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 1-9 of 9 results.

A198679 Decimal expansion of the absolute minimum of sin(x)+sin(2x)+sin(3x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 29 2011

Keywords

Comments

See A198677 for a guide to related sequences.

Examples

			x=5.615894235655113730430510613029770525768423...
min=-2.499607604320056604368126285618043218243...

Crossrefs

Cf. A198677.

Programs

  • Mathematica
    n = 3; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198679 *)
RealDigits[v] (* A198680 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
(* Second program: *)
Root[-108 + 2124x^2 - 4892x^4 + 729x^6, 1] // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Feb 19 2013 *)

A198729 Decimal expansion of the absolute minimum of sin(x)+sin(2x)+sin(3x)+sin(4x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 29 2011

Keywords

Comments

See A198677 for a guide to related sequences.

Examples

			x=5.764569445617181108633794876455734774725510...
min=-3.232384871111452330884269520493428181450...

Crossrefs

Cf. A198677.

Programs

  • Mathematica
    n = 4; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198729 *)
RealDigits[v]  (* A198730 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
RealDigits[Minimize[Plus@@Sin[x*Range[4]],x],10,100][[1,1]] (* Harvey P. Dale, Dec 14 2020 *)

A198730 Decimal expansion of the least x>0 that gives the absolute minimum of sin(x)+sin(2x)+sin(3x)+sin(4x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 29 2011

Keywords

Comments

See A198677 for a guide to related sequences.

Examples

			x=5.764569445617181108633794876455734774725510...
min=-3.232384871111452330884269520493428181450...

Crossrefs

Cf. A198677.

Programs

  • Mathematica
    n = 4; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198729 *)
RealDigits[v]  (* A198730 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]

A198731 Decimal expansion of the absolute minimum of sin(x)+sin(2x)+sin(3x)+sin(4x)+sin(5x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 29 2011

Keywords

Comments

See A198677 for a guide to related sequences.

Examples

			x=5.8590231016815427641022649970323299904568...
min=-3.96217336350328560023197791110649098988...

Crossrefs

Cf. A198677.

Programs

  • Mathematica
    n = 5; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198731 *)
RealDigits[v] (* A198732 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]

A198732 Decimal expansion of the least x>0 that gives the absolute minimum of sin(x)+sin(2x)+sin(3x)+sin(4x)+sin(5x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 29 2011

Keywords

Comments

See A198677 for a guide to related sequences.

Examples

			x=5.8590231016815427641022649970323299904568...
min=-3.9621733635032856002319779111064909898...

Crossrefs

Cf. A198677.

Programs

  • Mathematica
    n = 5; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198731 *)
RealDigits[v] (* A198732 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]

A198733 Decimal expansion of the absolute minimum of sin(x)+sin(2x)+sin(3x)+sin(4x)+sin(5x)+sin(6x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 29 2011

Keywords

Comments

See A198677 for a guide to related sequences.

Examples

			x=5.924356380456046882352862563376763567376562...
min=-4.690361803953941719169567767677307460891...

Crossrefs

Cf. A198677.

Programs

  • Mathematica
    n = 6; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198733 *)
RealDigits[v] (* A198734 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
RealDigits[FindMinimum[Plus@@Table[Sin[n*x],{n,6}],{x,.1}, WorkingPrecision -> 100][[1]]][[1]] (* Harvey P. Dale, Dec 08 2015 *)

A198734 Decimal expansion of the least x>0 that gives the absolute minimum of sin(x)+sin(2x)+sin(3x)+sin(4x)+sin(5x)+sin(6x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 29 2011

Keywords

Comments

See A198677 for a guide to related sequences.

Examples

			x=5.924356380456046882352862563376763567376562...
min=-4.6903618039539417191695677676773074608911...

Crossrefs

Cf. A198677.

Programs

  • Mathematica
    n = 6; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198733 *)
RealDigits[v] (* A198734 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]

A198678 Decimal expansion of the least x>0 that gives the absolute minimum of sin(x)+sin(2x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 29 2011

Keywords

Comments

See A198677 for a guide to related sequences.

Examples

			x=2*pi-2*arctan(r), where r=sqrt(6-sqrt(33))
x=5.347255851518260503318727031180159764862...
min=-1.760172593046086919405184649699273192...

Crossrefs

Cf. A198677.

Programs

  • Mathematica
    f[t_] := Sin[t]; x = Minimize[f[t] + f[2 t], t]
x = N[Minimize[f[t] + f[2 t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198677 *)
RealDigits[v]  (* A198678 *)
Plot[f[t] + f[2 t], {t, -3 Pi, 3 Pi}]

A198728 Decimal expansion of the least x>0 that gives the absolute minimum of sin(x)+sin(2x)+sin(3x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 29 2011

Keywords

Comments

See A198677 for a guide to related sequences.

Examples

			x=5.615894235655113730430510613029770525768423...
min=-2.499607604320056604368126285618043218243...

Crossrefs

Cf. A198677.

Programs

  • Mathematica
    n = 3; f[t_] = Sin[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = t /. Part[x, 2]
RealDigits[u]  (* A198679 *)
RealDigits[v] (* A198680 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
(* Second program: *)
RealDigits[3*Pi/2 + ArcSin[Root[-1 - 4x + 2x^2 + 6x^3, 3]], 10, 99] // First (* Jean-François Alcover, Feb 15 2013 *)
Showing 1-9 of 9 results.

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