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A188727 Decimal expansion of (e+sqrt(16+e^2))/4.

%I A188727 #14 Sep 08 2022 08:45:56
%S A188727 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 A188727 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 A188727 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 A188727 Decimal expansion of (e+sqrt(16+e^2))/4.
%C A188727 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 A188727 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 A188727 G. C. Greubel, <a href="/A188727/b188727.txt">Table of n, a(n) for n = 1..10000</a>
%e A188727 1.88862628964821616707581942532177092442419527...
%p A188727 evalf((exp(1)+sqrt(16+exp(2)))/4,140); # _Muniru A Asiru_, Nov 01 2018
%t A188727 r = e/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
%t A188727 N[t, 130]
%t A188727 RealDigits[N[t, 130]][[1]] (* A188727 *)
%t A188727 ContinuedFraction[t, 120] (* A188728 *)
%o A188727 (PARI) default(realprecision, 100); (exp(1) + sqrt(16 + exp(2)))/4 \\ _G. C. Greubel_, Oct 31 2018
%o A188727 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (Exp(1) + Sqrt(16 + Exp(2)))/4; // _G. C. Greubel_, Oct 31 2018
%Y A188727 Cf. A188640, A188727, A001113.
%K A188727 nonn,cons
%O A188727 1,2
%A A188727 _Clark Kimberling_, Apr 10 2011