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 14 results. Next

A197002 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/4) orthogonally.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 09 2011

Keywords

Comments

See the Mathematica program for a graph.
xo=0.36954256660758032082765604383693...
yo=0.40397275329951720931896174006631...
m=1.093169744985016922088153214160579...
|OP|=0.547499492185436214325204150357...

Links

Crossrefs

Cf. A197003, A196996, A197000, A003957.

Programs

  • Maple
    evalf(solve(cos(x)=x,x)/2, 140);  # Alois P. Heinz, Feb 20 2024
  • Mathematica
    c = Pi/4;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197002 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197003 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, 1}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]
  • PARI
    solve(x=0,1,cos(x)-x)/2  \\ Gleb Koloskov, Jun 16 2021

Formula

Equals d/2 = A003957/2, where d is the Dottie number. - Gleb Koloskov, Jun 16 2021

A197000 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=1+cos(x) orthogonally.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 09 2011

Keywords

Comments

See the Mathematica program for a graph.

Examples

			xo=1.2488014367215508560475125020128381535587614... <------ this constant
yo=1.3164595537507515212878992732671186100622603...
m=1.05417844265684217515747734305673483746142104...
|OP|=1.81454423617045980814297669595599066552030...

Crossrefs

Cf. A197001, A196996, A197002.

Programs

  • Mathematica
    c = 1;
xo = x /.
  FindRoot[x == Sin[x] (c + Cos[x]), {x, 1, 1.3}, WorkingPrecision -> 100]
RealDigits[xo] (* A197000 *)
m = 1/Sin[xo]
RealDigits[m]  (* A197001 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{c + Cos[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi}],  ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 2}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]
  • PARI
    solve(x=1,2, sin(x)*(cos(x)+1)-x) \\ Charles R Greathouse IV, Feb 03 2025

A196997 Decimal expansion of m, where y=m*x is the line through (0,0) which meets the curve y=cos(3*x) orthogonally at a point (x,y) satisfying 0 < x < 2*Pi.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 09 2011

Keywords

Comments

See the Mathematica program for a graph.
xo=0.9350272884749678361451944275323...
yo=0.3301955980451199836007253971727...
m=0.35314006565912096755666111412785...
|OP|=0.99161744799152518925689622748...

Crossrefs

Cf. A196996, A197000, A197002.

Programs

  • Mathematica
    c = 3;
xo = x /. FindRoot[0 == x + c*Sin[c*x] Cos[c*x], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A196996 *)
m = Sin[c*xo]/xo
RealDigits[m]  (* A196997 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Sin[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi/c}], ContourPlot[{y == m*x}, {x, 0, 1.5}, {y, -.1, 1}], PlotRange -> All, AspectRatio -> Automatic]

A197006 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/6) orthogonally.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

See the Mathematica program for a graph.
xo=0.460885580860965976987981282513698...
yo=0.553292712300593256734925495541442...
m=1.2004990723879979061250465124427113...
|OP|=0.7201030093885853693640956082816...

Crossrefs

Cf. A197007, A197002, A196996, A197000.

Programs

  • Mathematica
    c = Pi/6;
xo = x /.  FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197006 *)
m = 1/Sin[xo + c]
RealDigits[m] (* A197007 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
AspectRatio -> Automatic, AxesOrigin -> Automatic]

A196998 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) at which a line y=m*x meets the curve y=cos(5x/2) orthogonally.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 09 2011

Keywords

Comments

See the Mathematica program for a graph.
xo=1.055537135075475249854148417892290354122...
yo=0.481836913462240473673427172075977637742...
m=0.4564850420234501281397606474354137170643...
|OP|=1.1603126538559168441096914160911620183...

Crossrefs

Cf. A196996, A197000, A197002.

