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A188614 Decimal expansion of (circumradius)/(inradius) of side-silver right triangle.

%I A188614 #14 Dec 17 2016 15:07:07
%S A188614 3,2,6,1,9,7,2,6,2,7,3,9,5,6,6,8,5,6,1,0,5,8,0,5,5,1,0,3,0,0,3,2,7,4,
%T A188614 6,5,2,2,1,4,5,0,5,1,2,7,1,0,4,2,3,3,0,4,5,4,0,6,8,7,5,2,0,0,5,5,1,8,
%U A188614 0,2,4,9,3,4,6,4,3,1,1,7,5,6,2,8,0,0,6,7,4,0,4,0,2,8,3,3,0,7,6,4,9,3,0,9,3,9,8,9,7,7,9,5,4,0,8,0,6,3,0,8,6,6,6,3,1,9,1,2,1,5
%N A188614 Decimal expansion of (circumradius)/(inradius) of side-silver right triangle.
%C A188614 This ratio is invariant of the size of the side-silver right triangle ABC. The shape of ABC is given by sidelengths a,b,c, where a=r*b, and c=sqrt(a^2+b^2), where r=(silver ratio)=(1+sqrt(2)). This is the unique right triangle matching the continued fraction [2,2,2,...] of r; i.e., under the side-partitioning procedure described in the 2007 reference, there are exactly 2 removable subtriangles at each stage. (This is analogous to the removal of 2 squares at each stage of the partitioning of the silver rectangle as a nest of squares.)
%H A188614 Clark Kimberling, <a href="http://www.heldermann.de/JGG/JGG11/JGG112/jgg11014.htm">Two kinds of golden triangles, generalized to match continued fractions</a>, Journal for Geometry and Graphics, 11 (2007) 165-171.
%F A188614 (circumradius)/(inradius) = abc(a+b+c)/(8*area^2), where area=area(ABC).
%e A188614 ratio=3.26197262739566856105805510300327465221450 approx.
%p A188614 a179260 := sqrt(2+sqrt(2)) ; a014176 := 1+sqrt(2) ; 1/(a014176/a179260-1) ; evalf(%) ; # _R. J. Mathar_, Apr 05 2011
%t A188614 r= 1+2^(1/2); b=1; a=r*b; c=(a^2+b^2)^(1/2); area=(1/4)((a+b+c)(b+c-a)(c+a-b)(a+b-c))^(1/2); RealDigits[N[a*b*c*(a+b+c)/(8*area^2),130]][[1]]
%Y A188614 Cf. A188543, A152149, A188614, A188618.
%K A188614 nonn,cons
%O A188614 1,1
%A A188614 _Clark Kimberling_, Apr 05 2011