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

Maximilian Hasler maximilian.hasler at gmail.com
Tue Jul 1 22:20:30 CEST 2008

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