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.

A197757 Duplicate of A197488.

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 19 2011

Keywords

Examples

			0.92188408801586078481996924886181063657299567...

Crossrefs

Cf. A197739

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

A197739 Decimal expansion of least x>0 having sin(2x)=3*sin(6x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 18 2011

Keywords

Comments

This constant is the least x>0 for which the function f(x)=(sin(x))^2+(cos(3x))^2 has its maximal value. Least positive solutions of the equations f(x)=m/2, f(x)=m/3, f(x)=1, and f(x)=1/2 are given by sequences shown in the guide below.
In general, suppose that b and c are distinct positive real numbers. Let f(x)=(sin(bx))^2+cos((cx))^2. The extrema of f are the solutions of b*sin(2bx)=c*sin(2cx).
In the following guide, constants x given by the sequences (or explicit number) listed for each b,c are, in this order:
(1) least x>0 such that f(x)=(its maximum, m)
(2) m, the maximum of f
(3) least x>0 having f(x)=m/2
(4) least x>0 having f(x)=m/3
(5) least x>0 having f(x)=1
(6) least x>0 having f(x)=1/2
...
(b,c)=(1,2): A195700, x=25/16, A197589, A197591,
A019670, A197592
(b,c)=(1,3): A197739, A197588, A197590, A197755,
A003881, A197488
(b,c)=(1,4): A197758, A197759, A197760, A197761,
A019692 (x=pi/5), A003881
(b,c)=(1,pi): A197821, A197822, A197823, A197824,
A197726, A197826
(b,c)=(1,2*pi): A197827, A197828, A197829, A197830,
A197700, A197832
(b,c)=(1,3*pi): A197833, A197834, A197835, A197836,
A197837, A197838

Examples

			0.47765830906225463908192855125787887712170734750500...

Links

Crossrefs

Cf. A197739, A197588.

Programs

  • Mathematica
    b = 1; c = 3;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .47, .48}, WorkingPrecision -> 110]
RealDigits[r]  (* A197739 *)
m = s[r]
RealDigits[m]  (* A197588 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, 0.7, 0.8}, WorkingPrecision -> 110]
RealDigits[t]  (* A197590 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, 0.8, 0.9}, WorkingPrecision -> 110]
RealDigits[t]  (* A197755 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, 0.7, 0.8}, WorkingPrecision -> 110]
RealDigits[t]  (* A003881 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, .9, .93}, WorkingPrecision -> 110]
RealDigits[t]  (* A197488 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
RealDigits[ ArcTan[ Sqrt[ 2-Sqrt[3] ] ], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

A197489 Decimal expansion of least x>0 having cos(6x)=(cos 8x)^2.

Original entry on oeis.org

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

Examples

			0.2411975494055632886154556975122827214210...

Links

Crossrefs

Cf. A197476.

Programs

  • Mathematica
    b = 6; c = 8; 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}]
Showing 1-4 of 4 results.

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