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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A197480 Decimal expansion of least x>0 having cos(2x)=(cos 4x)^2.

Original entry on oeis.org

5, 6, 8, 8, 7, 1, 9, 6, 6, 4, 5, 2, 7, 2, 7, 7, 7, 8, 8, 9, 4, 7, 2, 4, 9, 3, 0, 0, 2, 7, 5, 0, 4, 1, 7, 4, 7, 9, 2, 4, 0, 2, 1, 4, 5, 1, 7, 4, 7, 8, 7, 6, 3, 6, 0, 0, 7, 5, 9, 1, 2, 6, 3, 3, 6, 8, 0, 4, 9, 1, 7, 3, 6, 7, 3, 6, 3, 6, 0, 8, 8, 9, 9, 4, 0, 1, 6, 4, 0, 2, 5, 8, 8, 2, 2, 3, 6, 3, 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.

Examples

			0.5688719664527277788947249300275041747924021...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = 4; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .55, .57}, WorkingPrecision -> 200]
RealDigits[t] (* A197480 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/4}]
RealDigits[ ArcCos[ Root[ -2 + 4#^2 - 4#^4 + #^6 & , 2]/2], 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)

A197482 Decimal expansion of least x>0 having cos(3x)=(cos 2x)^2.

Original entry on oeis.org

1, 8, 4, 3, 7, 6, 8, 1, 7, 6, 0, 3, 1, 7, 2, 1, 5, 6, 9, 6, 3, 9, 9, 3, 8, 4, 9, 7, 7, 2, 3, 6, 2, 1, 2, 7, 3, 1, 4, 5, 9, 9, 1, 3, 5, 1, 6, 5, 3, 9, 9, 3, 0, 9, 3, 2, 5, 4, 2, 7, 2, 3, 0, 7, 6, 3, 8, 2, 4, 4, 1, 3, 0, 1, 5, 3, 3, 2, 5, 3, 8, 9, 7, 4, 9, 9, 4, 1, 8, 9, 9, 1, 0, 2, 9, 9, 9, 1, 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.

Examples

			1.843768176031721569639938497723621273145...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 3; c = 2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.8, 1.9}, WorkingPrecision -> 200]
RealDigits[t] (* A197482 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 2.5}]

A197483 Decimal expansion of least x>0 having cos(3x)=(cos 4x)^2.

Original entry on oeis.org

4, 8, 2, 3, 9, 5, 0, 9, 8, 8, 1, 1, 1, 2, 6, 5, 7, 7, 2, 3, 0, 9, 1, 1, 3, 9, 5, 0, 2, 4, 5, 6, 5, 4, 4, 2, 8, 4, 2, 0, 7, 8, 7, 1, 4, 4, 9, 5, 2, 9, 7, 2, 8, 3, 0, 9, 9, 1, 3, 5, 2, 3, 9, 6, 5, 1, 4, 0, 9, 1, 0, 6, 5, 4, 5, 6, 0, 9, 7, 1, 3, 1, 6, 8, 1, 7, 2, 4, 8, 9, 8, 7, 7, 6, 9, 3, 5, 0, 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.

Examples

			0.48239509881112657723091139502456544284207...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 3; c = 4; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .47, .5}, WorkingPrecision -> 200]
RealDigits[t] (* A197483 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 0.6}]

A197485 Decimal expansion of least x>0 having cos(4x)=(cos(6x))^2.

Original entry on oeis.org

3, 4, 0, 5, 4, 4, 9, 0, 9, 1, 2, 1, 4, 3, 7, 0, 0, 3, 0, 9, 2, 5, 2, 6, 4, 0, 8, 1, 6, 3, 9, 1, 4, 2, 6, 2, 4, 6, 2, 5, 9, 2, 9, 2, 8, 1, 2, 8, 7, 6, 1, 2, 7, 9, 8, 1, 1, 4, 8, 7, 9, 0, 7, 7, 4, 0, 6, 1, 7, 7, 1, 9, 6, 6, 4, 6, 4, 6, 4, 0, 7, 1, 1, 3, 2, 7, 6, 1, 3, 6, 8, 9, 3, 2, 9, 1, 6, 0, 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.

Examples

			0.34054490912143700309252640816391426246259292812...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 4; c = 6; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .34, .35}, WorkingPrecision -> 100]
RealDigits[t] (* A197485 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 1/2}]
RealDigits[ ArcTan[ Sqrt[ Root[7#^4 - 68#^3 + 106#^2 - 68# + 7&, 1] ] ], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

A197486 Decimal expansion of least x>0 having cos(4x)=(cos(8x))^2.

Original entry on oeis.org

2, 8, 4, 4, 3, 5, 9, 8, 3, 2, 2, 6, 3, 6, 3, 8, 8, 9, 4, 4, 7, 3, 6, 2, 4, 6, 5, 0, 1, 3, 7, 5, 2, 0, 8, 7, 3, 9, 6, 2, 0, 1, 0, 7, 2, 5, 8, 7, 3, 9, 3, 8, 1, 8, 0, 0, 3, 7, 9, 5, 6, 3, 1, 6, 8, 4, 0, 2, 4, 5, 8, 6, 8, 3, 6, 8, 1, 8, 0, 4, 4, 4, 9, 7, 0, 0, 8, 2, 0, 1, 2, 9, 4, 1, 1, 1, 8, 1, 7
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.

Examples

			0.284435983226363889447362465013752087396201072...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 4; c = 8; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .28, .29}, WorkingPrecision -> 100]
RealDigits[t] (* A197486 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 0.4}]
RealDigits[ 1/2*ArcTan[ Sqrt[ Root[#^3 - 5#^2 + 19# - 7&, 1]]], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

A197490 Decimal expansion of least x > 0 having cos(x) = cos(2*Pi*x)^2.

Original entry on oeis.org

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

			0.564425476062659099384003228937788297677...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = 2 Pi; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .56, .57}, WorkingPrecision -> 110]
RealDigits[t] (* A197490 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/4}]

A197491 Decimal expansion of least x > 0 having cos(x) = cos(3*Pi*x/2)^2.

Original entry on oeis.org

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

			0.57854892542571838320407367324880211828681701...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = 3 Pi/2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .56, .58}, WorkingPrecision -> 110]
RealDigits[t] (* A197491 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/3}]

A197492 Decimal expansion of least x > 0 having cos(x) = cos(Pi*x)^2.

Original entry on oeis.org

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

			0.811493321502496430232169554116613810...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = Pi; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .81, .82}, WorkingPrecision -> 110]
RealDigits[t] (* A197492 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/3}]

A197493 Decimal expansion of least x > 0 having cos(x) = cos(Pi*x/2)^2.

Original entry on oeis.org

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

			1.32698009211327464157967233383038042664300...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = Pi/2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.32, 1.33},
   WorkingPrecision -> 110]
RealDigits[t] (* A197493 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 2}]

A197494 Decimal expansion of least x>0 having cos(x)=(cos(Pi*x/3))^2.

Original entry on oeis.org

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

			1.566023613622289702303820823948946110500...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = Pi/3; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.5, 1.6}, WorkingPrecision -> 110]
RealDigits[t] (* A197494 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 2}]
Previous Showing 11-20 of 52 results. Next

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