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A201752 Decimal expansion of the greatest x satisfying -x^2+2 = e^x.

%I A201752 #11 May 20 2025 19:25:22
%S A201752 5,3,7,2,7,4,4,4,9,1,7,3,8,5,6,6,0,4,2,5,6,7,6,2,9,8,9,7,7,9,6,7,5,3,
%T A201752 8,1,4,2,7,5,2,4,0,1,4,0,0,0,1,0,4,1,0,7,7,7,6,6,8,1,9,9,6,5,4,7,3,3,
%U A201752 7,7,3,2,7,5,1,1,3,7,7,2,9,9,1,5,2,4,7,5,6,9,1,5,5,4,3,6,8,4,2
%N A201752 Decimal expansion of the greatest x satisfying -x^2+2 = e^x.
%C A201752 See A201741 for a guide to related sequences. The Mathematica program includes a graph.
%H A201752 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201752 least:  -1.3159737777962901878871773873012710...
%e A201752 greatest:  0.53727444917385660425676298977967...
%t A201752 a = -1; b = 0; c = 2;
%t A201752 f[x_] := a*x^2 + b*x + c; g[x_] := E^x
%t A201752 Plot[{f[x], g[x]}, {x, -2, 1}, {AxesOrigin -> {0, 0}}]
%t A201752 r = x /. FindRoot[f[x] == g[x], {x, -1.4, -1.3}, WorkingPrecision -> 110]
%t A201752 RealDigits[r]    (* A201751 *)
%t A201752 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201752 RealDigits[r]    (* A201752 *)
%t A201752 RealDigits[x/.FindRoot[2-x^2==E^x,{x,5},WorkingPrecision->120],10,120][[1]] (* _Harvey P. Dale_, May 20 2025 *)
%Y A201752 Cf. A201741.
%K A201752 nonn,cons
%O A201752 0,1
%A A201752 _Clark Kimberling_, Dec 05 2011