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 64 results. Next

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 13 2011

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

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 *)
(* 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 *)

A201752 Decimal expansion of the greatest x satisfying -x^2+2 = e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 05 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -1.3159737777962901878871773873012710...
greatest:  0.53727444917385660425676298977967...

Links

Crossrefs

Cf. A201741.

Programs

  • Mathematica
    a = -1; b = 0; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -2, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.4, -1.3}, WorkingPrecision -> 110]
RealDigits[r]    (* A201751 *)
r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
RealDigits[r]    (* A201752 *)
RealDigits[x/.FindRoot[2-x^2==E^x,{x,5},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, May 20 2025 *)

A201897 Decimal expansion of the least x satisfying x^2+3x+2=e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -2.1093569955710161272316992470592578841155...
nearest to 0:  -0.608989103010165494835043701926011...
greatest:  2.99223487205393686509331145278388262181...

Links

Crossrefs

Cf. A201741, A201898, A201899.

Programs

  • Mathematica
    a = 1; b = 3; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 3.1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -2.2, -2.1}, WorkingPrecision -> 110]
RealDigits[r]     (* A201897, least *)
r = x /. FindRoot[f[x] == g[x], {x, -.7, -.6}, WorkingPrecision -> 110]
RealDigits[r]     (* A201898, nearest 0  *)
r = x /. FindRoot[f[x] == g[x], {x, 2.9, 3.0}, WorkingPrecision -> 110]
RealDigits[r]     (* A201899 greatest *)

Extensions

Name corrected by Sean A. Irvine, Jan 12 2025

A201898 Decimal expansion of the x nearest 0 that satisfies x^2+3x+2=e^x, negated.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -2.1093569955710161272316992470592578841155...
nearest to 0:  -0.608989103010165494835043701926011...
greatest:  2.99223487205393686509331145278388262181...

Links

Crossrefs

Cf. A201741, A201897, A201899.

Programs

  • Mathematica
    a = 1; b = 3; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 3.1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -2.2, -2.1}, WorkingPrecision -> 110]
RealDigits[r]     (* A201897, least *)
r = x /. FindRoot[f[x] == g[x], {x, -.7, -.6}, WorkingPrecision -> 110]
RealDigits[r]     (* A201898, nearest 0  *)
r = x /. FindRoot[f[x] == g[x], {x, 2.9, 3.0}, WorkingPrecision -> 110]
RealDigits[r]     (* A201899 greatest *)

Extensions

Name corrected by Sean A. Irvine, Jan 12 2025

A201899 Decimal expansion of the greatest x satisfying x^2+3x+2=e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -2.1093569955710161272316992470592578841155...
nearest to 0:  -0.608989103010165494835043701926011...
greatest:  2.99223487205393686509331145278388262181...

Links

Crossrefs

Cf. A201741, A201897, A201898.

Programs

  • Mathematica
    a = 1; b = 3; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 3.1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -2.2, -2.1}, WorkingPrecision -> 110]
RealDigits[r]     (* A201897, least *)
r = x /. FindRoot[f[x] == g[x], {x, -.7, -.6}, WorkingPrecision -> 110]
RealDigits[r]     (* A201898, nearest 0  *)
r = x /. FindRoot[f[x] == g[x], {x, 2.9, 3.0}, WorkingPrecision -> 110]
RealDigits[r]     (* A201899 greatest *)

Extensions

Name corrected by Sean A. Irvine, Jan 12 2025

A201905 Decimal expansion of the least x satisfying x^2+4x+2=e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -3.425667410202877373265626064725816697827357...
nearest to 0:  -0.35687491913863648565066705875991244...
greatest:  3.2349232177760663670327961327304430448478...

Links

Crossrefs

Cf. A201741.

Programs

  • Mathematica
    a = 1; b = 4; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -4, 3.3}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -3.5, -3.4}, WorkingPrecision -> 110]
RealDigits[r]     (* A201905 *)
r = x /. FindRoot[f[x] == g[x], {x, -.36, -.35}, WorkingPrecision -> 110]
RealDigits[r]     (* A201906 *)
r = x /. FindRoot[f[x] == g[x], {x, 3.2, 3.3}, WorkingPrecision -> 110]
RealDigits[r]     (* A201907 *)

A201906 Decimal expansion of the x nearest 0 that satisfies x^2 + 4*x + 2 = e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -3.425667410202877373265626064725816697827357...
nearest to 0:  -0.35687491913863648565066705875991244...
greatest:  3.2349232177760663670327961327304430448478...

Links

Crossrefs

Cf. A201741.

Programs

  • Mathematica
    a = 1; b = 4; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -4, 3.3}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -3.5, -3.4}, WorkingPrecision -> 110]
RealDigits[r]     (* A201905 *)
r = x /. FindRoot[f[x] == g[x], {x, -.36, -.35}, WorkingPrecision -> 110]
RealDigits[r]     (* A201906 *)
r = x /. FindRoot[f[x] == g[x], {x, 3.2, 3.3}, WorkingPrecision -> 110]
RealDigits[r]     (* A201907 *)

Extensions

a(84) onwards corrected by Georg Fischer, Aug 03 2021

A201907 Decimal expansion of the greatest x satisfying x^2+4x+2=e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -3.425667410202877373265626064725816697827357...
nearest to 0:  -0.35687491913863648565066705875991244...
greatest:  3.2349232177760663670327961327304430448478...

Links

Crossrefs

Cf. A201741.

Programs

  • Mathematica
    a = 1; b = 4; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -4, 3.3}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -3.5, -3.4}, WorkingPrecision -> 110]
RealDigits[r]     (* A201905 *)
r = x /. FindRoot[f[x] == g[x], {x, -.36, -.35}, WorkingPrecision -> 110]
RealDigits[r]     (* A201906 *)
 r = x /. FindRoot[f[x] == g[x], {x, 3.2, 3.3}, WorkingPrecision -> 110]
RealDigits[r]     (* A201907 *)

A201924 Decimal expansion of the least x satisfying x^2+4x+3=e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -3.024014501135293784775589627797395351659...
nearest to 0:  -0.79522661386054079889626155638871...
greatest:  3.2986275628038651802559413164923413431...

Links

Crossrefs

Cf. A201741.

Programs

  • Mathematica
    a = 1; b = 4; c = 3;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3.5, 3.5}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -3.1, -3.0}, WorkingPrecision -> 110]
RealDigits[r]     (* A201924 *)
r = x /. FindRoot[f[x] == g[x], {x, -.8, -.7}, WorkingPrecision -> 110]
RealDigits[r]     (* A201925 *)
 r = x /. FindRoot[f[x] == g[x], {x, 3.2, 3.3}, WorkingPrecision -> 110]
RealDigits[r]     (* A201926 *)

A201925 Decimal expansion of the x nearest 0 that satisfies x^2+4x+3=e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 06 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -3.024014501135293784775589627797395351659...
nearest to 0:  -0.79522661386054079889626155638871...
greatest:  3.2986275628038651802559413164923413431...

Links

Crossrefs

Cf. A201741.

Programs

  • Mathematica
    a = 1; b = 4; c = 3;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3.5, 3.5}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -3.1, -3.0}, WorkingPrecision -> 110]
RealDigits[r]     (* A201924 *)
r = x /. FindRoot[f[x] == g[x], {x, -.8, -.7}, WorkingPrecision -> 110]
RealDigits[r]     (* A201925 *)
 r = x /. FindRoot[f[x] == g[x], {x, 3.2, 3.3}, WorkingPrecision -> 110]
RealDigits[r]     (* A201926 *)
Showing 1-10 of 64 results. Next

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