[seqfan] Re: Iterates of n - A002828(n) ?

Tue Aug 20 16:26:33 CEST 2013

On Mon, Aug 19, 2013, *L. Edson Jeffery* <lejeffery2 at gmail.com>
<seqfan%40list.seqfan.eu?Subject=Re%3A%20%5Bseqfan%5D%20Re%3A%20Fwd%3A%20Iterates%20of%20n%20-%20A002828%28n%29%20%3F&In-Reply-To=%3CCAGRLqMgJ1BunNuV3BuK1Os91Dm_L-%3Djy_7PnNuNQOAm_Hx1yiw%40mail.gmail.com%3E>
 wrote:

>
>
> *in *http://list.seqfan.eu/pipermail/seqfan/2013-August/011586.html
>

>
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>
> ------------------------------
>
> AK>...will the trajectory visit every square-1 (A005563) < k before it
> reaches zero?
>
> Is it sufficient to prove, for any n, that max(square-1 < n) is in the
> trajectory?
>
> Yes, just that.

>
> AK>Which would imply that there would be only one infinite sequence a_n
> such that a(n-1) = a(n) - the least number of squares that add up to a(n).
>
> This should be essentially the first column in the first triangle below.
>
> AK>Then, would there be any regularity in the derived sequence counting the
> number of iterations needed to hop from A005563(n+1) to A005563(n)?
>
> According to the second triangle below, it appears that your hopping
> sequence should be {1,1,2,2,4,4,5,...}, unless I misunderstood or
> miscalculated.
>
> Yes, exactly that, the first differences of the row lengths 1, 1, 3, 5, 9,
13,18, ...
of the table given below.

 Not sure exactly what you mean by "regularity."
>
>
Something like what is present in these https://oeis.org/A213709 &
https://oeis.org/A226060 & https://oeis.org/A218543
Some regularity (e.g. partial scale-invariance) mixed with some "random" or
not-so-easily computable aspects.

> Nice ideas, Antti.
>
>
Not wholly my idea. I just tried to find a  variant with a more
"number-theoretical taste to it" than Carl White's original "binary
beanstalk" sequence https://oeis.org/A179016 (to which the above three
seq-links are related to) and other examples of such sequences that I have
submitted in other bases.
However, while it is easy to guarantee the existence of "choking points"
(through which all the iterated paths will come through) in base-related
beanstalk sequences, it seems not that easy when trying to construct such
phenomena from sequences related to the traditional number theory.
But what I glean from Pari-code at https://oeis.org/A002828
it seems actually to be a some kind of "crypto-base" sequence.
(Or at least involving modulus some power of 2. I have to re-read my
Burton.)

-- Antti

PS: If anybody comments on this, please add CC: to my gmail-account, so I
will get the reply even before the mailing list digest is ready.

>
>  n     Seq of trajectories for 0 <= n <= 48
> ---   -----------------------------------------
>  0     0
>  1     0
>  2     0
>  3     0
>  4     3,0
>  5     3,0
>  6     3,0
>  7     3,0
>  8     6,3,0
>  9     8,6,3,0
> 10     8,6,3,0
> 11     8,6,3,0
> 12     9,8,6,3,0
> 13     11,8,6,3,0
> 14     11,8,6,3,0
> 15     11,8,6,3,0
> 16     15,11,8,6,3,0
> 17     15,11,8,6,3,0
> 18     16,15,11,8,6,3,0
> 19     16,15,11,8,6,3,0
> 20     18,16,15,11,8,6,3,0
> 21     18,16,15,11,8,6,3,0
> 22     19,16,15,11,8,6,3,0
> 23     19,16,15,11,8,6,3,0
> 24     21,18,16,15,11,8,6,3,0
> 25     24,21,18,16,15,11,8,6,3,0
> 26     24,21,18,16,15,11,8,6,3,0
> 27     24,21,18,16,15,11,8,6,3,0
> 28     24,21,18,16,15,11,8,6,3,0
> 29     27,24,21,18,16,15,11,8,6,3,0
> 30     27,24,21,18,16,15,11,8,6,3,0
> 31     27,24,21,18,16,15,11,8,6,3,0
> 32     30,27,24,21,18,16,15,11,8,6,3,0
> 33     30,27,24,21,18,16,15,11,8,6,3,0
> 34     32,30,27,24,21,18,16,15,11,8,6,3,0
> 35     32,30,27,24,21,18,16,15,11,8,6,3,0
> 36     35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 37     35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 38     35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 39     35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 40     38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 41     38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 42     39,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 43     40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 44     41,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 45     43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 46     43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 47     43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
> 48     45,43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
>
>  n   n^2-1    Trajectories for n^2 - 1, 1 <= n <= 7
> --- -------  -----------------------------------------
>  1     0      0
>  2     3      0
>  3     8      6,3,0
>  4    15      11,8,6,3,0
>  5    24      21,18,16,15,11,8,6,3,0
>  6    35      32,30,27,24,21,18,16,15,11,8,6,3,0
>  7    48      45,43,40,38,35,32,30,27,24,21,18,16,15,11,8,6,3,0
>
>

> Ed Jeffery
>
> -------------------------------------------------------------
>
AK wrote earlier:

> Just wondering:
>
> If we iterate the function a(n) = n - A002828(n) (
> https://oeis.org/A002828 ) from any starting value k, will the trajectory
> visit every square-1 (A005563) < k before it reaches zero?
> Which would imply that there would be only one infinite sequence a_n such
> that a(n-1) = a(n) - the least number of squares that add up to a(n).
>
> Then, would there be any regularity in the derived sequence counting the
> number of iterations needed to hop from A005563(n+1) to A005563(n) ?
>
>
>
> Cheers,
>
> Antti
>
>
>