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 1-4 of 4 results.

A197016 Decimal expansion of the radius of the circle tangent to the curve y=cos(x) and to the positive x and y axes.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 08 2011

Keywords

Comments

Let (x,y) denote the point of tangency. Then
x=0.65099256993050253383380179140902527170294599...
y=0.79548271667012269646991174255794794798663548...
slope=-0.6059762763335882427824587356062000...
(The Mathematica program includes a graph.)

Examples

			radius=0.428778536030612863613691741048999...

Crossrefs

Cf. A197017, A197018, A197019, A197020.

Programs

  • Mathematica
    r = .428;
Show[Plot[Cos[x], {x, 0, Pi}],
ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
f[x_] := (x - Sin[x] Cos[x])/(1 - Sin[x]);
t = x /.FindRoot[Cos[x] == f[x] + Sqrt[2*f[x]*x - x^2], {x, .5, 1}, WorkingPrecision -> 100]
x1 = Re[t]  (* x coordinate of tangency point *)
y = Cos[x1] (* y coordinate of tangency point *)
radius = f[x1]
RealDigits[radius] (* A197016 *)
slope = -Sin[x1]   (* slope at tangency point *)

A197017 Decimal expansion of the radius of the circle tangent to the curve y=cos(2x) and to the positive x and y axes.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 08 2011

Keywords

Comments

Let (x,y) denote the point of tangency. Then
x=0.556627409764774263651183045638839616840052780212...
y=0.441743828977740325730277185387438343947805907493...
slope=-0.5283257380737094443139057566841614427843590...
(The Mathematica program includes a graph.)

Examples

			radius=0.2971056352748227167122144365263161994072960710...

Crossrefs

Cf. A197016, A197018, A197019, A197020.

Programs

  • Mathematica
    r = .297; c = 2;
Show[Plot[Cos[c*x], {x, 0, Pi}],
ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}],PlotRange -> All, AspectRatio -> Automatic]
f[x_] := (x - c*Sin[c*x] Cos[c*x])/(1 - c*Sin[c*x]);
t = x /. FindRoot[Cos[c*x] == f[x] + Sqrt[2*f[x]*x - x^2], {x, .5, 1}, WorkingPrecision -> 100]
x1 = Re[t]    (* x coordinate of tangency point *)
y = Cos[c*x1] (* y coordinate of tangency point *)
radius = f[x1]
RealDigits[radius] (* A197017 *)
slope = -Sin[x1] (* slope at tangency point *)

A197018 Decimal expansion of the radius of the circle tangent to the curve y=cos(3x) and to the positive x and y axes.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 08 2011

Keywords

Comments

Let (x,y) denote the point of tangency. Then
x=0.4252834568497833490618545391964703664552948...
y=0.2906881405190418936802785128662388404186594...
slope=-0.41257900534470955829852211550705870735...
(The Mathematica program includes a graph.)

Examples

			radius=0.218729488803644065897285223268121049303636199...

Crossrefs

Cf. A197016, A197017, A197019, A197020.

Programs

  • Mathematica
    r = .219; c = 3;
Show[Plot[Cos[c*x], {x, 0, Pi}],
ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
f[x_] := (x - c*Sin[c*x] Cos[c*x])/(1 - c*Sin[c*x]);
t = x /. FindRoot[Cos[c*x] == f[x] + Sqrt[2*f[x]*x - x^2], {x, .5, 1}, WorkingPrecision -> 100]
x1 = Re[t]      (* x coordinate of tangency point *)
y = Cos[c*x1]   (* y coordinate of tangency point *)
radius = f[x1]
RealDigits[radius] (* A197018 *)
slope = -Sin[x1] (* slope at tangency point *)

A345644 Decimal expansion of the radius of the circle tangent to the curves y=cos(x), y=-cos(x) and to the y-axis for x in [0,Pi/2].

Original entry on oeis.org

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

Views

Author

Gleb Koloskov, Jun 21 2021

Keywords

Comments

Let r and (x,y) denote the radius of the circle and the point of tangency in the first quadrant, respectively.
Then r in [0,1] is the root of equation cos(r+sqrt(r^2-1+sqrt(1-r^2)))^2 = 1-sqrt(1-r^2),
r = 0.642707872546532445779211778468607918285047824...,
x = r+sqrt(r^2-1+sqrt(1-r^2)) = 1.066010072972971718857583783392083793389510385...,
y = sqrt(1-sqrt(1-r^2)) = 0.483620364074368181073730094271148302685427120...

Examples

			0.642707872546532445779211778468607918285047824...

Crossrefs

Cf. A197016, A197017, A197018, A197019, A197020.

Programs

  • Mathematica
    r = r /. FindRoot[Cos[r + Sqrt[-1 + r^2 + Sqrt[1 - r^2]]]^2 == 1 - Sqrt[1 - r^2], {r, 1/2}]; Show[Plot[Cos[x], {x, 0, Pi/2}], Plot[-Cos[x], {x, 0, Pi/2}], Graphics[Circle[{r, 0}, r]], PlotRange -> All, AspectRatio -> Automatic] (* Vaclav Kotesovec, Jul 01 2021 *)
  • PARI
    solve(r=0,1,cos(r+sqrt(r^2-1+sqrt(1-r^2)))^2-1+sqrt(1-r^2))
Showing 1-4 of 4 results.

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