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%I #13 Sep 09 2021 20:02:46
%S 3,6,8,9,3,1,2,7,4,9,4,7,8,0,5,8,4,2,6,5,1,9,1,1,2,7,2,6,8,8,6,4,0,8,
%T 5,7,1,8,6,8,3,4,4,2,8,8,3,5,2,6,1,9,0,6,5,9,8,5,4,6,2,1,2,1,1,1,1,7,
%U 6,5,9,8,7,5,6,8,4,9,0,6,0,6,7,0,1,2,1,0,6,0,4,8,5,9,8,4,2,8,4,2,8,0,9,9,1,1,8,2,1,8,8,5,9,6,9,8,4,4,2,2,9,4,8,7,3,4,6,8
%N Decimal expansion of the angle B in the doubly e-ratio triangle ABC.
%C There is a unique (shape of) triangle ABC that is both side-e-ratio and angle-e-ratio. Its angles are B, t*B and pi-B-t*B, where t=e. "Side-e-ratio" and "angle-e-ratio" refer to partitionings of ABC, each in a manner that matches the continued fraction [2,1,2,1,1,4,1,1,6,...] of t. For doubly golden and doubly silver triangles, see A152149 and A188543. 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.
%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*e^2)=e*sin(B).
%e B=0.36893127494780584265191127268864 approximately.
%e B=21.1382 degrees approximately.
%t r = E; Clear[t]; RealDigits[FindRoot[Sin[r*t + t] == r*Sin[t], {t, 1}, WorkingPrecision -> 120][[1, 2]]][[1]]
%Y Cf. A152149, A188543.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Apr 03 2011
%E a(127) corrected by _Sean A. Irvine_, Sep 08 2021
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