Fibonacci based sum that is b-normal on binary numbers

Discussion in 'Mathematica' started by Roger Bagula, Nov 11, 2004.

  1. Roger Bagula

    Roger Bagula Guest

    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
     
    Roger Bagula, Nov 11, 2004
    #1
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  2. Roger Bagula

    Peter Pein Guest

    ....just another way to compute

    sfib = 1/10*(5*Log[4] + Sqrt[5]*Log[1/2*(7 - 3*Sqrt[5])])
     
    Peter Pein, Nov 12, 2004
    #2
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  3. Roger Bagula

    Roger Bagula Guest

    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).
     
    Roger Bagula, Nov 13, 2004
    #3
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