A198677 -id:A198677

A198679

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A198677 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=5.615894235655113730430510613029770525768423...

min=-2.499607604320056604368126285618043218243...

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 *)

CROSSREFS

Cf. A198677.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 29 2011

STATUS

approved

A198729

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A198677 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=5.764569445617181108633794876455734774725510...

min=-3.232384871111452330884269520493428181450...

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 *)

CROSSREFS

Cf. A198677.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 29 2011

STATUS

approved

A198730

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A198677 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=5.764569445617181108633794876455734774725510...

min=-3.232384871111452330884269520493428181450...

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}]

CROSSREFS

Cf. A198677.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 29 2011

STATUS

approved

A198731

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A198677 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=5.8590231016815427641022649970323299904568...

min=-3.96217336350328560023197791110649098988...

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}]

CROSSREFS

Cf. A198677.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 29 2011

STATUS

approved

A198732

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A198677 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=5.8590231016815427641022649970323299904568...

min=-3.9621733635032856002319779111064909898...

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}]

CROSSREFS

Cf. A198677.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 29 2011

STATUS

approved

A198733

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A198677 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..106.

EXAMPLE

x=5.924356380456046882352862563376763567376562...

min=-4.690361803953941719169567767677307460891...

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 *)

CROSSREFS

Cf. A198677.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 29 2011

STATUS

approved

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).

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A198677 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=5.924356380456046882352862563376763567376562...

min=-4.6903618039539417191695677676773074608911...

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}]

CROSSREFS

Cf. A198677.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 29 2011

STATUS

approved

A198678

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

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A198677 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=2*pi-2*arctan(r), where r=sqrt(6-sqrt(33))

x=5.347255851518260503318727031180159764862...

min=-1.760172593046086919405184649699273192...

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}]

CROSSREFS

Cf. A198677.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 29 2011

STATUS

approved

A198728

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

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A198677 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

x=5.615894235655113730430510613029770525768423...

min=-2.499607604320056604368126285618043218243...

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 *)

CROSSREFS

Cf. A198677.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 29 2011

STATUS

approved