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

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

1,1

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

(diagonal/shortest side) = 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).

Sequence in context: A021348 A346833 A192938 * A020817 A245217 A133768

Adjacent sequences: A188616 A188617 A188618 * A188620 A188621 A188622

Keyword

nonn,cons,easy

Author

Clark Kimberling, Apr 06 2011

Status

approved