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A188618 Decimal expansion of (diagonal)/(shortest side) of 1st electrum rectangle.

%I A188618 #17 Jun 12 2025 08:33:24
%S A188618 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 A188618 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 A188618 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 A188618 Decimal expansion of (diagonal)/(shortest side) of 1st electrum rectangle.
%C A188618 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 A188618 golden ratio: L/W=[1,1,1,1,1,1,1,1,1,1,1,...]
%C A188618 silver ratio: L/W=[2,2,2,2,2,2,2,2,2,2,2,...]
%C A188618 1st electrum ratio: L/W=[1,2,1,2,1,2,1,2,...]
%C A188618 2nd electrum ratio: L/W=[2,1,2,1,2,1,2,1,...].
%C A188618 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 A188618 Clark Kimberling, <a href="http://www.jstor.org/stable/27963362">A Visual Euclidean Algorithm</a>, The Mathematics Teacher 76 (1983) 108-109.
%F A188618 Equals sqrt(2+(1/2)sqrt(3)).
%e A188618 1.6929339632083818...
%t A188618 h=(1+3^(1/2))/2; (* continued fraction: h=[1,2,1,2,...]. *)
%t A188618 r=(1+h^2)^(1/2);
%t A188618 RealDigits[N[r, 130]][[1]]
%Y A188618 Cf. A188593 (golden), A121601 (silver), A188619 (2nd electrum).
%K A188618 nonn,cons
%O A188618 1,2
%A A188618 _Clark Kimberling_, Apr 06 2011