A188618 -id:A188618

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A188619

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

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

REFERENCES

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

FORMULA

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

EXAMPLE

(diagonal/shortest side) = 2.9093129111764094646 approximately.

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 *)

PROG

(PARI) default(realprecision, 100); sqrt(5+2*sqrt(3)) \\ G. C. Greubel, Nov 02 2018

(Magma) SetDefaultRealField(RealField(100)); Sqrt(5+2*Sqrt(3)); // G. C. Greubel, Nov 02 2018

CROSSREFS

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

KEYWORD

nonn,cons,easy,changed

AUTHOR

Clark Kimberling, Apr 06 2011

STATUS

approved

A188614

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.)

LINKS

Table of n, a(n) for n=1..130.

Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

FORMULA

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

EXAMPLE

ratio=3.26197262739566856105805510300327465221450 approx.

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]]

CROSSREFS

Cf. A188543, A152149, A188614, A188618.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Apr 05 2011

STATUS

approved

A188639

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

Table of n, a(n) for n=1..130.

Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

MATHEMATICA

t=1+3^(1/2); r=(t+(t^2+4)^(1/2))/2

FullSimplify[r]

N[r, 130]

CROSSREFS

Cf. A188619, A188618, A188638.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Apr 06 2011

STATUS

approved

A188638

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

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

Table of n, a(n) for n=1..130.

Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

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]]

CROSSREFS

Cf. A188618, A188619, A188639.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Apr 06 2011

STATUS

approved

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