       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>
• 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;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
>>
>>
>>
>...just another way to compute
>
>sfib = 1/10*(5*Log + Sqrt*Log[1/2*(7 - 3*Sqrt)])
>
>
>

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