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A201906 Decimal expansion of the x nearest 0 that satisfies x^2 + 4*x + 2 = e^x.

%I A201906 #10 Feb 07 2025 16:44:07
%S A201906 3,5,6,8,7,4,9,1,9,1,3,8,6,3,6,4,8,5,6,5,0,6,6,7,0,5,8,7,5,9,9,1,2,4,
%T A201906 4,0,9,5,9,9,2,0,0,5,2,6,2,0,8,0,4,2,0,9,9,6,8,1,8,4,5,7,7,9,2,0,7,4,
%U A201906 7,0,6,1,9,1,8,6,6,5,3,2,2,5,4,6,3,2,9,0,5,7,9,7,6,8,9,3,3,7,2,8
%N A201906 Decimal expansion of the x nearest 0 that satisfies x^2 + 4*x + 2 = e^x.
%C A201906 See A201741 for a guide to related sequences. The Mathematica program includes a graph.
%H A201906 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201906 least:  -3.425667410202877373265626064725816697827357...
%e A201906 nearest to 0:  -0.35687491913863648565066705875991244...
%e A201906 greatest:  3.2349232177760663670327961327304430448478...
%t A201906 a = 1; b = 4; c = 2;
%t A201906 f[x_] := a*x^2 + b*x + c; g[x_] := E^x
%t A201906 Plot[{f[x], g[x]}, {x, -4, 3.3}, {AxesOrigin -> {0, 0}}]
%t A201906 r = x /. FindRoot[f[x] == g[x], {x, -3.5, -3.4}, WorkingPrecision -> 110]
%t A201906 RealDigits[r]     (* A201905 *)
%t A201906 r = x /. FindRoot[f[x] == g[x], {x, -.36, -.35}, WorkingPrecision -> 110]
%t A201906 RealDigits[r]     (* A201906 *)
%t A201906 r = x /. FindRoot[f[x] == g[x], {x, 3.2, 3.3}, WorkingPrecision -> 110]
%t A201906 RealDigits[r]     (* A201907 *)
%Y A201906 Cf. A201741.
%K A201906 nonn,cons
%O A201906 0,1
%A A201906 _Clark Kimberling_, Dec 06 2011
%E A201906 a(84) onwards corrected by _Georg Fischer_, Aug 03 2021