cp's OEIS Frontend

A201936 Decimal expansion of the least number x satisfying 2*x^2=e^(-x).

%I A201936 #8 Feb 11 2025 13:56:00
%S A201936 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,
%T A201936 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,
%U A201936 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
%N A201936 Decimal expansion of the least number x satisfying 2*x^2=e^(-x).
%C A201936 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:
%C A201936 a.... b.... c.... x
%C A201936 1.... 0.... 0.... A126583
%C A201936 2.... 0.... 0.... A201936, A201937, A201938
%C A201936 1.... 0... -1.... A201940
%C A201936 1.... 1.... 0.... A201941
%C A201936 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 A201936 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.
%H A201936 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A201936 least x:  -2.617866613066812769178978059143202...
%e A201936 greatest negative x:  -1.487962065498177156254...
%e A201936 greatest x:  0.5398352769028200492118039083633...
%t A201936 a = 2; b = 0; c = 0;
%t A201936 f[x_] := a*x^2 + b*x + c; g[x_] := E^-x
%t A201936 Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
%t A201936 r = x /. FindRoot[f[x] == g[x], {x, -3, -2}, WorkingPrecision -> 110]
%t A201936 RealDigits[r]  (* A201936 *)
%t A201936 r = x /. FindRoot[f[x] == g[x], {x, -2, -1}, WorkingPrecision -> 110]
%t A201936 RealDigits[r]   (* A201937 *)
%t A201936 r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
%t A201936 RealDigits[r]   (* A201938 *)
%t A201936 (* Program 2: implicit surface of u*x^2+v=e^(-x) *)
%t A201936 f[{x_, u_, v_}] := u*x^2 + v - E^-x;
%t A201936 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, -4, 0}, {u, 1,10}];
%t A201936 ListPlot3D[Flatten[t, 1]]  (* for A201936 *)
%Y A201936 Cf. A201741 [a*x^2+b*x+c=e^x].
%K A201936 nonn,cons
%O A201936 1,1
%A A201936 _Clark Kimberling_, Dec 13 2011