A124677 Minimal total number of multiplications by single letters needed to generate all words of length n in the free monoid on two generators.

0, 2, 6, 13, 27

Offset

0,2

Links

Table of n, a(n) for n=0..4.

Example

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.

Crossrefs

See A075099, A075100 for a different way of counting multiplications. Here we only grow the words one letter at a time.

Sequence in context: A353232 A275970 A374148 * A034465 A182614 A288901

Adjacent sequences: A124674 A124675 A124676 * A124678 A124679 A124680

Keyword

more,nonn

Author

N. J. A. Sloane, Dec 25 2006

Extensions

Definition clarified by Benoit Jubin, Jan 24 2009

Status

approved