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Decimal expansion of (e+sqrt(16+e^2))/4.
2

%I #14 Sep 08 2022 08:45:56

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

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

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

%N Decimal expansion of (e+sqrt(16+e^2))/4.

%C Decimal expansion of shape of an (e/2)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r.

%C An (e/2)-extension rectangle matches the continued fraction A188728 of the shape L/W = (r+sqrt(4+r^2))/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/2)-extension rectangle, 1 square is removed first, then 1 square, then 7 squares, then 1 square, then 46 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

%H G. C. Greubel, <a href="/A188727/b188727.txt">Table of n, a(n) for n = 1..10000</a>

%e 1.88862628964821616707581942532177092442419527...

%p evalf((exp(1)+sqrt(16+exp(2)))/4,140); # _Muniru A Asiru_, Nov 01 2018

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

%t N[t, 130]

%t RealDigits[N[t, 130]][[1]] (* A188727 *)

%t ContinuedFraction[t, 120] (* A188728 *)

%o (PARI) default(realprecision, 100); (exp(1) + sqrt(16 + exp(2)))/4 \\ _G. C. Greubel_, Oct 31 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (Exp(1) + Sqrt(16 + Exp(2)))/4; // _G. C. Greubel_, Oct 31 2018

%Y Cf. A188640, A188727, A001113.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Apr 10 2011