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.
%I A124677 #5 Jul 12 2015 20:36:08 %S A124677 0,2,6,13,27 %N A124677 Minimal total number of multiplications by single letters needed to generate all words of length n in the free monoid on two generators. %e A124677 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 A124677 Layers 0 through 3 of the tree are as follows (the edges, which just join one layer to the next, have been omitted): %e A124677 .............0................. %e A124677 .......x...........y........... %e A124677 ..xx.....xy.....yx....yy....... %e A124677 xxx xxy xyx yxx xyy yxy yyx yyy %e A124677 a(n) is the minimal number of edges in a subtree that includes the root and all 2^n nodes at level n. %e A124677 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 A124677 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 A124677 See A075099, A075100 for a different way of counting multiplications. Here we only grow the words one letter at a time. %K A124677 more,nonn %O A124677 0,2 %A A124677 _N. J. A. Sloane_, Dec 25 2006 %E A124677 Definition clarified by _Benoit Jubin_, Jan 24 2009