A188619 - OEIS

%I #25 Oct 20 2024 00:27:31

%S 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,

%T 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,

%U 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

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

%C 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:

%C golden ratio: L/W=[1,1,1,1,1,1,1,1,1,1,1,...]

%C silver ratio: L/W=[2,2,2,2,2,2,2,2,2,2,2,...]

%C 1st electrum ratio: L/W=[1,2,1,2,1,2,1,2,...]

%C 2nd electrum ratio: L/W=[2,1,2,1,2,1,2,1,...].

%C 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.

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

%H G. C. Greubel, <a href="/A188619/b188619.txt">Table of n, a(n) for n = 1..10000</a>

%H Clark Kimberling, <a href="http://www.jstor.org/stable/27963362">A Visual Euclidean Algorithm</a>, The Mathematics Teacher 76 (1983) 108-109.

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

%e (diagonal/shortest side) = 2.9093129111764094646 approximately.

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

%t FullSimplify[r]

%t N[r, 130] (* ratio of diagonal h to shortest side; h=[1,2,1,2,1,2,...] *)

%t RealDigits[N[r, 130]][[1]]

%t RealDigits[Sqrt[5 + 2*Sqrt[3]], 10, 100][[1]] (* _G. C. Greubel_, Nov 02 2018 *)

%o (PARI) default(realprecision, 100); sqrt(5+2*sqrt(3)) \\ _G. C. Greubel_, Nov 02 2018

%o (Magma) SetDefaultRealField(RealField(100)); Sqrt(5+2*Sqrt(3)); // _G. C. Greubel_, Nov 02 2018

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

%K nonn,cons,easy,changed

%O 1,1

%A _Clark Kimberling_, Apr 06 2011