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



r3769
Posts:
30
Registered:
12/12/04


Re: Average length of random 1expansions
Posted:
Nov 4, 2000 12:27 PM


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 1expansion 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 1expansion of x[0] is n where x[n]=0. >> >> What is the average length of all the random 1expansions of 1/7? > >Let f(q) represent the expected length of the random 1expansion 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



