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 1-10 of 14 results. Next

A202348 Decimal expansion of x satisfying x = exp(x-2).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 20 2011

Keywords

Comments

For many choices of u and v, there is just one value of x satisfying x = exp(u*x+v). Guide to related sequences, with graphs included in Mathematica programs:
u v x
----- -- -------
1 -2 A202348
1 -3 A202494
-1 -1 A202357
-1 -2 A202496
-2 -2 A202497
-2 0 A202498
-3 0 A202499
-Pi 0 A202500
-Pi/2 0 A202501
-2*Pi -1 A202495
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 this sequence, take f(x,u,v) = x - exp(u*x+v) 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.
Actually there are two solutions to x = exp(x-2). This sequence gives the lesser one, x = -LambertW(-exp(-2)), and A226572 gives the greater one, x = -LambertW(-1,-exp(-2)) = 3.14619322062... - Jianing Song, Dec 30 2018

Examples

			x = 0.158594339563039362153395341987513893949...

Links

Crossrefs

Cf. A202320, A226572.

Programs

  • Mathematica
    (* Program 1: A202348 *)
u = 1; v = -2;
f[x_] := x; g[x_] := E^(u*x + v)
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .15, .16}, WorkingPrecision -> 110]
RealDigits[r]  (* A202348 *)
(* Program 2: implicit surface of x=e^(ux+v) *)
f[{x_, u_, v_}] := x - E^(u*x + v);
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, .3}]}, {v, 1, 5}, {u, -5, -.1}];
ListPlot3D[Flatten[t, 1]] (* for A202348 *)
RealDigits[-ProductLog[-1/E^2], 10, 99] // First (* Jean-François Alcover, Feb 26 2013 *)
  • PARI
    solve(x=0,1,exp(x-2)-x) \\ Charles R Greathouse IV, Feb 26 2013

Formula

Equals -LambertW(-exp(-2)) = 2 - A202320. - Jianing Song, Dec 30 2018

Extensions

Digits from a(93) on corrected by Jean-François Alcover, Feb 26 2013

A202322 Decimal expansion of x satisfying x+2=exp(-x).

Original entry on oeis.org

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, 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, 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
Offset: 0

Views

Author

Clark Kimberling, Dec 18 2011

Keywords

Comments

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:
u.... v.... x
1.... 2.... A202322
1.... 3.... A202323
2.... 2.... A202353
2.... e.... A202354
1... -1.... A202355
1.... 0.... A030178
2.... 0.... A202356
e.... 0.... A202357
3.... 0.... A202392
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 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.

Examples

			x=-0.442854401002388583141327999999336819716262...

Links

Crossrefs

Cf. A202320.

Programs

  • Mathematica
    (* Program 1:  A202322 *)
u = 1; v = 2;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.45, -.44}, WorkingPrecision -> 110]
RealDigits[r]  (* A202322 *)
(* Program 2: implicit surface of u*x+v=e^(-x) *)
f[{x_, u_, v_}] := u*x + v - E^-x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 1, 3}, {u, 1, 3}];
ListPlot3D[Flatten[t, 1]] (* for A202322 *)
RealDigits[ ProductLog[E^2] - 2, 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)
  • PARI
    lambertw(exp(2)) - 2 \\ G. C. Greubel, Jun 10 2017

Formula

x(u,v) = W(e^(v/u)/u) - v/u, where W = ProductLog = LambertW. - Jean-François Alcover, Feb 14 2013
Equals A226571 - 2 = LambertW(exp(2))-2. - Vaclav Kotesovec, Jan 09 2014

Extensions

Digits from a(84) on corrected by Jean-François Alcover, Feb 14 2013

A202351 Decimal expansion of least x satisfying 3*x = exp(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 17 2011

Keywords

Comments

See A202320 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.61906128673594511215232699402092223330147...
greatest:  1.51213455165784247389673967807203870460...

Links

Crossrefs

Cf. A202320.

Programs

  • Mathematica
    u = 3; v = 0;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 0.6, 0.7}, WorkingPrecision -> 110]
RealDigits[r] (* A202351 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.5, 1.6}, WorkingPrecision -> 110]
RealDigits[r] (* A202352 *)
RealDigits[ -ProductLog[-1/3], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)
  • PARI
    solve(x=0, 1, 3*x-exp(x)) \\ Michel Marcus, Nov 09 2017

Extensions

Offset corrected by Georg Fischer, Aug 02 2021

A202352 Decimal expansion of greatest x satisfying 3*x = exp(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 17 2011

Keywords

Comments

See A202320 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  0.61906128673594511215232699402092223330147...
greatest:  1.51213455165784247389673967807203870460...

Links

Crossrefs

Cf. A202320.

Programs

  • Mathematica
    u = 3; v = 0;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 0.6, 0.7}, WorkingPrecision -> 110]
RealDigits[r] (* A202351 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.5, 1.6}, WorkingPrecision -> 110]
RealDigits[r] (* A202352 *)
RealDigits[ -ProductLog[-1, -1/3], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)
  • PARI
    solve(x=1, 2, 3*x-exp(x)) \\ Michel Marcus, Nov 09 2017

Formula

Equals -LambertW(-1,-1/3). - Gleb Koloskov, Jun 12 2021

A104689 Decimal expansion of solution to exp(x) = e + x.

