Lehmer Cotangent Expansion

Author

Eric W. Weisstein
September 21, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/LehmerCotangentExpansion.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Expansions

Lehmer's Constant ξ

c[0]:=0
c[n_]:=c[n]=c[n-1]^2+c[n-1]+1

l=c/@Range[0,10]

FindSequenceFunction[l]

N[Cot[Fold[Plus,0,(ArcCot[c[#]](-1)^#&/@Range[0,20])]],20]//Timing

ϕ==cot((∑_(k=0)^∞)(((-1))^k) (cot^(-1))((LucasL[(3^k)])))

Developer`SetSystemOptions["SimplificationOptions"->{"SimplifyMaxExponents"->1000}];

LucasL/@(3^Range[0,5])//Simplify

cs=LucasL/@(3^Range[0,5])//Expand

FindSequenceFunction[{1,4,76,439204,84722519070079276,608130213374088941214747405817720942127490792974404}]

N[Cot[Fold[Plus,0,MapIndexed[ArcCot[#1](-1)^(#2[[1]]+1)&,cs]]],50]

N[GoldenRatio,50]

Sum[ArcCot[LucasL[3^n]](-1)^n,{n,0,5}]//N[#,50]&

ArcCot[GoldenRatio]//N[#,50]&

V7

Cot[Sum[(((-1))^k)ArcCot[LucasL[(3^k)]],{k,0,∞}]]

FullSimplify[%]

E

N[Cot[Fold[Plus,0,MapIndexed[ArcCot[#1](-1)^(#2[[1]]+1)&,{2,8,75,8949,119646723,15849841722437093,708657580163382065836292133774995,529026553215766321676623343348414600292754204772300344704877695232}]]],50]

FindSequenceFunction[{2,8,75,8949,119646723,15849841722437093,708657580163382065836292133774995,529026553215766321676623343348414600292754204772300344704877695232}]

N[E,50]

π

N[Cot[Fold[Plus,0,MapIndexed[ArcCot[#1](-1)^(#2[[1]]+1)&,{3,73,8599,400091364,371853741549033970,253461181173408820488703379557217678}]]],50]

N[Pi,50]

FindSequenceFunction[3,73,8599,400091364,371853741549033970,253461181173408820488703379557217678]

(Sqrt[2])

N[Cot[Fold[Plus,0,MapIndexed[ArcCot[#1](-1)^(#2[[1]]+1)&,{1,5,36,3406,14694817,727050997716715,2074744506784679417243551677046,46045625970633183674340934067371917846908361831894602280765110}]]],50]

FindSequenceFunction[{1,5,36,3406,14694817,727050997716715,2074744506784679417243551677046,46045625970633183674340934067371917846908361831894602280765110}]

N[Sqrt[2],50]

Non-Lehmer Expansion

ϕ==cot((∑_(k=0)^∞)(((-1))^k) (cot^(-1))((Fibonacci[2 k+2])))

Cot[Sum[(((-1))^k)ArcCot[Fibonacci[2k+2]],{k,0,∞}]]