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%I #22 Mar 01 2020 05:42:37
%S 6,5,7,4,0,5,4,8,2,9,7,6,5,3,2,5,9,2,3,8,0,9,6,8,5,4,1,5,2,9,3,9,7,1,
%T 2,6,5,4,1,4,9,5,9,4,6,4,8,7,8,3,9,3,7,0,7,8,2,0,9,2,8,0,8,5,8,8,5,3,
%U 9,5,0,6,1,3,8,1,7,7,3,5,0,7,0,1,7,1,5,1,6,5,4,4,0,5,2,2,7,8,0,5,2,8,1,2,6
%N Decimal expansion of the angle B in the doubly golden triangle ABC.
%C There is a unique (shape of) triangle ABC that is both side-golden and angle-golden. Its angles are B, C=t*B and A=pi-B-t*B, where t is the golden ratio. "Angle-golden" and "side-golden" refer to partitionings of ABC, each in a manner that matches the continued fraction [1,1,1,...] of t. (The partitionings are analogous to the partitioning of the golden rectangle into squares by the removal of exactly 1 square at each stage.)
%C For doubly silver and doubly e-ratio triangles, see A188543 and A188544.
%C For the side partitioning and angle partitioning (i,e, constructions) which match arbitrary continued fractions (of sidelength ratios and angle ratios), see the 2007 reference.
%D Clark Kimberling, "A new kind of golden triangle," in Applications of Fibonacci Numbers, Proc. Fourth International Conference on Fibonacci Numbers and Their Applications, Kluwer, 1991.
%H Iain Fox, <a href="/A152149/b152149.txt">Table of n, a(n) for n = 0..20000</a>
%H Jordi Dou, Clark Kimberling and Laurence Kuipers, <a href="http://www.jstor.org/stable/2321392">A Fibonacci sequence of nested triangles</a>, Problem S29, Amer. Math. Monthly 89 (1982) 696-697; proposed 87 (1980) 302.
%H 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 B is the number in [0,Pi] such that sin(B*t^2)=t*sin(B), where t=(1+5^(1/2))/2, the golden ratio.
%e The number B begins with 0.65740548 (equivalent to 37.666559... degrees).
%t r = (1 + 5^(1/2))/2; Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]
%o (PARI) t=(1+5^(1/2))/2; solve(b=.6, .7, sin(b*t^2)-t*sin(b)) \\ _Iain Fox_, Feb 11 2020
%Y Cf. A000045, A188543, A188544.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Nov 26 2008
%E Keyword:cons added and offset corrected by _R. J. Mathar_, Jun 18 2009
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