%I
%S 5,6,1,4,6,6,8,5,6,0,0,4,9,0,5,3,4,3,9,2,5,4,7,8,2,8,3,3,1,8,6,3,3,7,
%T 3,6,0,2,3,9,8,2,0,5,6,4,1,7,1,1,3,3,9,9,6,3,2,0,4,7,8,1,4,6,4,7,2,9,
%U 3,9,2,5,6,4,2,3,9,0,0,2,6,5,0,9,8,0,4,8,4,2,8,5,5,3,4,1,5,3,5,1,3,3,7,3,7,6,0,7,6,8,8,0,8,7,8,3,3,6,0,7,7,0,0,4,0,1,8,2,9,9
%N Decimal expansion of e + sqrt(1+e^2).
%C The shape of a rectangle WXYZ, denoted by [WXYZ], is defined by length/width: [WXYZ]=max{WX/YZ, YZ/WX}. Consider the following configuration of rectangles AEFD, EBCF, ABCD, where AEFD is not a square:
%C D................F....C
%C .......................
%C .......................
%C .......................
%C A................E....B
%C Let r=[AEFD]. The rextension rectangle of AEFD is here introduced as the rectangle ABCD for which [AEFD]=[EBCF] and AE<>EB. That is, AEFD has the prescribed shape r, and AEFD and EBCF are similar without being congruent.
%C We extend the definition of rextension rectangle to the case that 0<r<1; in this case, [AEFD]=1/r and ABCD is defined as above.
%C Then for all r>0, it is easy to prove that [ABCD] = (r+sqrt(4+r^2))/2.
%C This here is the length/width ratio for the (2e)extension rectangle.
%C A (2e)extension rectangle matches the continued fraction A188796 for the shape L/W=(e+sqrt(1+e^2). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (2e)extension rectangle, 5 squares are removed first, then 1 square, then 1 square, then 1 square, then 1 square, then 2 squares..., so that the original rectangle is partitioned into an infinite collection of squares.
%C Shapes of other rextension rectangles, partitionable into a collection of squares in accord with the continued fraction of the shape [ABCD], are approximated at A188635A188639, A188655A188659, and A188720A188737.
%C For (related) rcontraction rectangles, see A188738 and A188739.
%H G. C. Greubel, <a href="/A188640/b188640.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) 108109.
%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) 165171.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e Length/width = 5.61466856004905343925478283318633736023982...
%p evalf(exp(1)+sqrt(1+exp(2)),140); # _Muniru A Asiru_, Nov 01 2018
%t r=2E; t=(r+(4+r^2)^(1/2))/2; FullSimplify[t]
%t N[t, 130]
%t RealDigits[N[t, 130]][[1]]
%o (PARI) exp(1)+sqrt(1+exp(2)) \\ _Charles R Greathouse IV_, Jun 16 2011
%o (MAGMA) SetDefaultRealField(RealField(100)); Exp(1) + Sqrt(1 + Exp(2)); // _G. C. Greubel_, Oct 31 2018
%Y Cf. A188796, A188738, A188739, A188720.
%K nonn,cons,easy
%O 1,1
%A _Clark Kimberling_, Apr 10 2011
