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