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 1-9 of 9 results.

A196915 Decimal expansion of the slope (negative) at the point of tangency of the curves y=1/(1+x^2) and y=c*cos(x), where c is given by A196914.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 07 2011

Keywords

Examples

			x=-0.60762223769686865859001002682026364322748...

Crossrefs

Cf. A196914, A196913.

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 *)

A196816 Decimal expansion of the least x>0 satisfying 1/(1+x^2)=cos(x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 06 2011

Keywords

Examples

			1.10250582440641604357105015502222407388481058200...

Links

Crossrefs

Cf. A196914, A196817, A196818, A196819, A196820, A196821.

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]
  • PARI
    solve(x=1, 1.5, cos(x)*(1+x^2) - 1) \\ Michel Marcus, Feb 10 2015

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 *)

A196625 Decimal expansion of the number c for which the curve y=1/x is tangent to the curve y=cos(x-c), and 0 < x < 2*Pi; c = sqrt(r) - arccsc(r), where r = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 05 2011

Keywords

Comments

Let r=(1+sqrt(5))/2, the golden ratio. Let u=sqrt(r) and v=1/x. Let c=sqrt(r)-arccsc(r). The curve y=1/x is tangent to the curve y=cos(x-c) at (u,v), and the slope of the tangent line is r-1.
Guide to constants c associated with tangencies:
A196610: 1/x and c*cos(x)
A196619: 1/x - c and cos(x)
A196774: 1/x + c and sin(x)
A196625: 1/x and cos(c-x)
A196772: 1/x and sin(x+c)
A196758: 1/x and c*sin(x)
A196765: c/x and sin(x)
A196823: 1/(1+x^2) and -c+cos(x)
A196914: 1/(1+x^2) and c*cos(x)
A196832: 1/(1+x^2) and c*sin(x)
A197016: x=0, y=0, and cos(x)

Examples

			c=0.60578021702155370914841756575969877104...

Crossrefs

Cf. A139339, A196772.

Programs

  • Mathematica
    Plot[{1/x, Cos[x - 0.60578]}, {x, 0, 2 Pi}]
r = GoldenRatio; xt = Sqrt[r];
x1 = N[xt, 100]
RealDigits[x1]     (* A139339 *)
c = Sqrt[r] - ArcCsc[r];
c1 = N[c, 100]
RealDigits[c1]     (* A196625 *)
slope = N[r - Sqrt[5], 100]
RealDigits[slope]  (* -1+A001622; -1+golden ratio *)

Extensions

a(99) corrected by Georg Fischer, Jul 19 2021
Showing 1-9 of 9 results.

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