(a+b+c)/(8*area^2),130]][[1]]

Formula

(circumradius)/(inradius) = abc(a+b+c)/(8*area^2), where area=area(ABC).

A188639 Decimal expansion of length/width of a 2nd electrum rectangle.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 06 2011

Keywords

Comments

See A188619 for the definition of 2nd electrum rectangle. Briefly, it is a rectangle such that the removal of a silver rectangle from one end leaves a golden rectangle.

Links

Crossrefs

Cf. A188619, A188618, A188638.

Programs

  • Mathematica
    t=1+3^(1/2); r=(t+(t^2+4)^(1/2))/2
FullSimplify[r]
N[r, 130]

A188638 Decimal expansion of length/width of a meta-1st electrum rectangle.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Apr 06 2011

Keywords

Comments

See A188618 for the definition of 1st electrum rectangle. Briefly, it is a rectangle such that the removal of a golden rectangle from one end leaves a silver rectangle.

Links

Crossrefs

Cf. A188618, A188619, A188639.

Programs

  • Mathematica
    t=(1+3^(1/2))/2; r=(t+(t^2+4)^(1/2))/2
FullSimplify[r]
N[r, 130]
RealDigits[N[r, 130]][[1]]
Showing 1-4 of 4 results.

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