[seqfan] Re: Two make a palindrome

[Neil Sloane]

I think one can show that every number must appear (in Eric's palindrome
sequence, in any base). Let me do the binary case
Remark: the only numbers that can't be rearranged to form
a palindrome are those with an odd number of 1's AND an odd number of 0's.

Suppose N is the smallest number that never appears.
Suppose N has i 1's and j 0's. There are 4 cases
depending on the parity of i and j, but the argument is the same in each
case.

It is easy to see that the sequence is infinite.

Let it run for ever. After a while, after T steps, say, all the numbers
less than N that will ever appear have appeared. From now on every number
is bigger than N.
Suppose say that i is even and j is odd.
>From now on, that is, from T steps on, every term must have an odd number
of 1's and an even number of 0's......... (*)
 (Otherwise we could take the next term to be N , getting a number which is
not of the form odd no of 1's AND odd even no of 0's.)

Take a large number M, a long way along in the sequence, of this form. The
next term after M could be - and so must be - a smallish number, just
bigger than N, with an even number of 0's and 1's. This will combine with M
to form a palindrome.
That contradicts (*).

I think the same argument works in any base.
Let N be the smallest missing number.
Then all remaining numbers must be "non-reactive"
with respect to N. But there are infinitely many numbers
that will combine with N and after a while one of them
will be a candidate for the next term, getting a contradiction.

Neil


On Mon, Nov 11, 2013 at 4:05 PM, Neil Sloane <njasloane at gmail.com> wrote:

> Concerning the binary version, to get the ball rolling I added A230891 and
> A230892.
> They need more terms, a b-file, some theorems, ...
>
>
> On Sun, Nov 10, 2013 at 9:12 AM, Neil Sloane <njasloane at gmail.com> wrote:
>
>> Maximilian said:
>> "I propose the sequence as https://oeis.org/draft/A228407 where I added a
>> link to Rob's post/calculations, and also a list of "records of
>> minima", i.e., (n,a(n)) where the least missing integers occur. Maybe
>> these could become sequences on their own (the values and the indices separately)
>> if further investigations in that sense are to be made."
>> Yes, that would be a very good idea - could you possibly add those two
>> sequences?
>> So here - like Recaman's A005132 - we have a sequence that may or may not
>> contain every number!
>> Thanks!  Neil
>>
>>
>> On Sat, Nov 9, 2013 at 7:54 PM, Maximilian Hasler <
>> maximilian.hasler at gmail.com> wrote:
>>
>>> Rob, Eric, SeqFans,
>>> I propose the sequence as https://oeis.org/draft/A228407 where I added
>>> a link to Rob's post/calculations, and also a list of "records of
>>> minima", i.e., (n,a(n)) where the least missing integers occur. Maybe
>>> these could become sequences on their own (the values and the indices
>>> separately) if further investigations in that sense are to be made.
>>> Regards,
>>> Maximilian
>>>
>>> > Le 9 nov. 2013 à 17:10, "Rob Arthan" <rda at lemma-one.com> a écrit :
>>> >
>>> >> Eric,
>>> >>
>>> >> That's a fun sequence and an interesting conjecture. As you say, it
>>> is not easy to calculate by hand. To get a feel
>>> >> for the conjecture I wrote an ML program to do it. This is what I got
>>> for the first 200 values:
>>> (...)
>>> >> My program is now in a loop printing out n, a(n) and m(n). The
>>> evidence currently supports your conjecture but m(n) is
>>> >> growing quite slowly:
>>> >>
>>> >>   a(5846) = 589, m(5846) = 598
>>> >>   a(5847) = 598, m(5846) = 679
>>> >>   ...
>>> >>   a(11539) = 1617, m(11539) = 679
>>> >>   a(11540) =  679 m(11540) = 697
>>> >>
>>> >> So 697 persisted as the smallest missing integer for more than 5,000
>>> stages. I will leave it running and report back if anything noteworthy
>>> occurs.
>>> >>
>>> >> Regards,
>>> >>
>>> >> Rob.
>>> >>
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>
>>
>>
>> --
>> Dear Friends, I have now retired from AT&T. New coordinates:
>>
>> Neil J. A. Sloane, President, OEIS Foundation
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>>
>>
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>


-- 
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com