Dave, using your recurrence for A139251 as a hint, and using A151550, A151551, A151552, ... as intermediate steps, I found nice g.f.'s for all the toothpick sequences. For example, that for A139251 is x + 2*x^2 + 4*x^2*(1+x)*(Prod(1+x^(2^k-1)+2*x^(2^k),k=1..oo)-1)/(1+2*x). New version of OEIS in a few minutes. Neil