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A201896 Decimal expansion of the greatest x satisfying x^2 + 3*x + 1 = e^x.

%I A201896 #13 Feb 07 2025 16:44:07
%S A201896 2,8,9,3,1,1,6,4,3,0,9,2,5,2,7,1,2,2,0,3,1,5,5,3,4,9,3,1,3,4,9,5,3,0,
%T A201896 8,8,5,3,0,4,0,7,9,0,9,1,5,4,6,9,7,7,4,0,1,8,2,1,6,3,4,9,2,8,1,6,6,5,
%U A201896 5,3,6,6,0,7,8,3,3,7,3,0,5,1,9,0,8,9,2,1,0,2,3,8,8,7,1,7,3,4,9
%N A201896 Decimal expansion of the greatest x satisfying x^2 + 3*x + 1 = e^x.
%C A201896 See A201741 for a guide to related sequences. The Mathematica program includes a graph.
%H A201896 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201896 least:  -2.649219887767292965348496137953408152796...
%e A201896 greatest:  2.8931164309252712203155349313495308853...
%t A201896 a = 1; b = 3; c = 1;
%t A201896 f[x_] := a*x^2 + b*x + c; g[x_] := E^x
%t A201896 Plot[{f[x], g[x]}, {x, -4, 4}, {AxesOrigin -> {0, 0}}]
%t A201896 r = x /. FindRoot[f[x] == g[x], {x, -2.7, -2.6}, WorkingPrecision -> 110]
%t A201896 RealDigits[r]     (* A201895 *)
%t A201896 r = x /. FindRoot[f[x] == g[x], {x, 2.9, 3.0}, WorkingPrecision -> 110]
%t A201896 RealDigits[r]     (* A201896 *)  (* NOTE 3 zeros *)
%Y A201896 Cf. A201741.
%K A201896 nonn,cons
%O A201896 1,1
%A A201896 _Clark Kimberling_, Dec 06 2011
%E A201896 a(98) onwards corrected by _Georg Fischer_, Aug 03 2021