[seqfan] Re: Please help me describe this sequence

[David Seal]

I'm afraid I cannot think of any good way to produce a description that is both short and complete in itself - it seems to me to be a choice between a reasonably short description that talks about something like "the iterative process described in the comments", or a full description of that process, which would need to be fairly long. I'd be more inclined to go for the former, partly because I can see at least three possible sequences associated with the process - the sequence of values of N for which it stops, the sequence of the number of steps it takes to stop (or -1 if it doesn't stop) and the sequence of the values of T when it stops (or -1 if it doesn't stop). Each of those requires a fair amount of extra text around the description of the process, so separating out the full description of the process would help to avoid long run-on sentences.

That said, I'm not familiar with what OEIS conventions there might be about such matters - so I can only say how I would be inclined to write it if given a free rein, and I obviously have to defer to the opinions of OEIS editors.

On the individual sequences themselves, some case analysis says that if the sequence ever gets down to N = 4, its fate is determined by the value of T modulo 60 at that point (where I've labelled the cases with upper- or lower-case letters according to whether the sequence stops or increases forever:

A) If T modulo 6 is 1, the subsequent values of (T modulo 6, N) are (5,3), (2,2), (4,3), (1,2), (3,1) and the sequence stops.
 
B) If T modulo 12 is 6 or 10, the subsequent values of (T modulo 12, N) are (10 or 2, 3), (1 or 5, 2), (3 or 7, 1) and the sequence stops.

c) If T modulo 6 is 3, the next four values of (T modulo 6, N) are (1,3), (4,2), (0,3), (3,4) and that sequence loops, with T increasing by 12 every 4 steps.

d) If T modulo 20 is 16, the next four values of (T modulo 20, N) are (0,5), (5,6), (11,5), (16,4) and that sequence loops, with T increasing by 20 every 4 steps. 

E) If T modulo 12 is 11, the next two values of (T modulo 12, N) are (3,3), (6,4) and the sequence subsequently stops by case A.
 
F) If T modulo 60 is 4, 28, 40 or 52, the next two values of (T modulo 60, N) are (8 or 32 or 44 or 56, 5), (13 or 37 or 49 or 61, 4) and the sequence subsequently stops by case A.

g) If T modulo 12 is 2, the next two values of (T modulo 12, N) are (6,3), (9,4) and the sequence enters loop c.

h) If T modulo 60 is 0, 12, 24 or 48, the next two values of (T modulo 60, N) are (4 or 16 or 28 or 52, 5) and (9 or 21 or 33 or 57, 4) and the sequence enters loop c.

i) If T modulo 60 is 5, 17, 41 or 53, the next two values of (T modulo 60, N) are (9 or 21 or 45 or 57, 3) and (12 or 24 or 48 or 0, 4) and the sequence enters loop c via case h.

j) If T modulo 60 is 8, 32 or 44, the next two values of (T modulo 60, N) are (12 or 36 or 48, 5) and (17 or 41 or 53) and the sequence enters loop c via cases i and h.

k) If T modulo 60 is 29, the next two values of (T modulo 60, N) are (33,3) and (36,4) and the sequence enters loop d.

l) If T modulo 60 is 20, the next two values of (T modulo 60, N) are (24,5) and (29,4) and the sequence enters loop d via case k.

These cover all cases for T modulo 60 as follows:

   | +0 +12 +24 +36 +48
---+-------------------
0  |  h   h   h   d   h
1  |  A   A   A   A   A
2  |  g   g   g   g   g
3  |  c   c   c   c   c
4  |  F   d   F   F   F
5  |  i   i   l   i   i
6  |  B   B   B   B   B
7  |  A   A   A   A   A
8  |  j   k   j   j   d
9  |  c   c   c   c   c
10 |  B   B   B   B   B  
11 |  E   E   E   E   E

I wonder whether there are any other possible fates for such sequences besides stopping and the ever-increasing loops involving N cycling through (4,3,2,4) and (6,5,4,5) identified in the above?

David


> On 10/11/2020 11:45 Christian Lawson-Perfect <christianperfect at gmail.com> wrote:
> 
>  
> Hi seqfans,
> I've come up with a new sequence, but I'm not sure how best to describe it
> in an OEIS entry.
> 
> It arises from a dynamical process. You start with a natural number N, and
> a total T that starts at 0. At each step, do this:
> 
> * if N is 1, stop
> * add N to T
> * if N divides T, add 1 to N, otherwise subtract 1 from N.
> 
> So the sequence of N and T might look like this, for starting N = 4:
> 
> N	T
> 4	0
> 5	4
> 4	9
> 3	13
> 2	16
> 3	18
> 4	21
> ...	...
> 
> 
> Or this, for starting N = 8:
> 
> N	T
> 8	0
> 9	8
> 8	17
> 7	25
> 6	32
> 5	38
> 4	43
> 3	47
> 2	50
> 3	52
> 2	55
> 1	57
> stop
> 
> I always have trouble filling in the short description field for OEIS
> entries. Can anyone help?
> 
> --
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