Fibonacci based sum that is b-normal on binary numbers

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]=0;fib[1] = fib[2] = 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]=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

, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email:
URL : http://home.earthlink.net/~tftn

 

....just another way to compute

sfib = 1/10*(5*Log[4] + Sqrt[5]*Log[1/2*(7 - 3*Sqrt[5])])

 

Dear Peter Pein,
I want to thank you for finding this number.
It appears to be algebraic and not transcendental.
Some other algebraics have been shown to be b-normal
according to Eric Weisstein's site Math World notation for normal. (
Sqrt[n] types)
It seems a strange misuse of the original meaning of "normal"
in noise theory which refered to the Exp[x^/2]/Sqrt[2*Pi]
type of distriubution which isn't normal in the b-normal sense
( I think).