Theodorus's Constant

Author

Eric W. Weisstein
January 9, 2017

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

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

(Sqrt[3])

Binary

RealDigits[Sqrt[3],2,20]

Decimal digits

N[Sqrt[3],20]

RealDigits[Sqrt[3],10,20]

Base-60

RealDigits[Sqrt[3],60,20]

Continued fraction

ContinuedFraction[Sqrt[3],20]

1+ContinuedFractionK[If[Mod[n,2]==0,2,1],{n,∞}]

1+ContinuedFractionK[If[EvenQ[n],2,1],{n,∞}]

Cube

Sort[{Norm[Subtract@@#],#}&/@Subsets[Flatten[Outer[List,Sequence@@Table[{0,1},{3}]],2],{2}]]

BBP-Type Formula

<<Utilities`Simplify`

Monitor[Do[If[(b=FindBasis[Sqrt[3],-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,14,16},{n,64},{p,3,3}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,1}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,3,16},{n,64},{p,3,3}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,15,16},{n,64},{p,2,2}],{3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Sqrt[3],3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,4,16},{n,64},{p,3,3}],{3^k,n,p}];Print["DONE"]

Spiral Constant

A226317

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.440.3922&rep=rep1&type=pdf

Sum

Sum[((k^(-1/2))/(k+1)),{k,∞}]

NSum[((k^(-1/2))/(k+1)),{k,∞}]

NSum[1/(k^(3/2)+k^(1/2)),{k,∞}]

(1/2)-Sum[(((-1))^m)(Zeta[m+(1/2)]-1),{m,∞}]

(1/2)-NSum[(((-1))^m)(Zeta[m+(1/2)]-1),{m,∞}]

Integral

(2/(Sqrt[π])) (∫_0^∞)(-exp((t^2)) log(1-exp(-(t^2)))-1)\[DifferentialD]t

2/Sqrt[Pi]Integrate[(-Exp[t^2])*Log[1-Exp[-t^2]]-1,{t,0,Infinity}]

Plot[(-Exp[t^2])*Log[1-Exp[-t^2]]-1,{t,1,10},WorkingPrecision->20]

LogPlot[(-Exp[t^2])*Log[1-Exp[-t^2]]-1,{t,1,10},WorkingPrecision->50]

2/Sqrt[Pi]NIntegrate[(-Exp[t^2])*Log[1-Exp[-t^2]]-1,{t,0,Infinity},PrecisionGoal->20,WorkingPrecision->100]

Series[(-Exp[t^2])*Log[1-Exp[-t^2]]-1,{t,0,20}]

N[%5]

SeriesCoefficient[(-Exp[t^2])*Log[1-Exp[-t^2]]-1,{t,0,n}]