Query on sequence of dynamics of iterating A113172 Scrabble value of English word for the number n.

[Maximilian Hasler]

On Tue, Jul 1, 2008 at 17:03, Jonathan Post <jvospost3 at gmail.com> wrote:
> Thank you, Maximilian Hasler.
>
> I wrestled with the same question of index (0 or 1 step to get to orbit).
>
> I DON'T know there aren't any other cycles than the orbit of 9. I
> don't even know that A113172(n) <= n for all sufficiently large n.
> Hence I don't know that there are not infinite orbits.

I "conjecture" that n=1..8 and the fixed point 12 are the only values for which
A113172(n) is not strictly smaller than n.
Thus, there cannot be a fixed point other than 12 nor a cycle other
than [4,7,8,9],
and any number ends at one of these two. (This would give rise to
another sequence, but I don't think it is worth submitting to OEIS, at
least not during Summer...)

> So I continue (by hand: someone coding this might be nice) and, for
> that matter, coding would allow someone to do an extension of A113172
> to answer some of these questions:

The following code prints out the orbit up to the moment where a point
is reached for the second time.
Note that you stopped either at 12 or at 9, but 8, 7 and 4 should also
be ending points.
Regards,
Maximilian

orbit(n,p=1)={ local(nn=n,t=0); until( bittest(t,n=A113172(n)),
t+=1<<n; p && print1(n" -> "));n }
for(n=31,72,print(orbit(n)))

31 -> 15 -> 13 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
32 -> 18 -> 12 -> 12
33 -> 20 -> 12 -> 12
34 -> 19 -> 8 -> 9 -> 4 -> 7 -> 8
35 -> 22 -> 18 -> 12 -> 12
36 -> 22 -> 18 -> 12 -> 12
37 -> 20 -> 12 -> 12
38 -> 21 -> 15 -> 13 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
39 -> 16 -> 14 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
40 -> 12 -> 12
41 -> 15 -> 13 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
42 -> 18 -> 12 -> 12
43 -> 20 -> 12 -> 12
44 -> 19 -> 8 -> 9 -> 4 -> 7 -> 8
45 -> 22 -> 18 -> 12 -> 12
46 -> 22 -> 18 -> 12 -> 12
47 -> 20 -> 12 -> 12
48 -> 21 -> 15 -> 13 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
49 -> 16 -> 14 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
50 -> 14 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
51 -> 17 -> 12 -> 12
52 -> 20 -> 12 -> 12
53 -> 22 -> 18 -> 12 -> 12
54 -> 21 -> 15 -> 13 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
55 -> 24 -> 19 -> 8 -> 9 -> 4 -> 7 -> 8
56 -> 24 -> 19 -> 8 -> 9 -> 4 -> 7 -> 8
57 -> 22 -> 18 -> 12 -> 12
58 -> 23 -> 20 -> 12 -> 12
59 -> 18 -> 12 -> 12
60 -> 15 -> 13 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
61 -> 18 -> 12 -> 12
62 -> 21 -> 15 -> 13 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
63 -> 23 -> 20 -> 12 -> 12
64 -> 22 -> 18 -> 12 -> 12
65 -> 25 -> 22 -> 18 -> 12 -> 12
66 -> 25 -> 22 -> 18 -> 12 -> 12
67 -> 23 -> 20 -> 12 -> 12
68 -> 24 -> 19 -> 8 -> 9 -> 4 -> 7 -> 8
69 -> 19 -> 8 -> 9 -> 4 -> 7 -> 8
70 -> 13 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
71 -> 16 -> 14 -> 11 -> 9 -> 4 -> 7 -> 8 -> 9
72 -> 19 -> 8 -> 9 -> 4 -> 7 -> 8

> n   trajectory of iterated A113172(n)
> 31              31,15,13,11,9
> 32              32,18, 12
> 33              33,20,12
> 34              34,19,8,9
> 35              35,22,18,12
> 36              36,22,18,12
> 37              37,20,12
> 38              38,21,15,13,11,9
> 39              39,16,14,11,9
> 40              40,11,9
> 41              41,14,11,9
> 42              42,17,12
> 43              43,19,8,9
> 44              44,18,12
> 45              45,21,15,13,11,9
> 46              46,21,15,13,11,9
> 47              47,19,8,9
> 48              48,20,12
> 49              49,15,13,11,9
> 50              50,14,11,9
> 51              51,17,12
> 52              52,20,12
> 53              53,22,18,12
> 54              54,21,15,13,11,9
> 55              55,24,19,8,9
> 56              56,24,19,8,9
> 57              57,22
> 58              58,23,20,12
> 59              59,18,12
> 60              60,15,13,11,9
> 61              61,18,12
> 62              62,21,15,13,11,9
> 63              63,23,20,12
> 64              64,22,18,12
> 65              65,5,22,18,12
> 66              66,25,22,18,12
> 67              67,23,20,12
> 68              68,24,19,8,9
> 69              69,19,8,9
> 70              70,13,11,9
> 71              71,16,14,11,9
> 72              71,19,8,9
>
> Length to orbit or fixed point is a record:
> 0,2,38, what next?
>
> You're right, though, that I should cease psychoanalysis of seqfans,
> as I am unqualified, and it's rude.
>
> On 7/1/08, Maximilian Hasler <maximilian.hasler at gmail.com> wrote:
>> On Tue, Jul 1, 2008 at 15:53, Jonathan Post <jvospost3 at gmail.com> wrote:
>>  > There must be a better way than this artifice:
>>  > a(n) = -k if k is the least number iterations of A113172(n) to reach a
>>  > fixed point
>>  > a(n) = 0 if n is a fixed point under A113172
>>  > a(n) = +j if j is the least number of iterations before entering a cycle.
>>  >
>>  > n  a(n)  comment
>>  > 0   3   because 0->ZERO->13-> THIRTEEN ->11->
>>  > ELEVEN->9->NINE->4->FOUR->7->SEVEN->8->EIGHT->9 looping
>>
>> > ...
>>
>> > 8   2  because  (8,9)
>>  > 9   1  because 9 begins the (9,4,7,8,9,4,7,8,9,4,7,8,9,...) loop
>>
>>
>> Q: why a(9)=1 ?
>>  Zero iterations are needed to get into the loop!
>>
>>
>>  > 17 -2 because (17,12) gets to a fixed point
>>
>>
>> again : ONE iteration takes to the f.p.
>>  else it is not reasonable to say that 12 is at distance 0 of a fixed point.
>>
>>  Maybe a better convention would be:
>>  a(n)>0 is the number of elements in orbit(n) ending at a fixed point,
>>  a(n)<=0 is - the number of steps needed to reach a point of a cycle
>>  (I suppose there aren't any others than the orbit of 9.)
>>
>>  Then:
>>  a(n)=1  <=>  n is a fixed point
>>  a(n)=0  <=>  n is element of a cycle
>>
>>  Alternative:
>>
>> > I'm getting a sense of who's in the equivalence class who hates "word"
>>  > sequences and who is the in equivalence class who finds them
>>  > interesting.
>>
>>
>> I don't think these are equivalence classes.
>>  I even think that each of the two "belongs to" relations are ill
>>  defined. At least for me, "hating"(?) or "finding interesting" is not
>>  a function of the "word" kw alone.
>>
>>  Regards,
>>
>> Maximilian
>>
>