

A202537


Decimal expansion of x satisfying e^xe^(2x)=1.


8



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

0,1


COMMENTS

If u>0 and v>0, there is a unique number x satisfying e^(ux)e^(vx)=1. Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 1.... A002390
1.... 2.... A202537
1.... 3.... A202538
2.... 1.... A202539
3.... 1.... A202540
2.... 2.... A202541
3.... 3.... A202542
1/2..1/2... A202543
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 A202537, take f(x,u,v)=e^(ux)e^(vx)1 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=0..98.


EXAMPLE

x=0.382245085840035641329358499184857393759416422...


MATHEMATICA

(* Program 1: A202537 *)
u = 1; v = 2;
f[x_] := E^(u*x)  E^(v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, 2, 2}, {AxesOrigin > {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .3, .4}, WorkingPrecision > 110]
RealDigits[r] (* A202537 *)
(* Program 2: implicit surface for e^(ux)e(vx)=1 *)
f[{x_, u_, v_}] := E^(u*x)  E^(v*x)  1;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, 1, 4}, {u, 2, 20}];
ListPlot3D[Flatten[t, 1]] (* for A202537 *)
First[ RealDigits[ Log[ Root[#^3  #^2  1 & , 1]], 10, 99]] (* JeanFrançois Alcover, Feb 26 2013 *)


PROG

(PARI) solve(x=0, 1, exp(x)exp(2*x)1) \\ Charles R Greathouse IV, Feb 26 2013


CROSSREFS

Cf. A002390.
Sequence in context: A280567 A280835 A195426 * A220516 A010627 A103712
Adjacent sequences: A202534 A202535 A202536 * A202538 A202539 A202540


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Dec 21 2011


EXTENSIONS

Digits from a(90) on corrected by JeanFrançois Alcover, Feb 26 2013


STATUS

approved



