Merci beaucoup ! à+ É. Catapulté de mon aPhone
Le 24 oct. 2019 à 19:29, Andy Latto <andy.latto@pobox.com> a écrit :
There is some relevant discussion in the "Variations" section of https://en.wikipedia.org/wiki/Look-and-say_sequence
Andy
On Thu, Oct 24, 2019 at 12:04 PM Allan Wechsler <acwacw@gmail.com> wrote:
I posted this query to math-fun and not to seqfan, because my intention was to collect a census of these tuples, not to suggest a new sequence. While there are sequences on OEIS that have a similar motivation, these sequences encrypt the basic concept by arbitrarily converting the tuples into base-10 integers.
I think I already covered the concept "K1 32 23 2K 1A 1B 1C ...", which includes your asterisked 51 and 61 cases, unless I'm missing something.
We can easily prove that "22" is the only one-pair example.
There can be no two-pair examples. Such an example could only use two numbers, and the two possibilities "AA BB" and "AB BA" are easily seen to be impossible.
I am pretty sure there can be no three-pair examples, though I don't have a fast proof.
I am getting more and more convinced that we have now covered all the possible examples. Let's adopt the simplification of simply leaving all the singletons (the pairs of the form 1K) out of our presentations -- just assume the are off to the right. So the examples we have found have the following non-singleton parts:
22 21 32 23 31 33 K1 32 23 2K
I am coming to the conclusion that this is all there is, and feel like a proof is "on the tip of my tongue." One lemma is: none of the numbers can exceed the number of pairs.
On Thu, Oct 24, 2019 at 11:29 AM Georg Fischer <dr.georg.fischer@gmail.com> wrote:
Dear Allan,
Neil's link to the OEIS index leads to A047841 and A109776 which are close to your "census" map. I removed their tuples with zeroes, and with duplicates, sorted them by increasing digits, and checked (by a program) whether they are fixed points under the census map.
If I got it right, then I think that the following ones marked with "*" are new:
2,1, 3,2, 2,3, 1,K (K>3) 2,2 3,1, 1,2, 3,3, 1,A (A>3) *3,1, 2,2, 3,3, 1,A, 1,B (A>3, B>A) *3,1, 3,3, 1,A, 1,B (A>3, B>A) 4,1, 3,2, 2,3, 2,4, 1,A, 1,B, 1,C (A>4, B>A, C>B *5,1, 3,2, 2,3, 1,4, 2,5, 1,A, 1,B, 1,C (A>5, B>A, C>B) 5,1, 3,2, 2,3, 2,5, 1,A, 1,B, 1,C, 1,D *6,1, 3,2, 2,3, 1,4, 1,5, 2,6, 1,7, 1,8, 1,9
I imagine that your definition is different from several existing OEIS sequences because it is base-independant and does not count zero. I can only speculate whether there are more "template" tuples. Regards - Georg
Am 23.10.2019 um 18:45 schrieb Neil Sloane: Allan,in the Index to the OEIS there is this entry
self-describing numbers, sequences related to : [edit < https://oeis.org/w/index.php?title=Index_to_OEIS:_Section_Se&action=edit&sec...
]self-describing numbers: Autobiographical numbers: A047841 <http://oeis.org/A047841> (A104784 <http://oeis.org/A104784> is an erroneous version), self-describing primes: A108810 <http://oeis.org/A108810>, semiprimes: A173101 <http://oeis.org/A173101 , not complete information: A059504 <http://oeis.org/A059504>, primes therein: A109775 <http://oeis.org/A109775>, self descriptive (possibly redundant) numbers: A109776 <http://oeis.org/A109776> It may be that your variant is new - please add it & update the Index entry too! Best regards Neil
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-- Andy.Latto@pobox.com
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