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

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

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

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

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