       Fibonacci based sum that is b-normal on binary numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg52117] Fibonacci based sum that is b-normal on binary numbers
• From: Roger Bagula <tftn at earthlink.net>
• Date: Thu, 11 Nov 2004 04:52:29 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```This sum and it's b-normal sequence is due
to work of a friend who doesn't like me to use his name here or elsewhere
.He came up with two very nice sums using Fibonacci numbers.
I used the Binet  function in them and got very good agreement.
So I tried them in a b-normal.
I had to modify the result some to get this result.
I get a new sum that appears irrational
and an iteration that is b-normal .
I think that using the Binet function in this makes it
a new sequence sum.
I thought that this was a very remarkable result.

Clear[x,a,digits,f,fib]
(* convergent sum based on Fibonacci sequence to make a binary b-normal
iteration *)
digits=200
fib[n_Integer?Positive] :=fib[n] =    fib[n-1]+fib[n-2]
fib=0;fib = fib = 1;
sfib=Sum[fib[n]/((n+1)*2^(n+1)),{n,0,digits}]
N[sfib,digits]
x[n_]:=x[n]=Mod[2*x[n-1]+fib[n-1]/(2*n),1]
x=0
a=Table[N[x[n],digits],{n,0,digits}]
ListPlot[a,PlotJoined->True,PlotRange->All]
b=Sort[Table[N[x[n],digits],{n,0,digits}]];
ListPlot[b,PlotJoined->True,PlotRange->All]
Respectfully, Roger L. Bagula

tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn at netscape.net