A201938 -id:A201938 - cp's OEIS frontend

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

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

Offset: 1

Clark Kimberling, Dec 13 2011

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.

			least x:  -2.617866613066812769178978059143202...
greatest negative x:  -1.487962065498177156254...
greatest x:  0.5398352769028200492118039083633...

Index entries for transcendental numbers.

Cf. A201741 [a*x^2+b*x+c=e^x].

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 *)

A201937 Decimal expansion of the greatest negative number x satisfying 2*x^2=e^(-x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 13 2011

See A201936 for a guide to related sequences. The Mathematica program includes a graph.

			least x:  -2.617866613066812769178978059143202...
greatest negative x:  -1.487962065498177156254...
greatest x:  0.5398352769028200492118039083633...

Cf. A201936.

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 *)

cp's OEIS Frontend

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

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