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Topic: Average length of random 1-expansions
Replies: 9   Last Post: Nov 20, 2000 4:55 AM

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r3769

Posts: 30
Registered: 12/12/04
Re: Average length of random 1-expansions
Posted: Nov 4, 2000 12:27 PM
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Hugo van der Sanden wrote in message
<39FE2BA7.1371F868@crypt0.demon.co.uk>...
>r3769 wrote:
>>
>> Given an initial value x[0] with 0<x[0]<1, define a random 1-expansion of
>> x[0] recursively as follows:
>>
>> Choose [the integer] a[i] (randomly) subject to floor(a[i]*x[i])=1,
>> and then set x[i+1]=a[i]*x[i]-1.
>>
>> The length of a random 1-expansion of x[0] is n where x[n]=0.
>>
>> What is the average length of all the random 1-expansions of 1/7?

>
>Let f(q) represent the expected length of the random 1-expansion of 1/7,
>and let a thru f represent f(1/7) through f(6/7) respectively.
>
>Then we get a set of six simultaneous equations:
>a = 1 + (0 + a + b + c + d + e + f)/7
>b = 1 + (a + c + e)/3
>c = 1 + (b + e)/2
>d = 1 + (a + e)/2
>e = 1 + c
>f = 1 + e
>
>.. and solving gives a = 67, as required.


If p is prime and 1<n<p, is f(1/p)>f(1/n)?

R. Burge









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