Programs

  • Mathematica
    c = 5/2;
xo = x /.  FindRoot[0 == x + c*Sin[c*x] Cos[c*x], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A196998 *)
m = Sin[c*xo]/xo
RealDigits[m]  (* A196999 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Sin[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi/c}],
 ContourPlot[{y == m*x}, {x, 0, Pi/c}, {y, -.1, 1}], PlotRange -> All, AspectRatio -> Automatic]

A196999 Decimal expansion of slope of the line y=mx which meets the curve y=cos(5x/2) orthogonally (as in A196998).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 09 2011

Keywords

Comments

See the Mathematica program for a graph.
xo=1.055537135075475249854148417892290354122...
yo=0.481836913462240473673427172075977637742...
m=0.4564850420234501281397606474354137170643...
|OP|=1.1603126538559168441096914160911620183...

Crossrefs

Cf. A196996, A196997, A197000, A197002.

Programs

  • Mathematica
    c = 5/2;
xo = x /.  FindRoot[0 == x + c*Sin[c*x] Cos[c*x], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A196998 *)
m = Sin[c*xo]/xo
RealDigits[m]  (* A196999 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Sin[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi/c}],
 ContourPlot[{y == m*x}, {x, 0, Pi/c}, {y, -.1, 1}], PlotRange -> All, AspectRatio -> Automatic]

A197001 Decimal expansion of the slope of the line y=mx which meets the curve y=1+cos(x) orthogonally over the interval [0, 2*Pi] (as in A197000).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 09 2011

Keywords

Comments

See the Mathematica program for a graph.
xo=1.2488014367215508560475125020128381535587614...
yo=1.3164595537507515212878992732671186100622603...
m=1.05417844265684217515747734305673483746142104...
|OP|=1.81454423617045980814297669595599066552030...

Crossrefs

Cf. A196700, A196996, A197002.

Programs

  • Mathematica
    c = 1;
xo = x /.
  FindRoot[x == Sin[x] (c + Cos[x]), {x, 1, 1.3}, WorkingPrecision -> 100]
RealDigits[xo] (* A197000 *)
m = 1/Sin[xo]
RealDigits[m]  (* A197001 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{c + Cos[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi}],  ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 2}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]

A197003 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/4) orthogonally over the interval [0, 2*Pi] (as in A197002).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 09 2011

Keywords

Comments

See the Mathematica program for a graph.
xo=0.3695425666075803208276560438369...
yo=0.4039727532995172093189617400663...
m=1.09316974498501692208815321416057...
|OP|=0.54749949218543621432520415035...

Crossrefs

Cf. A003957, A197002, A196996, A197000.

Programs

  • Mathematica
    c = Pi/4;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197002 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197003 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, 1}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]
  • PARI
    my(d=solve(x=0,1,cos(x)-x)); sqrt(2-2*sqrt(1-d^2))/d \\ Gleb Koloskov, Jun 16 2021

Formula

Equals sqrt(2-2*sqrt(1-d^2))/d where d = A003957. - Gleb Koloskov, Jun 16 2021

A197004 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/3) orthogonally.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

See the Mathematica program for a graph.
xo=0.255465286103853596695882696613320272654788...
yo=0.264932084602776862434116494762571068650190...
m=1.0370570837365150046614795837584277605222343...
|OP|=0.3680373919265496189530095416155881110455...

Crossrefs

Cf. A197002, A196996, A197000.

Programs

  • Mathematica
    c = Pi/3;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197004 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197005 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}],
ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
AspectRatio -> Automatic, AxesOrigin -> Automatic]

A197005 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/3) orthogonally over the interval [0, 2*Pi] (as in A197004).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 10 2011

Keywords

Comments

See the Mathematica program for a graph.
xo=0.255465286103853596695882696613320272654788...
yo=0.264932084602776862434116494762571068650190...
m=1.0370570837365150046614795837584277605222343...
|OP|=0.3680373919265496189530095416155881110455...

Crossrefs

Cf. A197004, A197002, A196996, A197000.

Programs

  • Mathematica
    c = Pi/3;
xo = x /.  FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197004 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197005 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}],
ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
AspectRatio -> Automatic, AxesOrigin -> Automatic]
Showing 1-10 of 14 results. Next

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