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.

A197476 Decimal expansion of least x>0 having cos(x) = cos(2*x)^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 15 2011

Keywords

Comments

The Mathematica program includes a graph. Guide for least x>0 satisfying cos(b*x) = cos(c*x)^2, for selected b and c:
b.....c......x
1.....2.......A197476
1.....3.......A197477
1.....4.......A197478
2.....1.......A000796, Pi
2.....3.......A197479
2.....4.......A197480
3.....1.......A019669, Pi/2
3.....2.......A197482
3.....4.......A197483
4.....1.......A168229, arctan(sqrt(7))
4.....2.......A019669, Pi/2
4.....3.......A019679, Pi/12
4.....6.......A197485
4.....8.......A197486
6.....2.......A003881, Pi/4
6.....3.......A019670, Pi/3, tangency point
6.....4.......A197488
6.....8.......A197489
1.....4*Pi....A197334
1.....3*Pi....A197335
1.....2*Pi....A197490
1.....3*Pi/2..A197491
1.....Pi......A197492
1.....Pi/2....A197493
1.....Pi/3....A197494
1.....Pi/4....A197495
1.....2*Pi/3..A197506
2.....3*Pi....A197507
2.....3*Pi/2..A197508
2.....2*Pi....A197509
2.....Pi......A197510
2.....Pi/2....A197511
2.....Pi/3....A197512
2.....Pi/4....A197513
2.....Pi/6....A197514
Pi....1.......A197515
Pi....2.......A197516
Pi....1/2.....A197517
2*Pi..1.......A197518
2*Pi..2.......A197519
2*Pi..3.......A197520
Pi/2..Pi/3....A197521
Pi/2..Pi/6....3
Pi/3..1.......A197582
Pi/3..2.......A197583
Pi/3..1/3.....A197584
See A197133 for a guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected b and c.

Examples

			1.137743932905455557789449860055008349584...

Links

Crossrefs

Cf. A197133.

Programs

  • Mathematica
    b = 1; c = 2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.1, 1.3}, WorkingPrecision -> 200]
RealDigits[t] (* A197476 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 2}]
(* or *)
RealDigits[ ArcCos[ ((19 - 3*Sqrt[33])^(1/3) + (19 + 3*Sqrt[33])^(1/3) - 2)/6], 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)

Extensions

Edited by Georg Fischer, Jul 28 2021

A038198 Numbers n such that n^2 + 7 is a power of 2.

Original entry on oeis.org

1, 3, 5, 11, 181
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The exponents of the corresponding powers of 2 are 3, 4, 5, 7, 15 (see Ramanujan). - N. J. A. Sloane, Jun 01 2014
The terms lead to identities resembling Machin's Pi/4 = arctan(1/1) = 4*arctan(1/5) - arctan(1/239), for example, arctan(sqrt(7)/1) = 5*arctan(sqrt(7)/11) + 2*arctan(sqrt(7)/181), which can also be expressed as arcsin(sqrt(7/2^3)) = 5*arcsin(sqrt(7/2^7)) + 2*arcsin(sqrt(7/2^15)) (cf. A168229). - Joerg Arndt, Nov 09 2012

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 181, p. 56, Ellipses, Paris 2008.
  • L. J. Mordell, Diophantine Equations, Academic Press, NY, 1969, p. 205.
  • S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. See Question 464, p. 327. - N. J. A. Sloane, Jun 01 2014

Links

Crossrefs

Cf. A060728, A002249, A107920.

Programs

  • Mathematica
    ok[n_] := Reduce[k>0 && n^2 + 7 == 2^k, k, Integers] =!= False; Select[Range[1000], ok] (* Jean-François Alcover, Sep 21 2011 *)
  • PARI
    [x | n<-[0..99], issquare(2^n-7,&x)] \\ M. F. Hasler, Mar 11 2024

A195699 Decimal expansion of arcsin(sqrt(1/8)) and of arccos(sqrt(7/8)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 23 2011

Keywords

Examples

			arcsin(sqrt(1/8)) = 0.3613671239067078055891886763206666...

Links

Crossrefs

Cf. A195700, A195703, A195704.

Programs

  • Magma
    [Arcsin(Sqrt(1/8))]; // G. C. Greubel, Nov 18 2017
  • Mathematica
    r = Sqrt[1/8];
N[ArcSin[r], 100]
RealDigits[%]  (* A195699 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A168229 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188615 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195704 *)
  • PARI
    asin(sqrt(1/8)) \\ G. C. Greubel, Nov 18 2017

Formula

From Peter Bala, Jan 14 2022: (Start)
Equals (1/2)*arccos(3/4) = arctan(sqrt(7)/7).
Equals sqrt(7)*Sum_{n >= 0} 1/((16*n + 8)*(2^n)*binomial(2*n,n)).
Equals sqrt(2)*Sum_{n >= 0} binomial(2*n,n)/((8*n + 4)*32^n). (End)

A195704 Decimal expansion of arccos(-sqrt(1/8)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 23 2011

Keywords

Examples

			arccos(-sqrt(1/8)) = 1.93216345070...

Links

Crossrefs

Cf. A195699.

Programs

  • Magma
    [Arccos(-Sqrt(1/8))]; // G. C. Greubel, Nov 18 2017
  • Mathematica
    r = Sqrt[1/8];
N[ArcSin[r], 100]
RealDigits[%]  (* A195699 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A168229 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188615 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195704 *)
  • PARI
    acos(-sqrt(1/8)) \\ G. C. Greubel, Nov 18 2017

Formula

Equals Pi - arcsin(sqrt(7/2)/2) = Pi - arctan(sqrt(7)). - Amiram Eldar, Jul 09 2023
Showing 1-4 of 4 results.

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