A202322 - OEIS

%I #23 Nov 21 2024 15:33:09

%S 4,4,2,8,5,4,4,0,1,0,0,2,3,8,8,5,8,3,1,4,1,3,2,7,9,9,9,9,9,9,3,3,6,8,

%T 1,9,7,1,6,2,6,2,1,2,9,3,7,3,4,7,9,6,8,4,7,1,7,7,3,3,0,7,6,9,8,2,0,1,

%U 5,9,9,2,1,4,2,0,0,4,0,7,8,4,9,0,8,6,5,9,2,4,8,1,7,8,7,3,9,5,5

%N Decimal expansion of x satisfying x+2=exp(-x).

%C For many choices of u and v, there is just one value of x satisfying u*x+v=e^(-x).  Guide to related sequences, with graphs included in Mathematica programs:

%C u.... v.... x

%C 1.... 2.... A202322

%C 1.... 3.... A202323

%C 2.... 2.... A202353

%C 2.... e.... A202354

%C 1... -1.... A202355

%C 1.... 0.... A030178

%C 2.... 0.... A202356

%C e.... 0.... A202357

%C 3.... 0.... A202392

%C 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.

%C For an example related to A202322, take f(x,u,v)=x+2-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.

%H G. C. Greubel, <a href="/A202322/b202322.txt">Table of n, a(n) for n = 0..5000</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F x(u,v) = W(e^(v/u)/u) - v/u, where W = ProductLog = LambertW. - _Jean-François Alcover_, Feb 14 2013

%F Equals A226571 - 2 = LambertW(exp(2))-2. - _Vaclav Kotesovec_, Jan 09 2014

%e x=-0.442854401002388583141327999999336819716262...

%t (* Program 1:  A202322 *)

%t u = 1; v = 2;

%t f[x_] := u*x + v; g[x_] := E^-x

%t Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]

%t r = x /. FindRoot[f[x] == g[x], {x, -.45, -.44}, WorkingPrecision -> 110]

%t RealDigits[r]  (* A202322 *)

%t (* Program 2: implicit surface of u*x+v=e^(-x) *)

%t f[{x_, u_, v_}] := u*x + v - E^-x;

%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 1, 3}, {u, 1, 3}];

%t ListPlot3D[Flatten[t, 1]] (* for A202322 *)

%t RealDigits[ ProductLog[E^2] - 2, 10, 99] // First (* _Jean-François Alcover_, Feb 14 2013 *)

%o (PARI) lambertw(exp(2)) - 2 \\ _G. C. Greubel_, Jun 10 2017

%Y Cf. A202320.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Dec 18 2011

%E Digits from a(84) on corrected by _Jean-François Alcover_, Feb 14 2013