Search: a201750 -id:a201750
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A201741
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Decimal expansion of the number x satisfying x^2+2=e^x.
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+10
65
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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
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OFFSET
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1,2
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COMMENTS
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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
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.
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LINKS
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EXAMPLE
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x=1.31907367685736535441789910952084846442196...
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MATHEMATICA
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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]
(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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