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

Author

Clark Kimberling, Dec 04 2011

Keywords

• nonn
• cons

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.

Examples

			x=1.31907367685736535441789910952084846442196...
		

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A201936.

Programs

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 *)