Pierce Expansion

Author

Eric W. Weisstein
April 17, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/PierceExpansion.nb.

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

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

Code

PierceExpansion[x_,n_]:=Floor[NestList[((#)/(FractionalPart[#]))&,1/x,n]] /; 0<x<1

Table

cons={1/Sqrt[2],Catalan,1/E,EulerGamma,1/Pi,1/GoldenRatio,Sin[1],Cos[1],1/Cosh[1],1/Sinh[1],Log[2]};

Block[{$MaxExtraPrecision=1000},
TableForm[{#,PierceExpansion[#,10]}&/@cons,TableDepth->2]
]//TraditionalForm

Recurrence

PierceExpansion[x_,n_]:=Module[{k,a=(x-(Sqrt[(x^2)-4]))/2},
Outer[Plus,{-1,1},Table[-2 Cos[(3^k) ArcCos[-(a/2)]],{k,Ceiling[n/2]}]]
]

RSolve[{c[k]==c[k-1]^3-3c[k-1],c[0]==a},c[k],k]

FullSimplify[Table[-2 Cos[(3^k) ArcCos[-(c/2)]],{c,3,6}],k∈Integers&&k>=0]

Predictable Expansions

Solve[x==((c-(Sqrt[(c^2)-4]))/2),c]

Table[((c-(Sqrt[(c^2)-4]))/2),{c,3,6}]//Simplify

Table[Table[-2 Cos[(3^k) ArcCos[-(c/2)]],{k,0,5}]//FunctionExpand,{c,3,6}]//ColumnForm

Table[Flatten[{#-1,#+1}&/@(Table[-2 Cos[(3^k) ArcCos[-(c/2)]],{k,0,5}]//FunctionExpand)],
{c,3,6}]//ColumnForm

Table[PierceExpansion[((c-(Sqrt[(c^2)-4]))/2),5],{c,3,6}]//ColumnForm

c==3

FullSimplify[-2 Cos[(3^k) ArcCos[-(3/2)]],k∈Integers&&k>0]

Table[2Fibonacci[2*3^n+1]-Fibonacci[2*3^n],{n,0,5}]

Table[-2 Cos[(3^n) ArcCos[-(3/2)]],{n,0,5}]//FunctionExpand