When (d(n+1)-d(n))*(-1)^n is positive (d(n) = A000005(n))

[Joshua Zucker]

On Mon, Mar 3, 2008 at 2:53 PM, Joshua Zucker <joshua.zucker at gmail.com> wrote:
> On Sun, Mar 2, 2008 at 1:55 PM, Maximilian Hasler
>  <maximilian.hasler at gmail.com> wrote:
>  >  >  So I wonder, can we determine the limit, as n ->
>  >  >  infinity,  of a(n)/n?
>
>  >  yes : it decreases steadily.
>
>  So it makes sense to me that this would be asymptotic to 0.  Though of
>  course I suspect that a(n) keeps increasing, just more and more
>  slowly.
>

Now wait a minute -- I got this exactly backward?  The decreasing
steadily means that there are MORE and MORE (proportion) of numbers
where d(even) is smaller than d(even +/- 1)?  Now that's surprising.
OK, can anyone help give me some intuition about why that makes sense?

Thanks,
--Joshua



"Russ Cox" <rsc at swtch.com> wrote:
:> Of particular interest is the pair (010, 212). I cannot be certain that
:> this yields an infinite string, since I am relying only on observation of
:
:You might be interested in 
:
:Narad Rampersad, ``A note on avoidable words in ternary strings''
:http://arxiv.org/abs/math/0307363v1
:
:The fact that you can build an infinite square-free
:string without 010 and 212 follows from the construction
:used in Theorem 2.  (The construction actually avoids 101 and 202.)
:
:> Alternatively expressed: it appears that any infinite squarefree string that
:> avoids the substrings 010 and 212 must necessarily contain every other
:> allowable substring, of any length.
:
:The paper has nothing to say about this question, though.

Thanks very much - interesting, and timely (dated just a month ago).
I'll try writing to the author.

I do wonder how the author derives in Theorem 2, "[...] then the fixed
point h^\omega(0) is squarefree" - I'm guessing this relies on results
in Thue's original paper.

Hugo