%I #21 Oct 02 2022 23:01:43

%S 1,9,0,6,2,3,6,0,4,1,4,7,3,3,0,6,1,4,2,5,9,4,2,8,2,5,6,5,4,1,5,5,5,2,

%T 6,8,6,6,3,0,2,2,2,0,2,0,9,8,3,5,6,4,6,1,7,3,5,2,7,3,3,7,6,8,0,9,7,0,

%U 9,0,8,8,4,4,9,2,2,1,1,4,1,7,5,2,8,9,1,5,0,6,9,9,1,0,3,7,0,9,9,6,5,5,4,3,2,5,2,3,9,9,5,5,8,1,9,8,4,7,5,9,5,9,2,6,2,9,9,7,0,2

%N Decimal expansion of e-sqrt(e^2-1).

%C Decimal expansion of the shape of a lesser 2e-contraction rectangle.

%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 Suppose that ABCD is given and that the shape r=[ABCD] exceeds 2. The "r-contraction rectangles" of ABCD are here introduced as the rectangles AEFD and EBCF for which [AEFD]=[EBCF] and |AE|<>|EB|. That is, ABCD has the prescribed shape r, and AEFD and EBCF are mutually similar without being congruent. It is easy to prove that [AEFD]=(1/2)(r-sqrt(-4+r^2)) or [AEFD]=(1/2)(r+sqrt(-4+r^2)); in the former case, we call AEFD the "lesser r-contraction rectangle", and the latter, the "greater r-contraction rectangle".

%C Both r-contraction rectangles match the continued fraction of [AEFD] in the following way. Write the continued fraction as [a(1),a(2),a(3),...]. Then, in the manner in which the continued fraction [1,1,1,...] matches the step-by-step removal of single squares from a golden triangle (as well as the manner in which the continued fraction [2,2,2,...] matches the step-by-step removal of 2 squares at a time from a silver triangle, etc.), remove a(1) squares at step 1, then remove a(2) squares at step 2, and so on, obtaining in the limit a partition of AEFD as an infinite set of squares.

%C For (related) r-extension rectangles, see A188640.

%H G. C. Greubel, <a href="/A188738/b188738.txt">Table of n, a(n) for n = 0..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.

%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.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 0.190623604147330614259428256541555268663022202.. = 1/A188739 with continued fraction 0, 5, 4, 15, 6, 1, 13, 2, 1, 1, 21, 3, 2, 16, 1, 4, 1, 1, 157,...

%p evalf(exp(1)-sqrt(exp(2)-1),140); # _Muniru A Asiru_, Nov 01 2018

%t r = 2 E; t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]

%t N[t, 130]

%t RealDigits[N[t, 130]][[1]]

%t ContinuedFraction[t, 120]

%o (PARI) default(realprecision, 100); exp(1) - sqrt(exp(2)-1) \\ _G. C. Greubel_, Nov 01 2018

%o (Magma) SetDefaultRealField(RealField(100)); Exp(1) - Sqrt(Exp(2)-1); // _G. C. Greubel_, Nov 01 2018

%Y Cf. A001113, A188739 (inverse), A188627 (continued fraction), A188640.

%K nonn,cons

%O 0,2

%A _Clark Kimberling_, Apr 11 2011