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

*To*: mathgroup at smc.vnet.net*Subject*: [mg52154] Re: Fibonacci based sum that is b-normal on binary numbers*From*: Peter Pein <petsie at arcor.de>*Date*: Fri, 12 Nov 2004 02:14:22 -0500 (EST)*References*: <cmve7h$sh9$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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])]) -- Peter Pein Berlin