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 1-7 of 7 results.

A198671 Decimal expansion of the absolute minimum of cos(x)+cos(2x)+cos(3x)+cos(4x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 28 2011

Keywords

Comments

See A196361 for a guide to related sequences.

Examples

			x=1.002750019227300659428346483723194...
min=-1.519557881642848251369426811329...

Crossrefs

Cf. A196361.

Programs

  • Mathematica
    n = 4; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = 2 Pi - t /. Part[x, 2]
RealDigits[u]  (* A198671 *)
RealDigits[v] (* A198672 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
(* Second program: *)
Root[-125 + 600x + 1227x^2 + 512x^3, 1] // RealDigits[#, 10, 99]& // First (* Jean-François Alcover, Feb 19 2013 *)

A198672 Decimal expansion of the least x>0 that gives the absolute minimum of cos(x)+cos(2x)+cos(3x)+cos(4x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 28 2011

Keywords

Comments

See A196361 for a guide to related sequences.

Examples

			x=1.002750019227300659428346483723194323862...
min(f(x))=-1.519557881642848251369426811329...

Crossrefs

Cf. A196361, A198671.

Programs

  • Mathematica
    n = 4; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = 2 Pi - t /. Part[x, 2]
RealDigits[u] (* A198671 *)
RealDigits[v] (* A198672 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]

A198673 Decimal expansion of the absolute minimum of cos(x)+cos(2x)+cos(3x)+cos(4x)+cos(5x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 28 2011

Keywords

Comments

See A196361 for a guide to related sequences.

Examples

			x=0.8192724134245426759067178580741783999...
min=-1.728768207393169918251228314555655767777...

Crossrefs

Cf. A196361, A198674.

Programs

  • Mathematica
    n = 5; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = 2 Pi - t /. Part[x, 2]
RealDigits[u] (* A198673 *)
RealDigits[v] (* A198674 *)
Plot[s[t], {t, -3 Pi, 3 Pi}, PlotRange -> {-2, 6}]

A198674 Decimal expansion of the least x>0 that gives the absolute minimum of cos(x)+cos(2x)+cos(3x)+cos(4x)+cos(5x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 28 2011

Keywords

Comments

See A196361 for a guide to related sequences.

Examples

			x=0.8192724134245426759067178580741783999...
min=-1.728768207393169918251228314555655767777...

Crossrefs

Cf. A196361.

Programs

  • Mathematica
    n = 5; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = 2 Pi - t /. Part[x, 2]
RealDigits[u]  (* A198673 *)
RealDigits[v] (* A198674 *)
Plot[s[t], {t, -3 Pi, 3 Pi}, PlotRange -> {-2, 6}]

A198675 Decimal expansion of the absolute minimum of cos(x)+cos(2x)+cos(3x)+cos(4x)+cos(5x)+cos(6x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 28 2011

Keywords

Comments

See A196361 for a guide to related sequences.

Examples

			x=0.6926737452063682445569036306595182001...
min=-1.940589216902488020962568639391387253030...

Crossrefs

Cf. A196361, A198676.

Programs

  • Mathematica
    n = 6; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = 2 Pi - t /. Part[x, 2]
RealDigits[u]  (* A198675 *)
RealDigits[v]  (* A198676 *)
Plot[s[t], {t, -3 Pi, 3 Pi}, PlotRange -> {-2, 6}]

A198676 Decimal expansion of the least x>0 that gives the absolute minimum of cos(x)+cos(2x)+cos(3x)+cos(4x)+cos(5x)+cos(6x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 28 2011

Keywords

Comments

See A196361 for a guide to related sequences.

Examples

			x=0.6926737452063682445569036306595182001...
min=-1.940589216902488020962568639391387253030...

Crossrefs

Cf. A196361, A198675.

Programs

  • Mathematica
    n = 6; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = 2 Pi - t /. Part[x, 2]
RealDigits[u]  (* A198675 *)
RealDigits[v]  (* A198676 *)
Plot[s[t], {t, -3 Pi, 3 Pi}, PlotRange -> {-2, 6}]

A198670 Decimal expansion of the least x>0 that gives the absolute minimum of cos(x)+cos(2x)+cos(3x).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 28 2011

Keywords

Comments

See A196361 for a guide to related sequences.

Examples

			x=1.2929430585054266652256311954691354...
min(f(x))=-1.3155651547204494123522707...

Crossrefs

Cf. A196361.

Programs

  • Mathematica
    n = 3; f[t_] := Cos[t]; s[t_] := Sum[f[k*t], {k, 1, n}];
x = N[Minimize[s[t], t], 110]; u = Part[x, 1]
v = 2 Pi - t /. Part[x, 2]
RealDigits[u]   (* A196361 *)
RealDigits[v]   (* A198670 *)
Plot[s[t], {t, -3 Pi, 3 Pi}]
RealDigits[ ArcSin[ Sqrt[7 + Sqrt[7]/2]/3], 10, 99] // First (* Jean-François Alcover, Feb 15 2013 *)
Showing 1-7 of 7 results.

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