This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
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.
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. Layers 0 through 3 of the tree are as follows (the edges, which just join one layer to the next, have been omitted): .............0................. .......x...........y........... ..xx.....xy.....yx....yy....... xxx xxy xyx yxx xyy yxy yyx yyy a(n) is the minimal number of edges in a subtree that includes the root and all 2^n nodes at level n. 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. 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.
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