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

Jonathan Post jvospost3 at gmail.com
Tue Jul 1 21:53:36 CEST 2008

```Per my comment questioning fixed points and cycles in
A113172(A113172(...(n)...)).

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
1   4  because (1,3,8,9)
2   6  because (2,6,10,3,8,9)
3   3  because (3,8,9)
4   4  because (4,7,8,9)
5   5  because (5,10,3,8,9)
6   5  because (6,10,3,8,9)
7   3  because (7,8,9)
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
10 4  because (10,3,8,9)
11 2  because (11,9)
12 0  because 12 is the least fixed point of the iteration
13 3  because (13,11,9)
14 3  because (14,11,9)
15 4  because (15,13,11,9)
16 4  because (16,14,11,9)
17 -2 because (17,12) gets to a fixed point
18 -2 because (18.12)
19 3  because (19,8,9)
20 -2 because (20,12)
21 5  because (21,15,13,11,9)
22 -3 because (22,18,12)
23 -3 because (23,20,12)
24 4  because (24,19,8,9)
25 -4 because (25,22,18,12)
26 -4 because (26,22,18,12)
27 -3 because (27,20,12)
28 6  because (28,21,15,13,11,9)
29 5  because (29,16,14,11,9)
30 -2 because (30,12)

and so on.  Again, my questions are open as to the equivalence classes
under iteration. What is the next fixed point after 12? What is the
smallest start of a cycle other than (9,4,7,8,9,4,7,8,9,4,7,8,9,...)?

Further, there are versions of this in many other languages.
see:
http://en.wikipedia.org/wiki/Scrabble_letter_distributions

which gives data for:
English, Afrikaans, Arabic, Bulgarian, Catalan, Croatian, Czech,
Danish, Dutch, Finnish, French, German, Greek, Hebrew, Hungarian,
Icelandic, Italian, Malaysian, Norwegian, Polish, Portuguese,
Romanian, Russian, Slovak, Slovenian, Spanish, Swedish, Turkish,
Welsh.

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.

```