Addition chains

[Colin Mallows]

Following on the recent interest in addition chains, may I draw
attention to related problems for a free group on two generators.  
Namely, given a free group with generators a and b,  how many multiplies 
are needed to compute all the words of length n?  (See A075099).   For
computing a single word of length n, what is the worst case word, and how 
many multiplies does it need?  (Clearly  at least A003313).   

And free groups with three generators, etc.

   Colin Mallows (colinm at research.avayalabs.com)