[seqfan] Re: A130011 and the definition of "slowest increasing".

[M. F. Hasler]

Dear SeqFans,

I revised the comment I added in A130011 three days ago in order to
avoid changes back and forth.
I think that any such sequence has an infinite number of terms
a(n) >= 3n-2 (if it starts with a(1)=1), namely for all the indices  n
= a(k)+1  for some k,
since we must have exactly a(k) terms <= 3 a(k) and not one more.

So there isn't a unique "slowest increasing" version if not the
"greedy" (lexicographically first) one*.
Specifically, the one given at present has a(22)=64 which implies
a(65)>= 193 which is exactly the same as we get for the greedy
(lex.smallest) sequence.
One can always have an arbitrary number of smaller terms, e.g. if you
let a(2)=1000 then up to a(1000)=1998 you can use a(n)=n+998 which is
"much slower increasing" than a(n)=3n-2, but then again a(1001) must
be > 3000.

Of course I may be wrong (somehow too busy to think longer about
this), so any confirmation or correction of my reasoning is
appreciated.

* which I have now added as draft/A260107 (proposed), adding
M.Johnson's b-file of 10 000 terms. Will add the variant starting with
0 later.

Regards,
Maximilian

On Mon, Jul 13, 2015 at 9:23 AM, Heinz, Alois
<alois.heinz at hs-heilbronn.de> wrote:
> Am 13.07.2015 um 08:19 schrieb Lars Blomberg:
>
>>
>> Could someone please define what "slowest increasing" means?
>>
>> And what is the difference between "slowest increasing" and
>> "lexicographically first"?
>>
>
> "lexicographically first" ist the greedy approach.  Use the
> smallest a(n) that satisfies the condition given a(1), ..., a(n-1).
> And do not change it later.  This is easy algorithmically.
>
> "slowest increasing" here means that you accept a larger than greedy
> a(n) if it is possible to get a smaller a(m) for a larger m>n.
> This is more complicated algorithmically.
>
> Please do not change the definition of A130011.
>
> If you want to have a new sequence with the greedy approach,
> please use a new A-number.
>
> Best, Alois
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-- 
Maximilian