# [seqfan] Re: Digit chaos in a window

Alex M timeroot.alex at gmail.com
Thu Aug 19 02:00:37 CEST 2010

```Would you please elaborate? From what I see, each column is generated by
taking the previous column and adding a number to the end of it... the
number of digits taken is equal to the number appended to the end of the
column ("window"?), and the number in the final S is equal to the size of
the window.

I do not see, however, ...

Oh wait! I think I understand. Let me ask if I have this correct:

The "6" window is a "6" because after appending a 5 to the end, the longest
window with no repeated digits would be "6 7 8 9 1 0 5", which is 7 digits
long. It doesn't work, because it has to include at least 5 digits.
Similarly, it cannot be 7, because that would result in "8 9 1 0 7", which
is 5 digits long. It doesn't work either, because using a size-6 window will
produce more digits. 6, however, has exactly 5 digits before it that are not
6's, so it works... okay... I think I get it now...

I think this sequence will have to repeat eventually; considering that all
numbers will be less than 11, each new number can only depend on the 10
previous, maximum. This leaves a maximum of 10^10 states to cycle through.
So, eventually, it will have to reach a cycle.

Now that I think I understand it, I think I found some errors - I believe it
should be

S = 1 2 3 4 5 6 7 8 9 10 6 5 7 4 8 3 9 6 5 7 4 3 8* ...

and then

8* 6 5 9 7 4 3 5 8 6 4 7

Some interesting things to check would be
-Cycle length
-Digit densities

And then, of course, cycle lengths in other bases would make another great
sequence. :-D

~6 out of 5 statisticians say that the number of statistics that either make
no sense or use ridiculous timescales at all has dropped over 164% in the
last 5.62474396842 years.

On Wed, Aug 18, 2010 at 10:50 AM, Eric Angelini <Eric.Angelini at kntv.be>wrote:

>
> Hello SeqFans,
> a(n) is the size of the largest possible window which includes a(n) itself
> and
> a(n) non-repeated digits (the "window" is under the stroke "|"):
>
> S = 1 2 3 4 5 6 7 8 9 10 6 5 7 4 8 3 9 6 5 7 4 3 2 8 6 5 7 4 3 2 6 5 4 3 2
> 1 7 8 9 6 ...
>    | | | | | | | | |  | | | | | | | | | | | | | | | | | | | | | | | | | | |
> | | | |
>    1 1 1 1 1 1 1 1 1  2 7 9 8 6 9 4 1 7 8 4 6 7 3 9 7 3 4 6 7 3 5 4 2 5 3 1
> 6 6 6 2
>      2 2 2 2 2 2 2 2  3 8 1 9 5 1 8 0 4 3 8 5 4 2 6 4 2 3 5 4 2 7 3 6 4 2
> 5 5 5 1
>        3 3 3 3 3 3 3  4 9 0 1 7 0 3 6 8 9 3 7 3   5 3 8 2 7 3   4 2 5 3
> 4 4 4 7
>          4 4 4 4 4 4  5 1 6 0 4 6   5 3 6 9 4     7 2 6 8 4     3 6 4
> 3 3 3 8
>            5 5 5 5 5  6 0 5 6   5   7 9 5 6       4 8 5 6       2 5
> 2 2 2 9
>              6 6 6 6  7 6   5   7   4 6   5       3 6   5       6
> 1 1 1 6
>                7 7 7  8     7   4   8     7       2     7
> 7 7 7
>                  8 8  9         8   3             8
>   8 8
>                    9  1             9
>     9
>                       0
>
> Unless I've made errors in computing this, I don't see any pattern yet:
> when will it
> arise?!
>
> Best,
> É.
>
>
>
>
>
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>

```