login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; constant; graph; refs; listen; history; text; internal format)
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+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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 31 22:27 EDT 2021. Contains 346377 sequences. (Running on oeis4.)