an [0,1] iterative based on the Farey tree that isn't b-normal

• To: mathgroup at smc.vnet.net
• Subject: [mg52089] an [0,1] iterative based on the Farey tree that isn't b-normal
• From: Roger Bagula <tftn at earthlink.net>
• Date: Wed, 10 Nov 2004 04:45:38 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```This is an iteration based on two transcental numbers E and Pi.
Any other pair of rational  transcendentals can be used.
The result is limited ( not very well here!) to the interval [0,1]
It isn't actually a very good way to manufacture transcendentals, but it
works most of the time.
The idea is to present a [0,1] iterative
that doesn't depend on the Modulo one fractional part
and isn't b-normal by the Bailey definition.

Clear[x,y,a,b,s,g,a0,f]
(* transcendental Farey tree recursive function*)
t1=Pi;
t2=E;
f[a_,b_]:=((a/b)/(1-a/b))/;0<=a/b<=1/2
f[a_,b_]:=((1-a/b)/(a/b))/;1/2<a/b<=1
g[n_]:=g[n]=If[( g[n-1]<1/2)&&(g[n-1]>0),N[g[n-1]/(1-g[n-1]),20],If[
g[n-1]<1,N[(1-g[n-1])/g[n-1],20],N[(E/Pi)^2,20]]]
g[0]=N[f[t2,t1],20];
a0=Table[If [g[n]<1,g[n],0],{n,0,400}]
ListPlot[a0,PlotJoined->True,PlotRange->All]
a1=Sort[a0]
ListPlot[a1,PlotJoined->True,PlotRange->All]

Respectfully, Roger L. Bagula