

A201936


Decimal expansion of the least number x satisfying 2*x^2=e^(x).


6



2, 6, 1, 7, 8, 6, 6, 6, 1, 3, 0, 6, 6, 8, 1, 2, 7, 6, 9, 1, 7, 8, 9, 7, 8, 0, 5, 9, 1, 4, 3, 2, 0, 2, 8, 1, 7, 3, 2, 0, 2, 7, 4, 3, 5, 9, 4, 1, 0, 4, 8, 2, 9, 1, 9, 2, 1, 0, 5, 0, 8, 1, 6, 1, 0, 4, 0, 3, 7, 0, 3, 2, 5, 3, 3, 2, 2, 7, 9, 6, 5, 9, 9, 6, 5, 0, 6, 3, 6, 1, 7, 0, 4, 5, 6, 3, 3, 0, 5
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OFFSET

1,1


COMMENTS

For some choices of a, b, c, there is a unique value of x satisfying a*x^2+bx+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.... 0.... A126583
2.... 0.... 0.... A201936, A201937, A201938
1.... 0... 1.... A201940
1.... 1.... 0.... A201941
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 A201936, take f(x,u,v)=u*x^2+ve^(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 singlevalued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.


LINKS

Table of n, a(n) for n=1..99.


EXAMPLE

least x: 2.617866613066812769178978059143202...
greatest negative x: 1.487962065498177156254...
greatest x: 0.5398352769028200492118039083633...


MATHEMATICA

a = 2; b = 0; c = 0;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, 3, 2}, {AxesOrigin > {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 3, 2}, WorkingPrecision > 110]
RealDigits[r] (* A201936 *)
r = x /. FindRoot[f[x] == g[x], {x, 2, 1}, WorkingPrecision > 110]
RealDigits[r] (* A201937 *)
r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision > 110]
RealDigits[r] (* A201938 *)
(* 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, 0, .3}]}, {v, 4, 0}, {u, 1, 10}];
ListPlot3D[Flatten[t, 1]] (* for A201936 *)


CROSSREFS

Cf. A201741 [a*x^2+b*x+c=e^x].
Sequence in context: A136766 A199501 A021386 * A019679 A104457 A155832
Adjacent sequences: A201933 A201934 A201935 * A201937 A201938 A201939


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Dec 13 2011


STATUS

approved



