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MathGroup Archive 2004

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Re: Fibonacci based sum that is b-normal on binary numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg52162] Re: Fibonacci based sum that is b-normal on binary numbers
  • From: Roger Bagula <tftn at earthlink.net>
  • Date: Sat, 13 Nov 2004 04:40:11 -0500 (EST)
  • References: <cmve7h$sh9$1@smc.vnet.net> <cn1p4v$f08$1@smc.vnet.net>
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com

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).
Peter Pein wrote:

>Roger Bagula wrote:
>  
>
>>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
>>
>>tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
>>alternative email: rlbtftn at netscape.net
>>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])])
>
>  
>

-- 
Respectfully, Roger L. Bagula
tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn at netscape.net
URL :  http://home.earthlink.net/~tftn



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