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 1-10 of 52 results. Next

A197133 Decimal expansion of least x>0 having sin(x) = sin(2*x)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 12 2011

Keywords

Comments

The Mathematica program includes a graph.
Guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected numbers b and c:
b.....c.......x
1.....2.......A197133
1.....3.......A197134
1.....4.......A197135
1.....5.......A197251
1.....6.......A197252
1.....7.......A197253
1.....8.......A197254
2.....1.......A105199, x=arctan(2)
2.....3.......A019679, x=Pi/12
2.....4.......A197255
2.....5.......A197256
2.....6.......A197257
2.....7.......A197258
2.....8.......A197259
3.....1.......A197260
3.....2.......A197261
3.....4.......A197262
3.....5.......A197263
3.....6.......A197264
3.....7.......A197265
3.....8.......A197266
4.....1.......A197267
4.....2.......A195693, x=arctan(1/(golden ratio))
4.....3.......A197268
1.....4*Pi....A197522
1.....3*Pi....A197571
1.....2*Pi....A197572
1.....3*Pi/2..A197573
1.....Pi......A197574
1.....Pi/2....A197575
1.....Pi/3....A197326
1.....Pi/4....A197327
1.....Pi/6....A197328
2.....Pi/3....A197329
2.....Pi/4....A197330
2.....Pi/6....A197331
3.....Pi/3....A197332
3.....Pi/6....A197375
3.....Pi/4....A197333
1.....1/2.....A197376
1.....1/3.....A197377
1.....2/3.....A197378
Pi....1.......A197576
Pi....2.......A197577
Pi....3.......A197578
2*Pi..1.......A197585
3*Pi..1.......A197586
4*Pi..1.......A197587
Pi/2..1.......A197579
Pi/2..2.......A197580
Pi/2..1/2.....A197581
Pi/3..Pi/4....A197379
Pi/3..Pi/6....A197380
Pi/4..Pi/3....A197381
Pi/4..Pi/6....A197382
Pi/6..Pi/3....A197383
Pi/6..Pi/4..........., x=1
Pi/3..1.......A197384
Pi/3..2.......A197385
Pi/3..3.......A197386
Pi/3..1/2.....A197387
Pi/3..1/3.....A197388
Pi/3..2/3.....A197389
Pi/4..1.......A197390
Pi/4..2.......A197391
Pi/4..3.......A197392
Pi/4..1/2.....A197393
Pi/4..1/3.....A197394
Pi/4..2/3.....A197411
Pi/4..1/4.....A197412
Pi/6..1.......A197413
Pi/6..2.......A197414
Pi/6..3.......A197415
Pi/6..1/2.....A197416
Pi/6..1/3.....A197417
Pi/6..2/3.....A197418
Cf. A197476 for a similar table for sin(b*x) = sin(c*x)^2.

Examples

			0.272971849236824950408616...

Links

Crossrefs

Cf. A002194, A197134, A197476 (cos), A333322.

Programs

  • Mathematica
    b = 1; c = 2; f[x_] := Sin[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .1, .3}, WorkingPrecision -> 100]
RealDigits[t] (* A197133 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi}]
(* Second program: *)
RealDigits[ ArcSec[ Root[16 - 16 x^2 + x^6, 3]], 10, 100] // First (* Jean-François Alcover, Feb 19 2013 *)
  • PARI
    asin(2*sin(asin(3*sqrt(3)/8)/3)/sqrt(3)) \\ Gleb Koloskov, Sep 15 2021
  • PARI
    asin(polrootsreal(4*x^3-4*x+1)[2]) \\ Charles R Greathouse IV, Feb 12 2025

Formula

From Gleb Koloskov, Sep 15 2021: (Start)
Equals arcsin(2*sin(arcsin(3*sqrt(3)/8)/3)/sqrt(3))
= arcsin(2*sin(arcsin(A333322)/3)/A002194). (End)

Extensions

Edited and a(99) corrected by Georg Fischer, Jul 28 2021

A019679 Decimal expansion of Pi/12.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Equals cone's volume (radius = 1/2, height = 1) and semi-sphere's volume (radius = 1/2). - Eric Desbiaux, Dec 08 2008
Decimal expansion of least x > 0 having cos(4x) = (cos 3x)^2. See A197476. - Clark Kimberling, Oct 15 2011
Multiplied by 10, decimal expansion of 5*Pi/6. - Alonso del Arte, Aug 19 2013
Volume between a cylinder and the inscribed sphere of diameter 1. - Omar E. Pol, Sep 25 2013

Examples

			Pi/12 = 0.2617993877991494365385536152732919070164307...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4, p. 492.

Links

Crossrefs

Cf. A000796, A003881, A019669, A019670, A019675, A019683, A244978.

Programs

Formula

A003881 - A019673. - Omar E. Pol, Sep 25 2013
Equals Integral_{x = 0..1} x^2*sqrt(1 - x^6) dx. - Peter Bala, Oct 27 2019
Equals Sum_{k>=0} binomial(2*k,k)/((2*k+1)*4^(2*k+1)). - Amiram Eldar, May 30 2021
Constant divided by 10 = Pi/120 = 0.0261799387... = Sum_{n = -oo..oo} 1/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)) (using the Eisenstein summation convention Sum_{n = -oo..oo} = lim_{N -> oo} Sum_{n = -N..N}). Note that 22/7 - Pi = 240*Sum_{n >= 1} 1/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)). - Peter Bala, Nov 28 2021

A197488 Decimal expansion of least x > 0 having cos(6x) = (cos 4x)^2.

