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%I #11 Sep 01 2014 08:29:05
%S 1,6,9,2,9,3,3,9,6,3,2,0,8,3,8,1,8,0,7,3,0,6,2,9,6,0,3,2,1,5,5,5,9,6,
%T 2,2,3,0,5,9,1,0,3,1,2,5,6,1,4,3,7,6,4,6,7,0,6,9,4,2,7,3,9,1,6,6,2,0,
%U 3,9,5,7,7,3,0,2,1,5,6,7,4,5,5,9,2,7,8,3,1,5,3,7,9,6,5,8,6,5,7,4,1,2,0,0,2,0,0,2,8,4,4,6,4,5,9,5,8,7,0,2,9,6,6,9,5,0,3,4,7,1
%N Decimal expansion of (diagonal)/(shortest side) of 1st electrum rectangle.
%C The 1st electrum rectangle is introduced here as a rectangle whose length L and width W satisfy L/W=(1+sqrt(3))/2. 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.
%H Clark Kimberling, <a href="http://www.jstor.org/stable/27963362">A Visual Euclidean Algorithm</a>, The Mathematics Teacher 76 (1983) 108-109.
%F (diagonal)/(shortest side) = sqrt(2+(1/2)sqrt(3)).
%e (diagonal)/(shortest side) = 1.6929339632083818 approximately.
%t h=(1+3^(1/2))/2; (* continued fraction: h=[1,2,1,2,...].
%t r=(1+h^2)^(1/2)
%t FullSimplify[r]
%t N[r,130]
%t RealDigits[N[r, 130]][[1]]
%Y Cf. A188593 (golden), A121601 (silver), A188619 (2nd electrum).
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Apr 06 2011
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