Original entry on oeis.org

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

Views

Author

Zak Seidov, Apr 23 2005

Keywords

Comments

For the negative solution, see A202347.

Examples

			1.42037011802008345845842...

Links

Crossrefs

Cf. A202320, A202347.

Programs

  • Mathematica
    RealDigits[x/.FindRoot[E^x==E+x,{x,1.4},WorkingPrecision->120], 10, 120] [[1]] (* Harvey P. Dale, Aug 01 2011 *)
RealDigits[-E - LambertW[-1, -Exp[-E]], 10, 100][[1]] (* G. C. Greubel, Nov 09 2017 *)
  • PARI
    solve(x=0, 2, exp(x)-exp(1)-x) \\ Michel Marcus, Nov 09 2017

A202321 Decimal expansion of x > 0 satisfying x + 2 = exp(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 16 2011

Keywords

Comments

See A202320 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x < 0: -1.841405660436960637846604658012486...
x > 0:  1.1461932206205825852370610285213682...

Links

Crossrefs

Cf. A202320.

Programs

  • Mathematica
    u = 1; v = 2;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.9, -1.8}, WorkingPrecision -> 110]
RealDigits[r]  (* A202320 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r]  (* A202321 *)
RealDigits[-ProductLog[-1, -1/E^2] - 2, 10, 99] // First (* Jean-François Alcover, Feb 26 2013 *)
  • PARI
    solve(x=1, 2, x+2-exp(x)) \\ Michel Marcus, Nov 09 2017

Formula

a(n) = -LambertW(-1, -exp(-2)) - 2. - Vaclav Kotesovec, Jan 09 2014

A202324 Decimal expansion of x < 0 satisfying x + 3 = exp(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 16 2011

Keywords

Comments

See A202320 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x < 0: -2.9475309025422851275901263887139816414...
x > 0:  1.50524149579288336699862443213735394007...

Links

Crossrefs

Cf. A202320.

Programs

  • Mathematica
    u = 1; v = 3;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -2, -1}, WorkingPrecision -> 110]
RealDigits[r]  (* A202324 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.5, 1.6}, WorkingPrecision -> 110]
RealDigits[r]  (* A202325 *)
RealDigits[-3 - LambertW[-Exp[-3]], 10, 100][[1]] (* G. C. Greubel, Nov 09 2017 *)
  • PARI
    solve(x=-3, 0, x+3-exp(x)) \\ Michel Marcus, Nov 09 2017

Formula

Equals -3 - LambertW(-exp(-3)). - G. C. Greubel, Nov 09 2017

A202325 Decimal expansion of x > 0 satisfying x + 3 = exp(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 16 2011

Keywords

Comments

See A202320 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x < 0:  -2.9475309025422851275901263887139816414...
x > 0:  1.50524149579288336699862443213735394007...

Links

Crossrefs

Cf. A202320.

Programs

  • Mathematica
    u = 1; v = 3;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -2, -1}, WorkingPrecision -> 110]
RealDigits[r]  (* A202324 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.5, 1.6}, WorkingPrecision -> 110]
RealDigits[r]  (* A202325 *)
RealDigits[-3 - LambertW[-1, -Exp[-3]], 10, 100][[1]] (* G. C. Greubel, Nov 09 2017 *)
  • PARI
    solve(x=0, 2, x+3-exp(x)) \\ Michel Marcus, Nov 09 2017

A202343 Decimal expansion of x > 0 satisfying 2*x + 1 = exp(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 17 2011

Keywords

Comments

See A202320 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x = 1.25643120862616967698273761660921...

Links

Crossrefs

Cf. A202320.

Programs

  • Mathematica
    u = 2; v = 1;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]  (* A202343 *)
RealDigits[-1/2 - ProductLog[-1, -1/(2*Sqrt[E])], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)
  • PARI
    solve(x=1, 2, 2*x+1-exp(x)) \\ Michel Marcus, Nov 09 2017

A202345 Decimal expansion of x < 0 satisfying 2*x + 2 = exp(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 17 2011

Keywords

Comments

See A202320 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x<0:  -0.76803904701346556525568352607754...
x>0:  1.678346990016660653412884512094523...

Links

Crossrefs

Cf. A202320.

Programs

  • Mathematica
    u = 2; v = 2;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.8, -.7}, WorkingPrecision -> 110]
RealDigits[r]  (* A202345 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.6, 1.7}, WorkingPrecision -> 110]
RealDigits[r] (* A202346 *)
RealDigits[-1 - LambertW[-Exp[-1]/2], 10, 100][[1]] (* G. C. Greubel, Nov 09 2017 *)
  • PARI
    solve(x=-1, 0, 2*x+2-exp(x)) \\ Michel Marcus, Nov 09 2017

Formula

Equals -1 - lambertw(-exp(-1)/2). - G. C. Greubel, Nov 09 2017
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