Original entry on oeis.org

9, 2, 1, 8, 8, 4, 0, 8, 8, 0, 1, 5, 8, 6, 0, 7, 8, 4, 8, 1, 9, 9, 6, 9, 2, 4, 8, 8, 6, 1, 8, 1, 0, 6, 3, 6, 5, 7, 2, 9, 9, 5, 6, 7, 5, 8, 2, 6, 9, 9, 6, 5, 4, 6, 6, 2, 7, 1, 3, 6, 1, 5, 3, 8, 1, 9, 1, 2, 2, 0, 6, 5, 0, 7, 6, 6, 6, 2, 6, 9, 4, 8, 7, 4, 9, 7, 0, 9, 4, 9, 5, 5, 1, 4, 9, 9, 5, 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.
Also the solution of the least x > 0 satisfying (cos(x))^2 + (sin(3x))^2 = 1/2. See A197739. - Clark Kimberling, Oct 19 2011

Examples

			0.9218840880158607848199692488618106365729956...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 6; c = 4; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .92, .93}, WorkingPrecision -> 100]
RealDigits[t] (* A197488 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 1}]
RealDigits[ ArcCos[ Root[ -2 + 8#^2 - 6#^4 + #^6 & , 5]/2], 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)

Extensions

Digits from a(92) on corrected by Jean-François Alcover, Feb 19 2013

A197515 Decimal expansion of least x>0 having cos(Pi*x)=(cos x)^2.

Original entry on oeis.org

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

			1.501524980455762550683947262886278157...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = Pi; c = 1; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.5, 1.502},
   WorkingPrecision -> 200]
RealDigits[t]  (* A197515 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 3}]

A197521 Decimal expansion of least x>0 having cos(Pi*x/2)=(cos Pi*x/3)^2.

Original entry on oeis.org

3, 5, 2, 1, 3, 3, 7, 8, 2, 9, 5, 7, 1, 7, 1, 5, 6, 9, 8, 6, 9, 1, 9, 8, 8, 5, 6, 4, 4, 5, 4, 9, 1, 7, 9, 7, 7, 3, 0, 9, 1, 8, 1, 3, 9, 4, 7, 3, 3, 6, 8, 8, 7, 1, 9, 5, 4, 9, 1, 8, 4, 8, 6, 2, 5, 1, 5, 5, 9, 0, 6, 0, 9, 6, 1, 0, 2, 5, 9, 8, 8, 8, 9, 7, 4, 9, 7, 5, 6, 9, 0, 0, 3, 9, 4, 9, 7, 1, 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.
Conjecture: the constant here, 3.52133782..., is 3 plus the constant in A197383, the latter being the least t>0 satisfying sin(Pi*t/6)=(sin Pi*t/3)^2.
This number is irrational. I cannot prove it to be algebraic or transcendental. - Charles R Greathouse IV, Feb 16 2025

Examples

			3.521337829571715698691988564454917977309181394...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = Pi/2; c = Pi/3; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 3.5, 3.53}, WorkingPrecision -> 200]
RealDigits[t] (* A197521, appears to be 3+A197383  *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 4}]
RealDigits[ 6*ArcCos[ Root[ -1 - 4# + 4#^3 & , 2]]/Pi, 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)

A197334 Decimal expansion of least x > 0 having cos(x) = cos(4*Pi*x)^2.

Original entry on oeis.org

2, 3, 6, 6, 5, 2, 1, 8, 6, 9, 3, 0, 3, 8, 8, 6, 0, 5, 2, 2, 1, 9, 2, 5, 4, 2, 2, 2, 0, 6, 5, 9, 8, 6, 0, 8, 3, 0, 7, 3, 3, 1, 1, 3, 0, 4, 1, 5, 5, 7, 1, 2, 4, 2, 7, 4, 7, 1, 0, 5, 2, 6, 5, 7, 4, 6, 3, 1, 3, 2, 7, 6, 9, 7, 0, 3, 3, 9, 9, 0, 0, 7, 6, 7, 0, 3, 1, 3, 1, 5, 9, 0, 2, 0, 5, 3, 0, 8, 2
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 15 2025

Examples

			0.23665218693038860522192542220659860830733113...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = 4 Pi; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .235, .237}, WorkingPrecision -> 110]
RealDigits[t] (* A197334 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/8}]

A197335 Decimal expansion of least x > 0 having cos(x) = cos(3*Pi*x)^2.

Original entry on oeis.org

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

			0.309981489070136910318016226860187194650...

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = 3 Pi; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 3, .31}, WorkingPrecision -> 110]
RealDigits[t] (* A197335 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/4}]

A197477 Decimal expansion of least x>0 having cos(x)=(cos 3x)^2.

Original entry on oeis.org

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

Examples

			0.8418355356143638074857326765621643076...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = 3; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .8, .9}, WorkingPrecision -> 200]
RealDigits[t] (* A197477 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi/2}]
RealDigits[ ArcCos[ Root[ 1 - 8# - 8#^2 + 16#^3 + 16#^4 &, 2]], 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)

A197478 Decimal expansion of least x>0 having cos(x)=(cos 4x)^2.

Original entry on oeis.org

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

Examples

			0.6653753198206945999410976241416973212944400...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 1; c = 4; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .6, .7}, WorkingPrecision -> 200]
RealDigits[t] (* A197478 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 1}]

A197479 Decimal expansion of least x>0 having cos(2x)=(cos 3x)^2.

Original entry on oeis.org

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

Examples

			0.681089818242874006185052816327828524925...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 2; c = 3; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .6, .8}, WorkingPrecision -> 200]
RealDigits[t] (* A197479 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 1}]
Showing 1-10 of 52 results. Next

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