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.

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A197495 Decimal expansion of least x > 0 having cos(x) = cos(Pi*x/4)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 15 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			1.2783983856744496808872957323068365766686442...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = Pi/4; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.27, 1.29},
   WorkingPrecision -> 110]
RealDigits[t] (* A197495 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/2}]

A197506 Decimal expansion of least x > 0 having cos(x) = cos(2*Pi*x/3)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 15 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			1.101969565544068669496912835886267221998...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = 2 Pi/3; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t] (* A197506 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/2}]

Extensions

Offset corrected by Georg Fischer, Jul 28 2021

A197507 Decimal expansion of least x > 0 having cos(2*x) = cos(3*Pi*x)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 15 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			0.3942682585355915905633091545751377409550...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = 3 Pi; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .3, .4}, WorkingPrecision -> 110]
RealDigits[t] (* A197507 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/6}]

A197508 Decimal expansion of least x > 0 having cos(2*x) = cos(3*Pi*x/2)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 15 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			0.50629789923405982675001156278369703252865816395828...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = 3 Pi/2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .5, .51}, WorkingPrecision -> 110]
RealDigits[t] (* A197508 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/6}]

Extensions

a(87) onward corrected by Sean A. Irvine, Sep 08 2021

A197509 Decimal expansion of least x > 0 having cos(2*x) = cos(2*Pi*x)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 15 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			0.40574666075124821511608477705830690...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = 2 Pi; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[t] (* A197509 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/6}]

A197510 Decimal expansion of least x > 0 having cos(2*x) = cos(Pi*x)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 15 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			0.66349004605663732078983616691519021...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = Pi; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .6, .7}, WorkingPrecision -> 110]
RealDigits[t]  (* A197510 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 1}]

A197511 Decimal expansion of least x > 0 having cos(2*x) = cos(Pi*x/2)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 15 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			0.63919919283722484044364786615341828833...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = Pi/2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .63, .64}, WorkingPrecision -> 110]
RealDigits[t]  (* A197511 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 1}]
  • PARI
    solve(x=.5,1, cos(2*x) - cos(Pi*x/2)^2) \\ Charles R Greathouse IV, Nov 26 2024

Extensions

a(98) corrected by Sean A. Irvine, Sep 08 2021

A197512 Decimal expansion of least x > 0 having cos(2*x) = cos(Pi*x/3)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 16 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			3.0813735992823264623177056994554111...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = Pi/3; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 3, 3.1}, WorkingPrecision -> 110]
RealDigits[t]   (* A197512 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 4}]

A197513 Decimal expansion of least x > 0 having cos(2*x) = cos(Pi*x/4)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 16 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			2.4052337038770365360381128025228272460264...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = Pi/4; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 2.4, 2.41}, WorkingPrecision -> 110]
RealDigits[t]  (* A197513 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 3}]

A197514 Decimal expansion of least x > 0 having cos(2*x) = cos(Pi*x/6)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 16 2011

Keywords

Comments

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = cos(c*x)^2 for selected b and c.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			2.40347697899992522545129646324811831083...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = Pi/6; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 2.4, 2.41}, WorkingPrecision -> 110]
RealDigits[t]  (* A197514 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 3}]
Previous Showing 21-30 of 52 results. Next

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