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 1-7 of 7 results.

A201397 Decimal expansion of x satisfying x^2 + 2 = sec(x) and 0 < x < Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 01 2011

Keywords

Comments

For many choices of a and c, there are exactly two values of x satisfying a*x^2 + c = sec(x) and 0 < x < Pi. Guide to related sequences, with graphs included in Mathematica programs:
a.... c.... x
1.... 1.... A196816
1.... 2.... A201397
1.... 3.... A201398
1.... 4.... A201399
1.... 5.... A201400
1.... 6.... A201401
1.... 7.... A201402
1.... 8.... A201403
1.... 9.... A201404
1.... 10... A201405
2.... 0.... A201406, A201407
3.... 0.... A201408, A201409
4.... 0.... A201410, A201411
5.... 0.... A201412, A201413
6.... 0.... A201414, A201415
7.... 0.... A201416, A201417
8.... 0.... A201418, A201419
9.... 0.... A201420, A201421
10... 0.... A201422, A201423
3... -1.... A201515, A201516
4... -1.... A201517, A201518
5... -1.... A201519, A201520
6... -1.... A201521, A201522
7... -1.... A201523, A201524
8... -1.... A201525, A201526
9... -1.... A201527, A201528
10.. -1.... A201529, A201530
2.... 3.... A201531
3.... 2.... A200619
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 A201397, take f(x,u,v) = u*x^2 + v = sec(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

			1.2954596464154787686299132707186415897672...

Links

Crossrefs

Cf. A201280, A200614.

Programs

  • Mathematica
    (* Program 1:  A201397 *)
a = 1; c = 2;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]    (* A201397 *)
(* Program 2: implicit surface of u*x^2+v=sec(x) *)
Remove["Global`*"];
f[{x_, u_, v_}] := u*x^2 + v - Sec[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, 1}]}, {v, 0, 1}, {u, 2 + v, 10}];
ListPlot3D[Flatten[t, 1]]  (* for A201397 *)

A196817 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=2*cos(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			x=1.401269207599957942927187243790834191530882865453360260...

Links

Crossrefs

Cf. A196914.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Cos[x], 2 Cos[x], 3 Cos[x], 4 Cos[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Cos[x], {x, 1, 1.5}, WorkingPrecision -> 100]
RealDigits[t]  (* A196816 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196817 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196818 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196819 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196820 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196821 *)

A196818 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=3*cos(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			1.46461147970142500501464804801002599781808481310...

Links

Crossrefs

Cf. A196914.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Cos[x], 2 Cos[x], 3 Cos[x], 4 Cos[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Cos[x], {x, 1, 1.5}, WorkingPrecision -> 100]
RealDigits[t]  (* A196816 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196817 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196818 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196819 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196820 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196821 *)

A196819 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=4*cos(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			1.4933195357382420192666761841798184096253499369741587866...

Links

Crossrefs

Cf. A196914.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Cos[x], 2 Cos[x], 3 Cos[x], 4 Cos[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Cos[x], {x, 1, 1.5}, WorkingPrecision -> 100]
RealDigits[t]  (* A196816 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196817 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196818 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196819 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196820 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196821 *)

A196820 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=5*cos(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			1.50977190047072688535549375350098659944863772756...

Links

Crossrefs

Cf. A196914.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Cos[x], 2 Cos[x], 3 Cos[x], 4 Cos[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Cos[x], {x, 1, 1.5}, WorkingPrecision -> 100]
RealDigits[t]  (* A196816 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196817 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196818 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196819 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196820 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196821 *)

A196821 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=6*cos(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			1.5204494508338163631474588208905639631389853055832784...

Links

Crossrefs

Cf. A196914.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), Cos[x], 2 Cos[x], 3 Cos[x], 4 Cos[x]}, {x, 0, 2}]
t = x /. FindRoot[1 == (1 + x^2) Cos[x], {x, 1, 1.5}, WorkingPrecision -> 100]
RealDigits[t]  (* A196816 *)
t = x /. FindRoot[1 == 2 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196817 *)
t = x /. FindRoot[1 == 3 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196818 *)
t = x /. FindRoot[1 == 4 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]   (* A196819 *)
t = x /. FindRoot[1 == 5 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196820 *)
t = x /. FindRoot[1 == 6 (1 + x^2) Cos[x], {x, 1, 1.6},
   WorkingPrecision -> 100]
RealDigits[t]  (* A196821 *)

A196913 Decimal expansion of the number x satisfying 0 < x < 2*Pi and 2x = (1 + x^2)*tan(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			x=0.7682171553153782504312122866979254095466915658...

Crossrefs

Cf. A196816, A196914.

Programs

  • Mathematica
    Plot[{1/(1 + x^2), 0.874*Cos[x]}, {x, .5, 1}]
t = x /. FindRoot[Tan[x] == 2 x/(1 + x^2), {x, .5, 1}, WorkingPrecision -> 100]
RealDigits[t]    (* A196913 *)
c = N[Sqrt[t^4 + 6 t^2 + 1]/(t^4 + 2 t^2 + 1), 100]
RealDigits[c]    (* A196914 *)
slope = N[-c*Sin[t], 100]
RealDigits[slope](* A196915 *)
Showing 1-7 of 7 results.

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