Trott Constant

Author

Eric W. Weisstein
October 9, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Constants conceived and discovered by
Michael Trott

Code

FromGeneralContinuedFraction[{a0_,l_List?MatrixQ}]:=a0+Fold[#2[[1]]/(#1+#2[[2]])&,0,Reverse[l]]

(Subscript[T, 1])==0.1084101…

A039662

x=0.10841015122311136151129081140641509112215809093909091;

d=RealDigits[x]

RealDigits[FromContinuedFraction[Prepend[d[[1]],0]],10,Floor[Precision[x]]]

Not regular:

ContinuedFraction[0.10841015122311136151129081140641509112215809093909091]

Scheonfeld extension

S=0.108410151223111361511290811406415091122158090939090906901909090905909051190222132199090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909089090489011907169062904163190909329090909071411986311321341129031311219041221115909090909090719034336219011112622211111473522141512211290113229051159051901390719090909062`639

N[FromContinuedFraction[Prepend[RealDigits[S][[1]],0]],640]

See A039662

While it is theoretically possible to extend this sequence arbitrarily far using the same procedure, it is impractical to do so; the agreement using 639 terms is so extremely close that the number of consecutive digit pairs "90" (i.e., term pairs (9, 0)) that would immediately follow the 639th term would exceed 5*10^301.

(Subscript[T, 1.5])==0.320622…

A114376

T15={3,2,0,6,2,2,4,1,1,3,3,4,1,9,0,9,0,6,1,4,2,9,5,9,0,6,1,1,3,2,1,7,1,9,0,9,0,4,4,2,2,4,1,1,2,1,8,1,2,2,1,1,4,2,3,1,2,1,2,1,1,1,1,2,1,9,4,8,1,2,7,1,2,9,0,2,1,6,1,1,2,1,2,3,2,9,0,4,2,9,0,2,1,6,8,3,1,5,2,1,2,1,1,1,2,1,4};

N[FromDigits[{T15,-Length[T15]},10],Length[T15]]

FromContinuedFraction[Prepend[T15,0]]

r=RealDigits[FromContinuedFraction[Prepend[T15,0]],10,Length[T15]][[1]]

Transpose[{T15,r}]

FirstPosition[%101,{x_,y_}/;y=!=x]

(Subscript[T, 2])==0.27…

A091694

Length[T2={0,2,7,3,9,4,4,1,9,5,7,3,9,2,7,1,6,1,7,1,7,1,4,5,9,1,5,2,7,2,4,2,8,5,9,1,9,2,7,3,7,2,5,1,8,7,7,2,9,8,8,1,9,8,6,2,9,1,9,1,7,3,8,3,7,5,5,2,8,1,7,1,7,7,4,1,8,1,9,6,9,4,6,1,9,1,7,3,8,2,8,3,6,2,5,1,6,1,5,4,8,5,9,3,6,4,7,1,9,2,5,8,9,4,9,8,9,1,5,1,7,2,7,3,9,1,9,6,7,6,9,2,8,1,9,4,5,3,5,1,6,3,8,1,6}]

StringJoin@@(ToString/@T2)

N[FromGeneralContinuedFraction[{0,Partition[Rest[T2],2]}],100]

r=Prepend@@RealDigits[FromGeneralContinuedFraction[{0,Partition[Rest[y],2]}],10,Length[y]]

Position[Equal@@@Transpose[{y,Take[r,Length[y]]}],False,1,1][[1,1]]

(Subscript[T, 3])==0.48…

A113307

Oct. 24, 2005

Length[T3={0,4,8,2,6,7,7,2,8,1,9,3,9,1,8,1,5,9,9,4,9,3,9,2,9,1,5,2,7,9,6,1,7,2,7,8,9,1,8,3,5,6,7,1,7,5,7,1,7,1,5,1,9,2,9,2,9,1,6,3,7,3,6,1,8,2,6,1,6,1,8,1,7,6,9,2,9,1,5,2,9,3,9,5,6,2,7,2,6,5,8,1,3,1,8,3,8,6,8,3,6,1,8,1,9,2,7,5,6,6,5,2,9,1,9,7,9,2,9,2,4,1,5,2,9,2,9,8,8,1,8,2,9,7,9,2,5,8,5,1,8,1,5,1,8,4,8,6,8,3,9,2,9,1,3,1,9,1,8,3,7,1,5,3,8,4,7,7,8,1,9,2,5,7,9,2,9,8,9,3,8,1,9,1,6,2,5,7,4,2,9,2,6,1,9,6,6,6,9,1,9,3,8,2,8,1,8,1,8,3,7,1,8,5,7,1,6,9,9,1,9,8,1,6,7,2,5,1,8,4,7,1,9,2,9,2,7,2,8,9,8,4,7,2,9,1,8,4,9,1,8,8,1,2,9,3,8,8,9,1,4,2,8,1,9,2,8,1,8,1,5,7,7,2,7,3,7,5,8,1,7,1,7,2,5,1,6,1,7,8,5,1,6,3,4,0,3,8,7,8,7,8,7,8,7,8,7,8,7,8,7,7,5,9,9}]

StringJoin@@(ToString/@T3)

N[FromGeneralContinuedFraction[{0,Partition[Rest[T3],2]}],100]

r=Prepend@@RealDigits[FromGeneralContinuedFraction[{0,Partition[Rest[T3],2]}],10,Length[y]]

Position[Equal@@@Transpose[{y,Take[r,Length[y]]}],False,1,1][[1,1]]