A124677 - OEIS

%I #5 Jul 12 2015 20:36:08

%S 0,2,6,13,27

%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 .............0.................

%e .......x...........y...........

%e ..xx.....xy.....yx....yy.......

%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.

%K more,nonn

%O 0,2

%A _N. J. A. Sloane_, Dec 25 2006

%E Definition clarified by _Benoit Jubin_, Jan 24 2009