A201890 -id:A201890 - cp's OEIS frontend

A201741 Decimal expansion of the number x satisfying x^2+2=e^x.

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

Offset: 1

Clark Kimberling, Dec 04 2011

For some choices of a, b, c, there is a unique value of x satisfying a*x^2+b*x+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.... 2.... A201741
1.... 0.... 3.... A201742
1.... 0.... 4.... A201743
1.... 0.... 5.... A201744
1.... 0.... 6.... A201745
1.... 0.... 7.... A201746
1.... 0.... 8.... A201747
1.... 0.... 9.... A201748
1.... 0.... 10... A201749
-1... 0.... 1.... A201750, (x=0)
-1... 0.... 2.... A201751, A201752
-1... 0.... 3.... A201753, A201754
-1... 0.... 4.... A201755, A201756
-1... 0.... 5.... A201757, A201758
-1... 0.... 6.... A201759, A201760
-1... 0.... 7.... A201761, A201762
-1... 0.... 8.... A201763, A201764
-1... 0.... 9.... A201765, A201766
-1... 0.... 10... A201767, A201768
1.... 1.... 0.... A201769
1.... 1.... 1.... ..(x=0), A201770
1.... 1.... 2.... A201396
1.... 1.... 3.... A201562
1.... 1.... 4.... A201772
1.... 1.... 5.... A201889
1.... 2.... 1.... ..(x=0), A201890
1.... 2.... 2.... A201891
1.... 2.... 3.... A201892
1.... 2.... 4.... A201893
1.... 2.... 5.... A201894
1.... 3.... 1.... A201895, ..(x=0), A201896
1.... 3.... 2.... A201897, A201898, A201899
1.... 3.... 3.... A201900
1.... 3.... 4.... A201901
1.... 3.... 5.... A201902
1.... 4.... 1.... A201903, A201904
1.... 4.... 2.... A201905, A201906, A201907
1.... 4.... 3.... A201924, A201925, A201926
1.... 4.... 4.... A201927, A201928, A201929
1.... 4.... 5.... A201930
1.... 5.... 1.... A201931, A201932
1.... 5.... 2.... A201933, A201934, A201935
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 A201741, 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.

			x=1.31907367685736535441789910952084846442196...

Index entries for transcendental numbers.

Cf. A201936.

(* Program 1:  A201741 *)
a = 1; b = 0; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]   (* A201741 *)
(* 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, 1, 5}]},
{v, 1, 3}, {u, 0, 5}];
ListPlot3D[Flatten[t, 1]] (* for A201741 *)

A226278 Decimal expansion of the number x > 1 defined by 2*log(x) = x - 1.

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Jun 02 2013

There are two solutions to the equation 2*log(x) = x - 1: {1, 3.51286...}.

Apart from the leading digit the same as A201890. - R. J. Mathar, Jun 05 2013

			x = 3.512862417252339353965475233218432653832833663402647422251789454...

Cf. A101314, A201890, A202343.

Digits := 100; evalf([solve(2*ln(n)=n-1,n)]);

RealDigits[x /. FindRoot[2*Log[x] == x - 1, {x, 3.5}, WorkingPrecision -> 110]][[1]]
RealDigits[N[Exp[-ProductLog[-1,-1/(2*Sqrt[E])]-1/2],110]][[1]] (* Natalia L. Skirrow, Jul 13 2025 *)

solve(x=3,4,2*log(x)-x+1) \\ Charles R Greathouse IV, Jun 05 2013

Equals 1 + A201890.
Equals exp(-LambertW_-1(-1/(2*sqrt(e)))-1/2). - Natalia L. Skirrow, Jul 13 2025
Equals 1/A101314 = exp(A202343). - Hugo Pfoertner, Jul 13 2025

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A201741 Decimal expansion of the number x satisfying x^2+2=e^x.

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