I need help proving something about my sequence

1. May 26, 2013

Jamin2112

My sequence is a function f:Z+→{H,T} to represent heads and tails. It is defined by

f(1),f(2),f(3)f(4),f(5)f(6), ...
= H,T,HH,HT,TH,TT,HHH,THH,HTH,....,

where you can see that I'm just taking all heads-tails sequences of length 1, then of length 2, etcetera. This ensures

(1) that the long term frequency of heads is 1/2.
(2) that every finite heads-tails sequence is contained the infinite sequence thus defined.

The proofs of (1) and (2) are trivial. There's one more property that I'm not sure I have in my sequence, so I'd like help proving it (if it's true ....). I need that every heads-tails sequence of length N occurs with equal long-term frequency. So, for example, I should see HT just as often as TH when I look left to right at

HTHHHTTHTTHHHTHHHTH.....

Thoughts?????

2. May 26, 2013

micromass

This will be helpful: http://www.mth.uea.ac.uk/~h720/teaching/dynamicalsystems/champernowne1933.pdf [Broken]

Last edited by a moderator: May 6, 2017
3. May 27, 2013

Jamin2112

Could just be my internet, but the link isn't working. Hmm.

Last edited by a moderator: May 6, 2017