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 A201741 Decimal expansion of the number x satisfying x^2+2=e^x. 65
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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: a.... b.... c.... x 1.... 0.... 2.... A201741 1.... 0.... 3.... A201742 1.... 0.... 4.... A201743 1.... 0.... 5.... A201744 1.... 0.... 6.... A201745 1.... 0.... 7.... A201746 1.... 0.... 8.... A201747 1.... 0.... 9.... A201748 1.... 0.... 10... A201749 -1... 0.... 1.... A201750, (x=0) -1... 0.... 2.... A201751, A201752 -1... 0.... 3.... A201753, A201754 -1... 0.... 4.... A201755, A201756 -1... 0.... 5.... A201757, A201758 -1... 0.... 6.... A201759, A201760 -1... 0.... 7.... A201761, A201762 -1... 0.... 8.... A201763, A201764 -1... 0.... 9.... A201765, A201766 -1... 0.... 10... A201767, A201768 1.... 1.... 0.... A201769 1.... 1.... 1.... ..(x=0), A201770 1.... 1.... 2.... A201396 1.... 1.... 3.... A201562 1.... 1.... 4.... A201772 1.... 1.... 5.... A201889 1.... 2.... 1.... ..(x=0), A201890 1.... 2.... 2.... A201891 1.... 2.... 3.... A201892 1.... 2.... 4.... A201893 1.... 2.... 5.... A201894 1.... 3.... 1.... A201895, ..(x=0), A201896 1.... 3.... 2.... A201897, A201898, A201899 1.... 3.... 3.... A201900 1.... 3.... 4.... A201901 1.... 3.... 5.... A201902 1.... 4.... 1.... A201903, A201904 1.... 4.... 2.... A201905, A201906, A201907 1.... 4.... 3.... A201924, A201925, A201926 1.... 4.... 4.... A201927, A201928, A201929 1.... 4.... 5.... A201930 1.... 5.... 1.... A201931, A201932 1.... 5.... 2.... A201933, A201934, A201935 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. 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. LINKS EXAMPLE x=1.31907367685736535441789910952084846442196... MATHEMATICA (* Program 1:  A201741 *) a = 1; b = 0; c = 2; f[x_] := a*x^2 + b*x + c; g[x_] := E^x Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}] r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110] RealDigits[r]   (* A201741 *) (* Program 2: implicit surface of u*x^2+v=E^x *) f[{x_, u_, v_}] := u*x^2 + v - E^x; t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 5}]}, {v, 1, 3}, {u, 0, 5}]; ListPlot3D[Flatten[t, 1]] (* for A201741 *) CROSSREFS Cf. A201936. Sequence in context: A221723 A082171 A164795 * A280192 A325375 A317202 Adjacent sequences:  A201738 A201739 A201740 * A201742 A201743 A201744 KEYWORD nonn,cons AUTHOR Clark Kimberling, Dec 04 2011 STATUS approved

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Last modified September 24 06:13 EDT 2021. Contains 347623 sequences. (Running on oeis4.)