cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 04 2011

Keywords

Comments

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.

Examples

			x=1.31907367685736535441789910952084846442196...

Links

Crossrefs

Cf. A201936.

Programs

  • Mathematica
    (* 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 *)

A201941 Decimal expansion of x>0 satisfying x^2+x=e^(-x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 13 2011

Keywords

Comments

See A201936 for a guide to related sequences. The Mathematica program includes a graph.
1/(x+1) is the solution for real valued z in the equations dW(z)/dz=W(z) and 1=z(W(z)+1) where W(z) is the 0 branch of the Lambert W function. - Colin Linzer, Nov 01 2024

Examples

			x=0.444130228823966590585466329490984667...

Crossrefs

Cf. A201936.

Programs

  • Mathematica
    a = 1; b = 1; c = 0;
f[x_] := a*x^2 + b*x + c; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[r]   (* A201941 *)

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

Views

Author

Clark Kimberling, Dec 13 2011

Keywords

Comments

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

Examples

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

Crossrefs

Cf. A201936.

Programs

  • Mathematica
    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 *)

A201938 Decimal expansion of the greatest number x satisfying 2*x^2=e^(-x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 13 2011

Keywords

Comments

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

Examples

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

Crossrefs

Cf. A201936.

Programs

  • Mathematica
    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 *)

Extensions

Offset corrected by Georg Fischer, Aug 10 2021

A201940 Decimal expansion of x>0 satisfying x^2-1=e^(-x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 13 2011

Keywords

Comments

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

Examples

			x=1.147757632144743493034371990610674766434...

Crossrefs

Cf. A201936.

Programs

  • Mathematica
    a = 1; b = 0; c = -1;
f[x_] := a*x^2 + b*x + c; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r]  (* A201940 *)
Showing 1-5 of 5 results.

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