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

%I A188619 #25 Oct 20 2024 00:27:31
%S A188619 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 A188619 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 A188619 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 A188619 Decimal expansion of (diagonal)/(shortest side) of 2nd electrum rectangle.
%C A188619 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 A188619 golden ratio: L/W=[1,1,1,1,1,1,1,1,1,1,1,...]
%C A188619 silver ratio: L/W=[2,2,2,2,2,2,2,2,2,2,2,...]
%C A188619 1st electrum ratio: L/W=[1,2,1,2,1,2,1,2,...]
%C A188619 2nd electrum ratio: L/W=[2,1,2,1,2,1,2,1,...].
%C A188619 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 A188619 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.13 Steinitz Constants, p. 241.
%H A188619 G. C. Greubel, <a href="/A188619/b188619.txt">Table of n, a(n) for n = 1..10000</a>
%H A188619 Clark Kimberling, <a href="http://www.jstor.org/stable/27963362">A Visual Euclidean Algorithm</a>, The Mathematics Teacher 76 (1983) 108-109.
%F A188619 Equals sqrt(5+2*sqrt(3)).
%e A188619 (diagonal/shortest side) = 2.9093129111764094646 approximately.
%t A188619 h = 1 + 3^(1/2); r = (1 + h^2)^(1/2)
%t A188619 FullSimplify[r]
%t A188619 N[r, 130] (* ratio of diagonal h to shortest side; h=[1,2,1,2,1,2,...] *)
%t A188619 RealDigits[N[r, 130]][[1]]
%t A188619 RealDigits[Sqrt[5 + 2*Sqrt[3]], 10, 100][[1]] (* _G. C. Greubel_, Nov 02 2018 *)
%o A188619 (PARI) default(realprecision, 100); sqrt(5+2*sqrt(3)) \\ _G. C. Greubel_, Nov 02 2018
%o A188619 (Magma) SetDefaultRealField(RealField(100)); Sqrt(5+2*Sqrt(3)); // _G. C. Greubel_, Nov 02 2018
%Y A188619 Cf. A188593 (golden), A121601 (silver), A188618 (1st electrum).
%K A188619 nonn,cons,easy
%O A188619 1,1
%A A188619 _Clark Kimberling_, Apr 06 2011