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A201741 Decimal expansion of the number x satisfying x^2+2=e^x.

%I A201741 #16 Jan 30 2025 10:39:10
%S A201741 1,3,1,9,0,7,3,6,7,6,8,5,7,3,6,5,3,5,4,4,1,7,8,9,9,1,0,9,5,2,0,8,4,8,
%T A201741 4,6,4,4,2,1,9,6,6,7,8,0,8,2,5,4,9,7,6,6,9,2,5,6,0,8,9,0,0,4,9,0,5,1,
%U A201741 2,7,0,7,6,3,4,6,1,0,7,3,1,6,7,2,5,1,0,4,0,6,3,8,4,4,9,4,0,2,7
%N A201741 Decimal expansion of the number x satisfying x^2+2=e^x.
%C A201741 For some choices of a, b, c, there is a unique value of x satisfying a*x^2+b*x+c=e^x, for other choices, there are two solutions, and for others, three.  Guide to related sequences, with graphs included in Mathematica programs:
%C A201741 a.... b.... c.... x
%C A201741 1.... 0.... 2.... A201741
%C A201741 1.... 0.... 3.... A201742
%C A201741 1.... 0.... 4.... A201743
%C A201741 1.... 0.... 5.... A201744
%C A201741 1.... 0.... 6.... A201745
%C A201741 1.... 0.... 7.... A201746
%C A201741 1.... 0.... 8.... A201747
%C A201741 1.... 0.... 9.... A201748
%C A201741 1.... 0.... 10... A201749
%C A201741 -1... 0.... 1.... A201750, (x=0)
%C A201741 -1... 0.... 2.... A201751, A201752
%C A201741 -1... 0.... 3.... A201753, A201754
%C A201741 -1... 0.... 4.... A201755, A201756
%C A201741 -1... 0.... 5.... A201757, A201758
%C A201741 -1... 0.... 6.... A201759, A201760
%C A201741 -1... 0.... 7.... A201761, A201762
%C A201741 -1... 0.... 8.... A201763, A201764
%C A201741 -1... 0.... 9.... A201765, A201766
%C A201741 -1... 0.... 10... A201767, A201768
%C A201741 1.... 1.... 0.... A201769
%C A201741 1.... 1.... 1.... ..(x=0), A201770
%C A201741 1.... 1.... 2.... A201396
%C A201741 1.... 1.... 3.... A201562
%C A201741 1.... 1.... 4.... A201772
%C A201741 1.... 1.... 5.... A201889
%C A201741 1.... 2.... 1.... ..(x=0), A201890
%C A201741 1.... 2.... 2.... A201891
%C A201741 1.... 2.... 3.... A201892
%C A201741 1.... 2.... 4.... A201893
%C A201741 1.... 2.... 5.... A201894
%C A201741 1.... 3.... 1.... A201895, ..(x=0), A201896
%C A201741 1.... 3.... 2.... A201897, A201898, A201899
%C A201741 1.... 3.... 3.... A201900
%C A201741 1.... 3.... 4.... A201901
%C A201741 1.... 3.... 5.... A201902
%C A201741 1.... 4.... 1.... A201903, A201904
%C A201741 1.... 4.... 2.... A201905, A201906, A201907
%C A201741 1.... 4.... 3.... A201924, A201925, A201926
%C A201741 1.... 4.... 4.... A201927, A201928, A201929
%C A201741 1.... 4.... 5.... A201930
%C A201741 1.... 5.... 1.... A201931, A201932
%C A201741 1.... 5.... 2.... A201933, A201934, A201935
%C A201741 Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
%C A201741 For an example related to A201741, take f(x,u,v)=u*x^2+v-e^x and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
%H A201741 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201741 x=1.31907367685736535441789910952084846442196...
%t A201741 (* Program 1:  A201741 *)
%t A201741 a = 1; b = 0; c = 2;
%t A201741 f[x_] := a*x^2 + b*x + c; g[x_] := E^x
%t A201741 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A201741 r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
%t A201741 RealDigits[r]   (* A201741 *)
%t A201741 (* Program 2: implicit surface of u*x^2+v=E^x *)
%t A201741 f[{x_, u_, v_}] := u*x^2 + v - E^x;
%t A201741 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 5}]},
%t A201741 {v, 1, 3}, {u, 0, 5}];
%t A201741 ListPlot3D[Flatten[t, 1]] (* for A201741 *)
%Y A201741 Cf. A201936.
%K A201741 nonn,cons
%O A201741 1,2
%A A201741 _Clark Kimberling_, Dec 04 2011