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A188720 Decimal expansion of (e+sqrt(4+e^2))/2. 4

%I

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

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

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

%N Decimal expansion of (e+sqrt(4+e^2))/2.

%C Decimal expansion of shape of an e-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles having shape r.

%C An e-extension rectangle matches the continued fraction A188721 of the shape L/W = (1/2) *(e+sqrt(4+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 an e-extension rectangle, 3 squares are removed first, then 21 squares, then 2 squares, then 40 squares, then 1 square,..., so that the original rectangle is partitioned into an infinite collection of squares.

%C (e+sqrt(4+e^2))/2 = [e,e,e,... ] (continued fraction). - _Clark Kimberling_, Sep 23 2013

%H G. C. Greubel, <a href="/A188720/b188720.txt">Table of n, a(n) for n = 1..10000</a>

%e 3.046524695333472471811401766587155243274607058879794774577422496312...

%p evalf((exp(1)+sqrt(4+exp(2)))/2,140); # _Muniru A Asiru_, Nov 01 2018

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

%t N[t, 130]

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

%t RealDigits[(E+Sqrt[4+E^2])/2,10,150][[1]] (* _Harvey P. Dale_, Jan 07 2015 *)

%o (PARI) default(realprecision, 100); (exp(1) + sqrt(4 + exp(2)))/2 \\ _G. C. Greubel_, Oct 31 2018

%o (MAGMA) SetDefaultRealField(RealField(100)); (Exp(1) +Sqrt(4+Exp(2)))/2; // _G. C. Greubel_, Oct 31 2018

%Y Cf. A188640, A188721, A188726, A188722, A188725.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Apr 09 2011

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Last modified April 23 13:56 EDT 2021. Contains 343204 sequences. (Running on oeis4.)