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 1-2 of 2 results.

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

A105199 Decimal expansion of arctan(2).

Original entry on oeis.org

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

Views

Author

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005

Keywords

Comments

arctan(2) + A073000 = Pi/2.
arctan(2) is the (minimal) central angle of a regular icosahedron, which is the platonic solid having 20 faces and 12 vertices. The (minimal) central angle is AOB, where A and B are any neighboring pair of vertices and O is the center. To evaluate AOB, it is helpful to start with 12 vertices: (0,c*t,d), (d,0,c*t), (c*t,d,0) where c=(1 or -1) and d=(1 or -1) and t is the golden ratio, (1+sqrt(5))/2. For neighboring vertices, one can select (t,1,0) and (0,t,1). - Clark Kimberling, Feb 10 2009
Lesser interior angle (in radians) of a golden rhombus; i.e., either of the angles bisected by the longer diagonal. A137218 is the greater interior angle. - Rick L. Shepherd, Apr 10 2017
The apex angle in the isosceles triangle that is the triangle with angles A, B and C in which the maximum values of sin(A) + sin(B)*sin(C) is attained. The maximum value is phi (A001622) (Rabinowitz, 2007). - Amiram Eldar, Aug 04 2022
Also <5_1> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024

Examples

			1.107148717794090503017065460...

Links

Crossrefs

Cf. A001622, A195693, A197292, A197376.
Cf. A137218 (larger interior angle of the golden rhombus).

Programs

  • Mathematica
    RealDigits[ArcTan[2], 10, 105][[1]] (* Indranil Ghosh, Apr 10 2017 *)
  • PARI
    default(realprecision, 120);
atan(2) \\ Rick L. Shepherd, Apr 10 2017

Formula

Equals Sum_{k>=1} arctan(8k/(4k^4+5)). [Boros and Moll, from R. J. Mathar, Apr 12 2010]
Equals 2*A195693. - Rick L. Shepherd, Apr 10 2017
Equals arcsin(2/sqrt(5)) = arccos(1/sqrt(5)). - Amiram Eldar, Aug 04 2022
Equals 2 - log(5) + (Integral_{x=0..2} log(1 + x^2) dx)/2. - Vaclav Kotesovec, Oct 06 2023
Equals 3*A197292 = A197376/2. - Hugo Pfoertner, Nov 06 2024

Extensions

Offset corrected by R. J. Mathar, Apr 12 2010
Showing 1-2 of 2 results.

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