# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a188616 Showing 1-1 of 1 %I A188616 #13 May 04 2016 12:11:10 %S A188616 5,9,1,0,6,7,7,9,9,7,0,5,1,6,4,8,7,9,7,6,3,2,3,2,3,7,4,1,9,6,6,2,1,7, %T A188616 2,3,6,0,5,4,9,7,8,5,3,1,4,6,5,8,3,4,0,5,9,0,5,0,3,1,3,2,9,0,3,6,5,9, %U A188616 4,6,1,4,7,0,8,5,5,8,0,0,1,2,5,4,3,4,3,8,2,2,5,8,1,9,1,6,4,3,1,2,6,6,0,3,6,8,6,5,6,4,1,3,8,1,5,7,7,8,3,7 %N A188616 Decimal expansion of angle B of unique side-golden and angle-silver triangle. %C A188616 Let r=(golden ratio)=(1+sqrt(5))/2 and u=(silver ratio)=1+sqrt(2). A triangle ABC with sidelengths a,b,c is side-golden if a/b=r and angle-silver if C/B=u. There is a unique triangle that has both properties. The quickest way to understand the geometric reasons for the names is by analogy to the golden and silver rectangles. For the former, exactly 1 square is available at each stage of the partitioning of the rectangle into a nest of squares, and for the former, exactly 2 squares are available. Analogously, for ABC, exactly one 1 triangle of a certain kind is available at each stage of a side-partitioning procedure, and exactly 2 triangles of another kind are available for angle-partitioning. For details, see the 2007 reference. %H A188616 Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171. %e A188616 B=0.59106779970516487976323237419662 approximately %t A188616 Remove["Global`*"]; r=1+2^(1/2); u=(1+5^(1/2))/2; RealDigits[FindRoot[Sin[r*t+t]==u*Sin[t],{t,1}, WorkingPrecision->120][[1,2]]][[1]] %Y A188616 Cf. A152149, A188543. %K A188616 nonn,cons %O A188616 0,1 %A A188616 _Clark Kimberling_, Apr 05 2011 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE