Benoit Jubin (Jan 24 2009) suggests replacing "monoid" in the definition by "semigroup".
I believe a(2n) = a(n) + 2^(2n). I guess a(7) = 156.
EXAMPLE
a(3)=11 because each of xxx,xxy,xyx,xyy,yxx,yxy,yyx,yyy can be obtained in one step from xx,xy,yy and it takes three multiplications to produce xx, xy, yy.
