0,3

The cited paper does not include the numerical sequence, but these numbers will be included in an addendum when the paper is reprinted this spring in my "Selected Papers on Discrete Mathematics".

Computed by a simple 20-line C program, which runs fast but needs exponential memory. The basic idea is: Include a new word m=j.k for all subwords j and k whose total weight is n-1, unless one of the following three conditions is true: (a) j>0 && k>0 && r[j]==l[k]; (b) k>0 && l[k]>0 && l[l[k]]==j; (c) j>0 && r[j]>0 && r[r[j]]==k; so that the first few words of positive weight are 1=0.0, 2=0.1, 3=1.0, 4=0.2, 5=3.0, 6=0.4, 7=0.5, 8=1.3, 9=2.1, 10=4.0, 11=5.0; words 6 through 11 are the ones of weight 3 listed in the example.

D. E. Knuth, ``Notes on central groupoids,'' Journal of Combinatorial Theory 8 (1970), 376-390. [Especially page 389.]

a(4)=6 because the following six elements have four multiplications: a.(a.(a.(a.a))), a.(((a.a).a).a), (a.a).((a.a).a), (a.(a.a)).(a.a), (a.(a.(a.a))).a, (((a.a).a).a).a

Sequence in context: A157253 A074933 A003178 this_sequence A131553 A094485 A021819

Adjacent sequences: A079491 A079492 A079493 this_sequence A079495 A079496 A079497

nonn

D. E. Knuth, Jan 20 2003