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%I #27 Apr 28 2021 04:27:20
%S 1,8,6,8,5,1,7,0,9,1,8,2,1,3,2,9,7,6,4,3,7,3,0,7,3,7,5,5,8,2,3,4,9,8,
%T 6,4,8,7,5,0,4,3,2,1,9,1,2,8,1,7,4,8,7,3,7,5,7,0,1,5,1,0,1,8,7,4,2,3,
%U 8,8,9,8,2,7,6,4,3,3,6,8,1,5,0,6,8,2,0,6,3,6,0,6,7,2,8,3,0,2,3,9,2,2,4,5,0,4,7,2,7,3,4,1,3,5,4,5,1,3,4,5,8,6,7,6,8,9,2,7,5,4
%N Decimal expansion of (2+sqrt(13))/3.
%C Decimal expansion of the length/width ratio of a (4/3)-extension rectangle.
%C See A188640 for definitions of shape and r-extension rectangle.
%C A (4/3)-extension rectangle matches the continued fraction [1,1,6,1,1,1,1,6,1,1,1,1,6,...] for the shape L/W= (2+sqrt(13))/3. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (4/3)-extension rectangle, 1 square is removed first, then 1 square, then 6 squares, then 1 square, then 1 square,..., so that the original rectangle is partitioned into an infinite collection of squares.
%H Daniel Starodubtsev, <a href="/A188655/b188655.txt">Table of n, a(n) for n = 1..10000</a>
%H Clark Kimberling, <a href="http://www.jstor.org/stable/27963362">A Visual Euclidean Algorithm</a>, The Mathematics Teacher 76 (1983) 108-109.
%e length/width = 1.868517091821329764373....
%t r = 4/3; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
%t RealDigits[(2 + Sqrt@ 13)/3, 10, 111][[1]] (* Or *)
%t RealDigits[Exp@ ArcSinh[2/3], 10, 111][[1]] (* _Robert G. Wilson v_, Aug 17 2011 *)
%Y Cf. A188640, A010122.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Apr 09 2011
%E a(130) corrected by _Georg Fischer_, Apr 01 2020
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