So B(s) is the maximum number of consecutive even numbers that will ever be hit iterating a tricot starting from s (or perhaps you are defining it as this +1)? And your claim is that this is finite for all starting values when using a homogenous tricot ex: (2k->3k ; 2k+1->3k+2)? How do we know it is finite? I would have guessed that it could be (and maybe even should be) unbounded!
Yes, but actually my proof is wrong… When I tried to write it more cleanly a few days ago I noticed an error at the “Replacing in (1)” line, and everything that follows crumbles down. That also means that perhaps I can find a number whose trajectory is 110100100010000…
There is a continued fraction for root(2).
root(2) = 1 + 1/(1+root(2)).
Substituting the left side into the right side we get:
root(2) = 1 + 1/(2 + 1/(1+root(2)))
Again we get root(2) = 1 + 1/(2 +1/(2 + 1/(…)))
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