Plouffe's Constants

Author

Eric W. Weisstein
August 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/PlouffesConstants.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PlouffesConstants.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plouffe (Subscript[a, n])

Table[Sin[2^n],{n,10}]//N

<<Utilities`Plot`

Block[{$MaxExtraPrecision=100},
IntegerListPlot[N[Table[Sin[2^n],{n,200}],100],AxesLabel->TraditionalForm/@{n,(Subscript[a, n])}]
]

Block[{$MaxExtraPrecision=300},
Timing[N[Plus@@Table[If[Sin[2^n]<0,1/2^(n+1),0],{n,0,#}],50]]&/@{10,100,1000}//ColumnForm
]

N[(1/(2π)),20]

V6

Sum[If[Sin[2^n]<0,1/2^(n+1),0],{n,0,250}]

N[%,20]

V7

Sum[If[Sin[2^n]<0,1/2^(n+1),0],{n,0,∞}]//InputForm

NSum[If[Sin[2^n]<0,1/2^(n+1),0],{n,0,∞}]//Timing

N[Sum[If[Sin[2^n]<0,1/2^(n+1),0],{n,0,10^3}],20]

Sum[If[Sin[2^n]<0,1/2^(n+1),0],{n,0,234}]

N[%]

Plouffe (Subscript[b, n])

Table[Cos[2^n],{n,10}]//N

Block[{$MaxExtraPrecision=100},
IntegerListPlot[N[Table[Cos[2^n],{n,200}],100],AxesLabel->TraditionalForm/@{n,(Subscript[b, n])}]
]

Block[{$MaxExtraPrecision=1000},
Timing[N[Plus@@Table[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,#}],110]]&/@{10,100,1000,3000}//ColumnForm
]

b=0.47562607673598855480497725290495471428196053135382275289581113420957404028749840047127387688562952320461698847111073662357672`110.;

With[{n=5000},FromDigits[BitXor@@@Transpose[First[RealDigits[#,2,n,-1]]&/@{(1/(2π)),(1/π)}],2]/2^n
]//N[#,100]&

Block[{$MaxExtraPrecision=1000},
Timing[N[Plus@@Table[If[Cos[2^n]Cos[2^(n-1)]<0,1/2^(n+1),0],{n,0,#}],110]]&/@{10,100,1000,3000}//ColumnForm
]

With[{n=5000},FromDigits[BitXor@@@Transpose[First[RealDigits[#,2,n,-1]]&/@{(1/(4π)),(1/π)}],2]/2^n
]//N[#,100]&

V6

Sum[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,∞}]//InputForm

TimeConstrained[NSum[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,∞}],30]

N[Sum[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,200}],20]

N[Sum[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,250}],20]

V7

Sum[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,∞}]//InputForm

TimeConstrained[NSum[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,∞}],30]

N[Sum[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,200}],20]

N[Sum[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,250}],20]

Plouffe (Subscript[c, n])

Block[{$MaxExtraPrecision=100},
IntegerListPlot[N[Table[Tan[2^n],{n,200}],100],
AxesLabel->TraditionalForm/@{n,(Subscript[c, n])}]
]

Block[{$MaxExtraPrecision=300},
Timing[N[Plus@@Table[If[Tan[2^n]<0,1/2^(n+1),0],{n,0,#}],50]]&/@{10,100,1000}//ColumnForm
]

N[1/Pi,20]

V6

Sum[If[Tan[2^n]<0,1/2^(n+1),0],{n,0,∞}]//InputForm

TimeConstrained[NSum[If[Cos[2^n]<0,1/2^(n+1),0],{n,0,∞}],30]

N[Sum[If[Tan[2^n]<0,1/2^(n+1),0],{n,0,200}],20]

N[Sum[If[Tan[2^n]<0,1/2^(n+1),0],{n,0,250}],20]

V7

Sum[If[Tan[2^n]<0,1/2^(n+1),0],{n,0,∞}]//InputForm

TimeConstrained[NSum[If[Tan[2^n]<0,1/2^(n+1),0],{n,0,∞}],30]

N[Sum[If[Tan[2^n]<0,1/2^(n+1),0],{n,0,200}],20]

N[Sum[If[Tan[2^n]<0,1/2^(n+1),0],{n,0,250}],20]

Plouffe (Subscript[α, n])

Block[{$MaxExtraPrecision=300},
Timing[N[Plus@@Table[If[Sin[2^n ArcSin[1/2]]<0,1/2^(n+1),0],{n,0,#}],50]]&/@{10,100,1000}//ColumnForm
]

N[1/12]

Plouffe (Subscript[β, n])

Block[{$MaxExtraPrecision=300},
Timing[N[Plus@@Table[If[Cos[2^n ArcCos[1/2]]<0,1/2^(n+1),0],{n,0,#}],50]]&/@{10,100,1000}//ColumnForm
]

Plouffe (Subscript[γ, n])

Block[{$MaxExtraPrecision=300},
Timing[N[Plus@@Table[If[Tan[2^n ArcTan[1/2]]<0,1/2^(n+1),0],{n,0,#}],50]]&/@{10,100,1000}//ColumnForm
]

N[ArcTan[1/2]/π,20]