%N Minimal total number of multiplications by single letters needed to generate all words of length n in the free monoid on two generators.
%e Form a tree with the empty word 0 as the root. Each node has potentially 4 children, corresponding to premultiplication by x or y and postmultiplication by x and y.
%e Layers 0 through 3 of the tree are as follows (the edges, which just join one layer to the next, have been omitted):
%e xxx xxy xyx yxx xyy yxy yyx yyy
%e a(n) is the minimal number of edges in a subtree that includes the root and all 2^n nodes at level n.
%e a(3) = 13 because each of xxx,xxy,xyx,xyy,yxx,yxy,yyx,yyy can be obtained in one step from xx,xy,yy; that is, we don't need yx. The corresponding subtree has 2 + 3 + 8 = 13 edges.
%e a(4) = 27 because one computes successively: 0, x,y, xx,xy,yy, xxx,xyx,xxy,yxy,yyx,yyy and then all 16 words of length 4.
%Y See A075099, A075100 for a different way of counting multiplications. Here we only grow the words one letter at a time.
%A _N. J. A. Sloane_, Dec 25 2006
%E Definition clarified by _Benoit Jubin_, Jan 24 2009