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 1-3 of 3 results.

A197682 Decimal expansion of Pi/(2 + 2*Pi).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 17 2011

Keywords

Comments

The number Pi/(2 + 2*Pi) is the least x > 0 such that sin(x) = cos(Pi*x).
If b and c are distinct real numbers, the solutions of sin(bx) = cos(cx) are x = (k - 1/2)*Pi/(b + c), where k runs through the integers. Thus, if b > 0 and c > 0, the least solution x > 0 is Pi/(2*b + 2*c), so that this is also the least x > 0 for which sin(c*x) = cos(b*x). Related sequences, each with a Mathematica program which includes a graph:
...
b.....c.......sequence........x
1.....2.......A019673........ x = Pi/6
1.....3.......A019678........ x = Pi/8
1.....4.......(A000796)/10... x = Pi/10
1.....Pi......A197682........ x = Pi/(2+2*Pi)
1.....2*Pi....A197683........ x = Pi/(2+4*Pi)
1.....1/Pi....A197684........ x = Pi^2/(2+2*Pi)
1.....2/Pi....A197685........ x = Pi^2/(4+2*Pi)
1.....Pi/2....A197686........ x = Pi/(2+Pi)
1.....Pi/3....A197687........ x = 3*Pi/(6+2*Pi)
1.....Pi/4....A197688........ x = 2*Pi/(4+Pi)
1.....Pi/6....A197689........ x = 3*Pi/(6+Pi)
2.....3.......(A000796)/10... x = Pi/10
2.....Pi......A197690........ x = Pi/(4+2*Pi)
2.....2*Pi....A197691........ x = Pi/(4+4*Pi)
2.....1/Pi....A197692........ x = Pi^2/(2+4*Pi)
2.....2/Pi....A197693........ x = Pi^2/(4+4*Pi)
2.....Pi/2....A197694........ x = Pi/(4+Pi)
3.....Pi......A197695........ x = Pi/(2+2*Pi)
3.....2*Pi....A197696........ x = Pi/(6+4*Pi)
3.....1/Pi....A197697........ x = Pi^2/(2+6*Pi)
3.....2/Pi....A197698........ x = Pi^2/(4+6*Pi)
3.....Pi/2....A197699........ x = Pi/(6+Pi)
1/2...Pi......A197700........ x = Pi/(1+2*Pi)
1/2...2*Pi....A197701........ x = Pi/(1+4*Pi)
1/2...1/Pi....A197724........ x = Pi^2/(2+Pi)
1/2...2/Pi....A197725........ x = Pi^2/(4+Pi)
1/2...Pi/2....A197726........ x = Pi/(1+Pi)
1/2...Pi/4....A197727........ x = 2*Pi/(2+Pi)
1/3...Pi/3....A197728........ x = 3*Pi/(2+2*Pi)
1/3...Pi/6....A197729........ x = 3*Pi/(2+Pi)
2/3...Pi/6....A197730........ x = 3*Pi/(4+Pi)
1/4...Pi......A197731........ x = 2*Pi/(1+4*Pi)
1/4...Pi/2....A197732........ x = 2*Pi/(1+2*Pi)
1/4...Pi/4....A197733........ x = 2*Pi/(1+Pi)
1/5...Pi/5....10*A197691..... x = 5*Pi/(2+2*Pi)
1/6...Pi/6....A197735........ x = 3*Pi/(1+Pi)
1/8...Pi/8....A197736........ x = 4*Pi/(1+Pi)

Examples

			0.37927349649738807267221534452244643...

Links

Crossrefs

Cf. A197683.

Programs

  • Mathematica
    b = 1; c = Pi;
t = x /. FindRoot[Sin[b*x] == Cos[c*x], {x, .3, .4}]
N[Pi/(2*b + 2*c), 110]
RealDigits[%]  (* A197682 *)
Simplify[Pi/(2*b + 2*c)]
Plot[{Sin[b*x], Cos[c*x]}, {x, 0, Pi}]
  • PARI
    1/(2/Pi+2) \\ Charles R Greathouse IV, Sep 27 2022

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 *)

A197827 Decimal expansion of least x > 0 having sin(2*x) = 2*Pi*sin(4*Pi*x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 19 2011

Keywords

Comments

For a discussion and guide to related sequences, see A197739.

Examples

			0.24405505512124668685356429784849535656...

Crossrefs

Cf. A197739.

Programs

  • Mathematica
    b = 1; c = 2 Pi;
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, .24, .25}, WorkingPrecision -> 110]
RealDigits[r]  (* A197827 *)
m = s[r]
RealDigits[m]  (* A197828 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, .4, .42}, WorkingPrecision -> 110]
RealDigits[t] (* A197829 *)
Plot[{s[x], d}, {x, 0, .7}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, .91, .92}, WorkingPrecision -> 110]
RealDigits[t] (* A197830 *)
Plot[{s[x], d}, {x, 0, Pi/2}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[t] (* A197700 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, .93, .94}, WorkingPrecision -> 110]
RealDigits[t] (* A197832 *)
Plot[{s[x], d}, {x, 0, 1}, AxesOrigin -> {0, 0}]

Extensions

Definition corrected by Georg Fischer, Aug 10 2021
Showing 1-3 of 3 results.

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