[seqfan] Re: Replacing chunks by their average

[Eric Angelini]

Hello Lars,
this is what I've just answered to Jim Nastos
who reacted exactly like you!
Best,
É.

>
Hi Jim,
Yes, you're right about 1,2,1,2,1,2,...
but there is even 1,1,1,1,1,1,1,... !
So I decided to start the seq with
the first non trivial "seed" which is
_not_ 1,2,2,1,3,... and _not_ 1,2,2,3,1,...
(those two starts lead to a quick
contradiction) but 1,2,2,2,1,3,... does
the job.
There are infinitely many such seq and
some others must be fun to build
(which I will try tomorrow), but I guess
the one I've suggested is the lexico-
first non trivial one with the stated
property...
(I hope!)
Best,
É.
Catapulté de mon aPhone

Le 28 mars 2015 à 22:30, "Jim Nastos" <nastos at gmail.com<mailto:nastos at gmail.com>> a écrit :


Hi Eric,
  Interesting idea, but the self-rebuilding also applies to 1,2,1,2,1,2,... And 1,2,3,4,5,6,....
  Why did you start with your S?
Jim

On Mar 28, 2015 12:28 PM, "Eric Angelini" <Eric.Angelini at kntv.be<mailto:Eric.Angelini at kntv.be>> wrote:

S=
1,2,2,2,1,3,2,2,1,2,4,3,1,2,1,2,1,7,2,4,1,2,1,2,1,2,12,3,1,4,1,2,1,2,1,2,1,23,
2,4,1,4,1,2,1,2,1,2,1,2,44,3,14,14,1,2,1,2,1,2,1,2,1,87,2,4,1,4,1,4,1,2,1,2,1,
2,1,2,1,2,172,3,1,4,1,4,1,4,1,2,1,2,1,2,1,2,1,2,1,343,...

Hello SeqFans,
Replace in S all chunks of same-parity integers by the average of the said chunk and you will get S back.

Example:
- the first chunk is made by 1, -- replace it with 1;
- the second chunk is 2,2,2, -- replace it with 2;
- the third chunk is 1,3, -- replace it with 2;
- the fourth chunk is 2,2, -- replace it with 2;
- the fifth chunk is made by 1, -- replace it with 1;
- the sixth chunk is made by 2,4, -- replace it by 3;
etc.
The replacements so far have been 1,2,2,2,3,1... which rebuilds S.

----
In considering S, one quickly notices this array where the rightmost diagonal is A023105:

1,
2,
2,
2,1,3,
2,2,1,2,4,
3,1,2,1,2,1,7,
2,4,1,2,1,2,1,2,12,
3,1,4,1,2,1,2,1,2,1,23,
2,4,1,4,1,2,1,2,1,2,1,2,44,
3,14,14,1,2,1,2,1,2,1,2,1,87,
2,4,1,4,1,4,1,2,1,2,1,2,1,2,1,2,172,
3,1,4,1,4,1,4,1,2,1,2,1,2,1,2,1,2,1,343,
...

Best,
É.


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Seqfan Mailing list -http://list.seqfan.eu/
Catapulté de mon aPhone

Le 29 mars 2015 à 09:03, "Lars Blomberg" <lars.blomberg at visit.se<mailto:lars.blomberg at visit.se>> a écrit :

Hi,
The sequence 1,2,1,2,1,2,... seems to also fulfill the stated requirements so I am probably missing something here.
/Lars

-----Ursprungligt meddelande-----
Från: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] För Eric Angelini
Skickat: den 28 mars 2015 20:28
Till: Sequence Discussion list
Ämne: [seqfan] Replacing chunks by their average


S=
1,2,2,2,1,3,2,2,1,2,4,3,1,2,1,2,1,7,2,4,1,2,1,2,1,2,12,3,1,4,1,2,1,2,1,2,1,23,
2,4,1,4,1,2,1,2,1,2,1,2,44,3,14,14,1,2,1,2,1,2,1,2,1,87,2,4,1,4,1,4,1,2,1,2,1,
2,1,2,1,2,172,3,1,4,1,4,1,4,1,2,1,2,1,2,1,2,1,2,1,343,...

Hello SeqFans,
Replace in S all chunks of same-parity integers by the average of the said chunk and you will get S back.

Example:
- the first chunk is made by 1, -- replace it with 1;
- the second chunk is 2,2,2, -- replace it with 2;
- the third chunk is 1,3, -- replace it with 2;
- the fourth chunk is 2,2, -- replace it with 2;
- the fifth chunk is made by 1, -- replace it with 1;
- the sixth chunk is made by 2,4, -- replace it by 3; etc.
The replacements so far have been 1,2,2,2,3,1... which rebuilds S.

----
In considering S, one quickly notices this array where the rightmost diagonal is A023105:

1,
2,
2,
2,1,3,
2,2,1,2,4,
3,1,2,1,2,1,7,
2,4,1,2,1,2,1,2,12,
3,1,4,1,2,1,2,1,2,1,23,
2,4,1,4,1,2,1,2,1,2,1,2,44,
3,14,14,1,2,1,2,1,2,1,2,1,87,
2,4,1,4,1,4,1,2,1,2,1,2,1,2,1,2,172,
3,1,4,1,4,1,4,1,2,1,2,1,2,1,2,1,2,1,343,
...

Best,
É.


_______________________________________________

Seqfan Mailing list - http://list.seqfan.eu/


_______________________________________________

Seqfan Mailing list - http://list.seqfan.eu/